Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 23, 2006, 95–150 \begin{matrix}c \\ ◯ \end{matrix} O. Krupková, P. Volný

O. Krupková and P. Volný

DIFFERENTIAL EQUATIONS WITH CONSTRAINTS IN JET BUNDLES: LAGRANGIAN AND HAMILTONIAN

SYSTEMS

(submitted by V.V. Lychagin)

Abstract. The paper is a survey of the theory of Lagrangian systems with non-holonomic constraints in jet bundles. The subject of the paper are systems of second-order ordinary and partial differential equations that arise as extremals of variational functionals in fibered manifolds. A geometric setting for Euler-Lagrange and Hamilton equations, based on the concept of Lepage class is presented. A constraint is modeled in the underlying fibered manifold as a fibered submanifold endowed with a distribution (the canonical distribution). A constrained system is defined by means of a Lepage class on the constraint submanifold. Constrained Euler-Lagrange equations and constrained Hamilton equations, and properties of the corresponding exterior differential systems, such as regularity, canonical form, or existence of a constraint Legendre transformation, are presented. The case of mechanics (ODEs) and field theory (PDEs) are investigated separately, however, stress is put on a unified exposition, so that a direct comparison of results and formulas is at hand.

1. Introduction

Since the 30’s of the last century when the pioneer paper by Chetaev was published [4], the study of non-holonomic constrained systems has been of growing interest in mechanics, control theory and geometry.

Namely during the past 15 years much effort has been devoted to developments of geometric methods and studies of geometric structures of nonholonomic mechanics; among the many contributions to the subject, let us mention here at least [3, 6, 9, 15, 17, 20, 26, 27, 28, 30, 31, 33, 34, 38], and references therein. Recently, several authors have started to study a more general situation of partial differential equations (field theories) with constraints given by systems of first-order partial differential equations [2, 22, 25, 37]. Since the geometric origin of these constraints is the same as in mechanics, it is natural also in this generalized situation to call such constraints “non-holonomic”.

The papers investigating non-holonomic systems differ in approaches, methods, geometric setting, kind of constraints studied, and many other aspects. Usually (and this is in no case specific for constrained systems), tools, structures and methods used in mechanics (i.e. ordinary differential equations) and field theory (partial differential equations) are essentially different. The aim of this paper is to present foundations of a general geometric theory of non-holonomic systems as a part of the calculus of variations on fibered manifolds. It is based on the theory of Lepage equivalents of Lagrangians (Krupka [10, 12]) and of dynamical forms (Krupková [14, 15, 16, 19]), and on study of exterior differential systems associated with variational equations (Krupková [14, 16, 19, 21]). It is also important to note that a constraint is modeled in the underlying fibered manifold as a fibered submanifold endowed with a distribution (called canonical distribution) [15, 22]. This structure plays a key role in studying the geometry of non-holonomic constrained systems, and represents a correct mathematical realization of the physical d’Alembert’s principle (that is ambiguous in case of velocity dependent constraints in mechanics, and completely unclear in field theory). The setting of [15]

37J60, 49Q99. tion.

2000 Mathematical Subject Classification. primary: 35A30, secondary:

Key words and phrases. Jet bundles, non-holonomic constraints, semiholonomic constraints, holonomic constraints, constrained Lagrangian systems, constrained Euler-Lagrange equations, Hamilton–De Donder equations, regularity of constrained systems, momenta, Hamiltonian, Legendre transforma-

Research supported by grants 201/06/0922 of the Czech Science Foundation, and MSM 6198959214 of the Czech Ministry of Education, Youth and Sports.

O. Krupková also would like to acknowledge the hospitality of the Department of Mathematics, La Trobe University, Melbourne.

and [22] brings a unified approach to mechanics and field theory, both unconstrained and with constraints, and can be directly transferred to higher-order situation [18].

There are basically two different approaches to systems with constraints:

• constrained system is modeled as a modified unconstrained system, defined on the same manifold as the unconstrained system (in mechanics this concerns so called “constraint forces” and dynamics governed by equations with Lagrange multipliers),

• constrained system is modeled as a system defined on the constraint submanifold (dynamics are modeled by the so called “reduced equations”, without Lagrange multipliers).

As shown in [15] and [22], both these approaches are equivalent. In this paper, however, we prefer the latter one, since it is more geometrical, and enables us to study constrained systems by the same tools as unconstrained systems. We focus on variational systems, i.e. such that their dynamics are given by differential equations that arise as equations for extremals of Lagrangians (Euler–Lagrange equations). First, we recall basic facts on unconstrained Lagrangian systems and their associated Hamiltonian systems in jet bundles. Then we turn to the concept of non-holonomic constraint structure. Finally we study Lagrangian systems subjected to non-holonomic constraints, namely constrained Euler– Lagrange equations and constrained Hamilton equations, where we devote our attention to such problems as regularity of constrained systems, or existence of an appropriate “constraint Legendre transformation”. The cases of mechanics (ordinary differential equations) and field theory (partial differential equations) are investigated separately, however, stress is put on a unified exposition, so that common features on one hand and differences on the other hand are transparent and their geometric origin becomes clear. We also tried to provide analogous results and formulas in such a way that the reader could compare them directly.

This work is basically a review paper, however, it contains also new, yet unpublished results (this concerns Sec. 4.5 and 4.6 on constrained Hamilton–De Donder equations and constraint Legendre transformation for general non-holonomic constraints in field theory).

2. Calculus in jet bundles

We start with a brief introduction of notations, basic structures and the corresponding calculus to be used. For more details we refer to the books by Saunders [32] and Krupková [16], and the papers by Krupka [10, 12].

We consider a fibered manifold π : Y → X with dim X = n ≥ 1, dim Y = m + n, and its jet prolongations π_{1} : J^{1}Y → X and π_{2} : J^{2}Y → X. All manifolds and mappings are smooth, and the summation convention on repeated indices applies throughout.

A mapping γ : U → Y, where U ⊂ X is an open set, is called a section of π if π ∘ γ = id_{U}. We denote by J^{1}γ and J^{2}γ the first and the second jet prolongation of γ, respectively. Note that J^{1}γ (resp. J^{2}γ) is a section of π_{1} (resp. π_{2}). A section δ of π_{1} is called holonomic if δ = J^{1}γ for a section γ of π.

A vector field ξ on Y is called π-vertical if T π ·ξ = 0, and π-projectable if T π · ξ = ξ_{0} ∘ π for a vector field ξ_{0} on X. Considering the projections π_{1} : J^{1}Y → X, π_{2} : J^{2}Y → X, π_{1,0} : J^{1}Y → Y, π_{2,1} : J^{2}Y → J^{1}Y and π_{2,0} : J^{2}Y → Y the concepts of the corresponding verticality and projectability are obtained quite similarly. For the module of vector fields (resp. π_{r}-vertical vector fields) on J^{r}Y, r = 1, 2, we shall use the notation X (J^{r}Y) (resp. V(J^{r}Y)).

Denote by Ω^{q}(J^{1}Y) the module of q-forms on J^{1}Y. A form η ∈ Ω^{q}(J^{1}Y) is called π_{1}-horizontal (resp. π_{1,0}-horizontal) if i_{ξ}η = 0 for every π_{1}-vertical (resp. π_{1,0}-vertical) vector field ξ on J^{1}Y; η ∈ Ω^{q}(J^{1}Y) is called contact if J^{1}γ^{∗}η = 0 for every section γ of π [10]. A contact form η ∈ Ω^{q}(J^{1}Y) is called 1-contact if for every π_{1}-vertical vector field ξ the form i_{ξ}η is π_{1}-horizontal; it is called k-contact, where 2 ≤ k ≤ q, if for every π_{1}-vertical vector field ξ the form i_{ξ}η is (k − 1)-contact [12]. We denote

• Ω_{X}^{q}(J^{1}Y) the module of π_{1}-horizontal q-forms on J^{1}Y,

• Ω^{q−i,i}(J^{1}Y) the module of i-contact q-forms on J^{1}Y,

• Ω_{Y}^{q−i,i}(J^{1}Y) the submodule of Ω^{q−i,i}(J^{1}Y) consisting of π_{1,0}-horizontal forms.

It is important to mention that every form η ∈ Ω^{q}(J^{1}Y) has a unique decomposition into contact components as follows (Krupka [12]):

π_{2,1}^{∗}η = hη + p_{1}η + ⋯ + p_{q−1}η + p_{q}η,  (2.1)

where h and p_{k} (k ≥ 1) denotes the horizontalization and k-contactization operators, respectively, assigning to η its horizontal (resp. k-contact, 1 ≤ k ≤ q) component.

Therefore, we shall also use the following notations:

• Ω^{q(≥k)}(J^{1}Y) = Ω^{q−k,k}(J^{1}Y) ⊕ Ω^{q−k−1,k+1}(J^{1}Y) ⊕ ⋯ ⊕ Ω^{0,q}(J^{1}Y), i.e. the module of q-forms on J^{1}Y that are at least k-contact,

• Ω_{Y}^{q(≥k)}(J^{1}Y) = Ω_{Y}^{q−k,k}(J^{1}Y) ⊕ Ω_{Y}^{q−k−1,k+1}(J^{1}Y) ⊕ ⋯ ⊕ Ω_{Y}^{0,q}(J^{1}Y), i.e. π_{1,0}-horizontal q-forms on J^{1}Y that are at least k-contact.

We denote by (x^{i}, y^{σ}), where 1 ≤ i ≤ n, 1 ≤ σ ≤ m, local fibered coordinates on Y, and by (x^{i}, y^{σ}, y_{j}^{σ}) and (x^{i}, y^{σ}, y_{j}^{σ}, y_{jk}^{σ}), where 1 ≤ j ≤ k ≤ n, associated coordinates on J^{1}Y and J^{2}Y, respectively. We put

ω_{0} = dx^{1} ∧ ⋯ ∧ dx^{n}, ω_{j} = i_{∂/∂x^{j}}ω_{0}.  (2.2)

In case that dim X = 1, we write (t, q^{σ}), where 1 ≤ σ ≤ m, to denote local fibered coordinates on Y, and (t, q^{σ}, q̇^{σ}) (resp. (t, q^{σ}, q̇^{σ}, q̈^{σ})) for associated coordinates on J^{1}Y (resp. J^{2}Y).

In calculations we use either a canonical basis of one forms, i.e. (dx^{i}, dy^{σ}, dy_{j}^{σ}) on J^{1}Y and (dx^{i}, dy^{σ}, dy_{j}^{σ}, dy_{jl}^{σ}) on J^{2}Y (alternatively, if dim X = 1, (dt, dq^{σ}, dq̇^{σ}) and (dt, dq^{σ}, dq̇^{σ}, dq̈^{σ})), or better a basis adapted to the contact structure, i.e. (dx^{i}, ω^{σ}, dy_{j}^{σ}) on J^{1}Y and (dx^{i}, ω^{σ}, ω_{j}^{σ}, dy_{jl}^{σ}), on J^{2}Y, where

ω^{σ} = dy^{σ} − y_{i}^{σ} dx^{i}, ω_{j}^{σ} = dy_{j}^{σ} − y_{ji}^{σ} dx^{i}  (2.3)

are local canonical contact 1-forms. Alternatively, if dim X = 1, adapted bases take the form (dt, ω^{σ}, dq̇^{σ}) and (dt, ω^{σ}, ω̇^{σ}, dq̈^{σ}), where

ω^{σ} = dq^{σ} − q̇^{σ} dt, ω̇^{σ} = dq̇^{σ} − q̈^{σ} dt.  (2.4)

In an adapted basis to the contact structure every k-contact component p_{k}η of a q-form η (where 1 ≤ k ≤ q) is expressed by means of a wedge product containing exactly k of the canonical contact 1-forms above.

If f is a function on J^{1}Y, we have by (2.1) the exterior derivative df canonically splitted into the horizontal and contact component,

π_{2,1}^{∗}df = hdf + pdf,  (2.5)

with

hdf = \frac{df}{dx^{j}} dx^{j},  (2.6)

where d/dx^{j}, 1 ≤ j ≤ n, denotes the j-th total derivative operator (also called j-th formal derivative operator),

\frac{d}{dx^{j}} = \frac{∂}{∂x^{j}} + y_{j}^{σ} \frac{∂}{∂y^{σ}} + y_{ij}^{σ} \frac{∂}{∂y_{i}^{σ}} .  (2.7)

For convenience of notations we also use the ‘cut’ total derivative operators,

\frac{d′}{dx^{j}} = \frac{∂}{∂x^{j}} + y_{j}^{σ} \frac{∂}{∂y^{σ}} = \frac{d}{dx^{j}} − y_{ij}^{σ} \frac{∂}{∂y_{i}^{σ}} , 1 ≤ j ≤ n.  (2.8)

If dim X = 1, these formulas take the following form:

hdf = \frac{df}{dt} dt,  (2.9)
\frac{d}{dt} = \frac{∂}{∂t} + q̇^{σ} \frac{∂}{∂q^{σ}} + q̈^{σ} \frac{∂}{∂q̇^{σ}} ,  (2.10)
\frac{d′}{dt} = \frac{∂}{∂t} + q̇^{σ} \frac{∂}{∂q^{σ}} = \frac{d}{dt} − q̈^{σ} \frac{∂}{∂q̇^{σ}} .  (2.11)

Definition 2.1. 1-contact (n+1)-forms on J^{r}Y, horizontal with respect to the projection π_{r,0}, are called dynamical forms of order r [16].

Horizontal n-forms on J^{r}Y are called Lagrangians of order r [10]. By a local Lagrangian (of order r) we shall mean a Lagrangian defined on an open subset of J^{r}Y.

Definition 2.2. [10] Let λ be a Lagrangian on J^{1}Y. An n-form ρ is called Lepage equivalent of λ if hρ = λ and p_{1}dρ is a dynamical form. The form p_{1}dρ is then called the Euler–Lagrange form of λ and denoted by E_{λ}.

As proved in [12], every Lagrangian has a Lepage equivalent. For a Lagrangian of order r Lepage equivalents are of order 2r − 1, and the Euler–Lagrange form is of order 2r. It should be stressed that while Lepage equivalent of a Lagrangian need not be unique, the Euler–Lagrange form always is unique.

3. Mechanical systems with constraints

Throughout this section we consider a fibered manifold π : Y → X, dim X = 1, and we assume dim Y = m + 1, where m ≥ 1. Main sources for our exposition are the following: [1, 8, 11, 12, 35, 36] for the inverse variational problem, [15, 16, 21] for a geometric approach to variational ordinary differential equations, [7, 10, 12, 14, 16] for (unconstrained) Lagrangian and Hamiltonian mechanics in jet bundles, [15, 28] for the model of the non-holonomic constraint structure, and [3, 4, 6, 9, 15, 17, 20, 27, 28, 29, 30, 31, 33, 38] for non-holonomic Lagrangian and Hamiltonian systems.

3.1. Dynamical forms. Let E be a dynamical form on J^{2}Y. A section γ of π is called a path of E if

E ∘ J^{2}γ = 0.  (3.1)

In fibered coordinates E reads

E = E_{σ}ω^{σ} ∧ dt,  (3.2)

where E_{σ} are functions of (t, q^{ν}, q̇^{ν}, q̈^{ν}), and the equation for paths of E takes the form of a system of m second-order ordinary differential equations for the components (γ^{ν}) of γ as follows:

E_{σ}(t, γ^{ν}(t), \frac{dγ^{ν}}{dt} , \frac{d^{2}γ^{ν}}{dt^{2}} ) = 0, 1 ≤ σ ≤ m.  (3.3)

We stress that these equations need not be “solvable with respect to the second derivatives”, meaning that they need not be expressible in a normal form,

\frac{d^{2}γ^{σ}}{dt^{2}} = F^{σ}(t, γ^{ν}(t), \frac{dγ^{ν}}{dt} ).  (3.4)

Equations for paths of dynamical forms can be represented by means of exterior differential systems locally generated by 1-forms [14].

Proposition 3.1. Let E be a dynamical form on J^{2}Y. A section γ of π is a path of E if and only if

J^{2}γ^{∗}i_{ξ}α = 0 ∀ξ ∈ V(J^{2}Y),  (3.5)

where α is any 2-form such that p_{1}α = E.

Proof. By a direct computation we immediately obtain that (3.1) is equivalent with the condition J^{2}γ^{∗}i_{ξ}E = 0 ∀ξ ∈ V(J^{2}Y). Now, since contraction by vertical vector fields is compatible with the decomposition of forms to contact components (2.1), and prolongations of sections annihilate contact forms, we can see that adding to E (which is 1-contact) any 2-contact form F gives us

J^{2}γ^{∗}i_{ξ}(E + F) = J^{2}γ^{∗}i_{ξ}E + J^{2}γ^{∗}i_{ξ}F = J^{2}γ^{∗}i_{ξ}E ∀ξ ∈ V(J^{2}Y). (3.6)
□

Definition 3.1. [15] Let E ∈ Ω_{Y}^{2} (J^{2}Y) be a dynamical form. The equivalence class of 2-forms (on an open subset U ⊂ J^{2}Y) defined by

α_{1} ∼ α_{2} iff p_{1}α_{1} = p_{1}α_{2} = E|_{U}  (3.7)

is called Lepage class of E on U. The family of all local Lepage classes of E will be referred to as Lepage class of E and will be denoted by [α]_{E}, or simply [α].

By the above proposition, the equation for paths of E (on U) coincides with equations for holonomic integral sections of the distribution

Δ_{α} = annih {i_{ξ}α | ∀ξ ∈ V(J^{2}Y)} = span {ζ ∈ X (J^{2}Y) | i_{ζ}α = 0}, (3.8)

where α is any representative of the Lepage class of E (on U).

Definition 3.2. [15] Let [α] be a Lepage class of E. Every representative α ∈ [α] is called a Hamiltonian system associated with E. The distribution Δ_{α} is called a dynamical distribution of E. Equations for (all) integral sections of Δ_{α} are called Hamilton equations associated with E.

In what follows we shall be interested in dynamical forms that can be represented by first-order Lepage classes. This means that the dynamics are described by dynamical distributions defined on (open subsets of) J^{1}Y.

Proposition 3.2. Let E be a dynamical form on J^{2}Y. The following conditions are equivalent:

(1) Around each point in J^{1}Y there exists a Lepage class of E.

(2) In every fibered chart, E takes the form (3.2), where the functions E_{σ} are affine in the second derivatives, i.e.,

E_{σ} = A_{σ}(t, q^{ν}, q̇^{ν}) + B_{σρ}(t, q^{ν}, q̇^{ν})q̈^{ρ}.  (3.9)

Proof. We have

\begin{matrix}α = E+F =E_{σ}ω^{σ}∧ dt+F_{σν}ω^{σ}∧ ω^{ν} + F_{σν}^{01}ω^{σ}∧ ω̇^{ν} + F_{σν}^{11}ω̇^{σ} ∧ ω̇^{ν} \\ = (E_{σ}−2F_{σν}q̇^{ν}−F_{σν}^{01}q̈^{ν})dq^{σ} ∧ dt + (F_{νσ}^{01}q̇^{σ}−2F_{σν}^{11}q̈^{ν})dq̇^{σ} ∧ dt \\ +F_{σν}dq^{σ} ∧ dq^{ν} + F_{σν}^{01}dq^{σ} ∧ dq̇^{ν} + F_{σν}^{11}dq̇^{σ} ∧ dq̇^{ν}. \end{matrix} (3.10)

Hence, α is projectable onto an open subset of J^{1}Y iff F_{σν} and F_{σν}^{01} do not depend on q̈^{ρ}, F_{σν}^{11} = 0, and

\frac{∂E_{σ}}{∂q̈^{ρ}} = F_{σρ}^{01};  (3.11)

consequently,

\frac{∂^{2}E_{σ}}{∂q̈^{ρ}∂q̈^{ν}} = 0.  (3.12)

The first-order Lepage class is represented by 2-forms

α = A_{σ}ω^{σ} ∧ dt + B_{σν}ω^{σ} ∧ dq̇^{ν} + F_{σν}ω^{σ} ∧ ω^{ν},  (3.13)

where F_{σν} are arbitrary functions of (t, q^{ρ}, q̇^{ρ}). □

Definition 3.3. A dynamical form on J^{2}Y that has a Lepage class around each point of J^{1}Y is called J^{1}Y -pertinent, or, a first-order mechanical system.

Finally, we recall the concept of a regular dynamical form.

Definition 3.4. [14, 15] A first-order mechanical system (respectively, a J^{1}Y -pertinent dynamical form) E is called regular if around each point of J^{1}Y there exists a dynamical distribution Δ_{α}, α ∈ [α]_{E} such that rank Δ_{α} = 1.

Proposition 3.3. [14, 15] The following conditions are equivalent: (1) A first-order mechanical system E is regular.

(2) The following condition holds:

det(\frac{∂E_{σ}}{∂q̈^{ν}} ) = det(B_{σν}) ≠ 0.  (3.14)

(3) Equations for paths of E have an equivalent normal form (3.4) where F^{σ} = −B^{σν}A_{ν}.

3.2. Variational ODE’s and related Hamiltonian systems. A dynamical form E ∈ Ω_{Y}^{2} (J^{2}Y) is called (globally) variational if there exists a Lagrangian λ such that (possibly up to a projection), E = E_{λ}. E is called locally variational if it is variational in a neighborhood of every point in J^{2}Y [11, 12]. In fibered coordinates this means that the components E_{σ} of E take the form of Euler–Lagrange expressions of λ = L dt, i.e.

E_{σ} = \frac{∂L}{∂q^{σ}} − \frac{d}{dt} \frac{∂L}{∂q̇^{σ}}  (3.15)

if λ is a first order Lagrangian.

It is known that a locally variational form need not be globally variational [35]: obstructions come from the topology of the manifold Y. Every (globally) variational form on J^{2}Y possesses a global secondorder Lagrangian. This Lagrangian is locally equivalent with first-order Lagrangians (we say that it can be locally reduced to first-order Lagrangians).

A dynamical form E ∈ Ω_{Y}^{2} (J^{2}Y) is locally variational if and only if its components E_{σ}, 1 ≤ σ ≤ m, satisfy the Helmholtz conditions [8]

\begin{aligned}\frac{∂E_{σ}}{∂q̈^{ν}} − \frac{∂E_{ν}}{∂q̈^{σ}} & = & 0, \\ \frac{∂E_{σ}}{∂q̇^{ν}} + \frac{∂E_{ν}}{∂q̇^{σ}} − 2\frac{d}{dt} \frac{∂E_{ν}}{∂q̈^{σ}} & = & 0, \\ \frac{∂E_{σ}}{∂q^{ν}} − \frac{∂E_{ν}}{∂q^{σ}} + \frac{d}{dt} \frac{∂E_{ν}}{∂q̇^{σ}} − \frac{d^{2}}{dt^{2}} \frac{∂E_{ν}}{∂q̈^{σ}} & = & 0. \end{aligned}  (3.16)

Local (second-order) Lagrangians then can be constructed using the following formula [36]

L = q^{σ} ∫_{0}^{1} E_{σ}(t, uq^{ν}, uq̇^{ν}, uq̈^{ν})du.  (3.17)

Notice that from the Helmholtz conditions one easily gets that every second-order locally variational form is J^{1}Y -pertinent, i.e., defines a firstorder mechanical system; it is called a first-order Lagrangian system.

Recall that every representative α of the Lepage class of E is called a Hamiltonian system associated with E.

As shown in [10], every first-order Lagrangian λ has a unique first-order Lepage equivalent, the Cartan form, denoted by Θ_{λ}. Consequently, local first-order Lepage classes of E are represented by 2-forms

α = dΘ_{λ} + F,  (3.18)

where F is an arbitrary 2-contact form on the domain of definition of dΘ_{λ}. Hence, if λ is (any) Lagrangian for E on U ⊂ J^{1}Y, the Lepage class of E on U is given by [α]|_{U} = [dΘ_{λ}]. To simplify notations, we write with an obvious inaccuracy,

[α] = [dΘ_{λ}] = dΘ_{λ} mod Ω_{Y}^{0,2}(J^{1}Y).  (3.19)

Moreover, we have the following stronger result:

Theorem 3.1. (Krupková [14, 15]). Every first-order Lepage class of E has a unique closed representative, defined on J^{1}Y.

The unique closed 2-form mentioned above is denoted by α_{E} and called the Lepage equivalent of E [14]. If λ is a Lagrangian for E (possibly local, of order r ≥ 1) then (up to a projection) α_{E}|_{U} = dΘ_{λ}; here U denotes the domain of definition of dΘ_{λ}.

In fibered coordinates, where E = E_{σ}ω^{σ} ∧ dt and λ = L dt, we have

α_{E} = E_{σ}ω^{σ} ∧ dt + \frac{1}{2} \frac{∂E_{σ}}{∂q̇^{ν}} ω^{σ} ∧ ω^{ν} + \frac{∂E_{σ}}{∂q̈^{ν}} ω^{σ} ∧ ω̇^{ν},  (3.20)
Θ_{λ} = L dt + \frac{∂L}{∂q̇^{σ}} ω^{σ},  (3.21)

and, on the domain of definition of λ,

E = E_{λ} = p_{1}dΘ_{λ} = E_{σ}dq^{σ} ∧ dt where E_{σ} = \frac{∂L}{∂q^{σ}} − \frac{d}{dt} \frac{∂L}{∂q̇^{σ}} . (3.22)

Since the functions E_{σ} are affine in the q̈’s, we write

E_{σ} = A_{σ} + B_{σν}q̈^{ν},  (3.23)

where A_{σ} and B_{σν} are functions of (t, q^{σ}, q̇^{σ}),

B_{σν} = −\frac{∂^{2}L}{∂q̇^{σ}∂q̇^{ν}} , A_{σ} = \frac{∂L}{∂q^{σ}} − \frac{∂^{2}L}{∂t ∂q̇^{σ}} − \frac{∂^{2}L}{∂q^{ν}∂q̇^{σ}} q̇^{ν}.  (3.24)

Remark 3.1. In what follows we shall always assume that E is defined on J^{2}Y and is everywhere nontrivially of order 2. This means that (B_{σν}) in (3.23) is everywhere a non-zero matrix, or, equivalently, for every Lagrangian λ the Cartan 2-form dΘ_{λ} is everywhere nontrivially of order one.

Definition 3.5. Paths of a locally variational form are called extremals. Equations for paths of a locally variational form (respectively, equations for holonomic integral sections of associated dynamical distributions) are called Euler–Lagrange equations. Equations for integral sections of the dynamical distributions are called Hamilton equations, their integral sections are then called Hamilton extremals. The dynamical distribution Δ_{α_{E}} is called the Euler–Lagrange distribution.

Note that locally for every Lagrangian λ of E, Δ_{α_{E}} = Δ_{dΘ_{λ}}.

A principal question in the theory of Lagrangian systems on fibered manifolds is the relationship between Hamilton equations on one side and Euler–Lagrange equations on the other side [13]. It is clear that every extremal prolonged to J^{1}Y is a Hamilton extremal. The converse, however need not hold: a Hamilton extremal need not be a solution of the Euler–Lagrange equations.

The problem of equivalence between the set of extremals and Hamilton extremals is solved by the following theorem.

Theorem 3.2. (Krupková [15]). If E is regular then the dynamical distributions Δ_{α}, where α belongs to the first-order Lepage class of E, coincide on the common domain of definition (and their rank equals to 1). Consequently, if E is regular then for every α ∈ [α]_{E} the Hamilton equations are equivalent with the Euler–Lagrange equations.

Notice that if E is regular then every dynamical distribution is locally spanned by the following semispray:

ζ = \frac{∂}{∂t} + q̇^{σ} \frac{∂}{∂q^{σ}} − B^{σρ}A_{ρ} \frac{∂}{∂q̇^{σ}} ,  (3.25)

where (B^{σρ}) is the inverse matrix to (B_{ρν}).

Definition 3.6. A Lagrangian λ is called regular if its Euler–Lagrange form E_{λ} is regular [14].

Proposition 3.3 and Theorem 3.1 easily imply that we have the following equivalent characterizations of a regular Lagrangian:

(1) rank Δ_{dΘ_{λ}} = corank dΘ_{λ} = 1.

(2) If λ is a first-order Lagrangian,

det (\frac{∂^{2}L}{∂q̇^{σ}∂q̇^{ν}} ) ≠ 0.  (3.26)

In view of Theorem 3.1 and a theorem on a canonical form of the Lepage equivalent of E ([14]) one obtains the following result:

Proposition 3.4. Let E be regular. Then in a neighborhood of every point in J^{1}Y there is a local coordinate transformation (t, q^{σ}, q̇^{σ}) → (t, q^{σ}, p_{σ}) such that every α belonging to the first-order Lepage class of E takes the canonical form

α = −dH ∧ dt + dp_{σ} ∧ dq^{σ} + F,  (3.27)

where F is a 2-contact π_{1,0}-horizontal 2-form.

This transformation is called Legendre transformation.

Functions H and p_{σ} above can be expressed in terms of a first-order Lagrangian λ for E; it holds

p_{σ} = \frac{∂L}{∂q̇^{σ}} ,  H = −L + p_{σ}q̇^{σ}.  (3.28)

In Legendre coordinates Hamilton equations take the “canonical form”

\frac{d(p_{σ} ∘ δ)}{dt} = −\frac{∂H}{∂q^{σ}} ,  \frac{d(q^{σ} ∘ δ)}{dt} = \frac{∂H}{∂p_{σ}} .  (3.29)

Summarizing, for a regular Lagrangian system (represented by a regular locally variational form E on J^{2}Y), all related Hamiltonian systems are (locally) equivalent and Hamilton equations are equivalent with Euler–Lagrange equations. Hamilton extremals coincide with prolongations of extremals, and are solutions of the canonical equations (3.29).

3.3. Nonholonomic constraints. Let us introduce the non-holonomic constraint structure in J^{1}Y, as defined in [15].

Definition 3.7. By a constraint submanifold or a non-holonomic constraint in J^{1}Y we shall understand a submanifold Q ⊂ J^{1}Y, fibered over Y, precisely speaking, a surjective submersion π_{1,0}|_{Q} : Q → Y.

We denote by k the codimension of Q and assume that 1 ≤ k ≤ m − 1. A nonholonomic constraint Q in J^{1}Y of codimension k can be locally

expressed by equations

f^{a}(t, q^{σ}, q̇^{σ}) = 0, 1 ≤ a ≤ k,  (3.30)

where

rank (\frac{∂f^{a}}{∂q̇^{σ}} ) = k,  (3.31)

or, equivalently, by equations in a normal form,

q̇^{m−k+a} − g^{a}(t, q^{σ}, q̇^{1}, …, q̇^{m−k}) = 0, 1 ≤ a ≤ k.  (3.32)

A section γ of π defined on an open set W ⊂ X is called a holonomic path in Q if for every x ∈ W

J^{1}γ(x) ∈ Q.  (3.33)

Given a constraint submanifold Q in J^{1}Y there naturally arise the following local distributions, defined on the domain U of definition of the functions f^{a}:

(1) D_{U} = annih {df^{a}, 1 ≤ a ≤ k}; rank D_{U} is constant on U due to (3.31) and equal to 2m + 1 − k.

(2) C̃_{U} = annih {ϕ^{a}, 1 ≤ a ≤ k}, where

ϕ^{a} = f^{a}dt + \frac{∂f^{a}}{∂q̇^{σ}} ω^{σ}, 1 ≤ a ≤ k.  (3.34)

(3) C_{U} = annih {ϕ^{a}, df^{a}, 1 ≤ a ≤ k}.

C̃_{U} is called extended constraint distribution; it has a constant rank equal to 2m + 1 − k. C_{U} is called constraint distribution related to the constraint submanifold Q on U, and its rank equals to 2m + 1 − 2k.

The following assertions hold ([15]):

Proposition 3.5. Q ∩ U is an integral submanifold of D_{U}. For every point x ∈ Q, the forms df^{a}(x), 1 ≤ a ≤ k, annihilate the tangent space T_{x}Q to the manifold Q at x, i.e. along Q, D = annih {df^{a}, 1 ≤ a ≤ k} = T Q.

Corollary 3.1. Let Q be a non-holonomic constraint of codimension k in J^{1}Y, and let f^{a} = 0 and f̄^{a} = 0, where 1 ≤ a ≤ k, be two sets of equations of Q on an open set U ⊂ J^{1}Y. Then there are functions γ_{j}^{a} on U such that at each point of U, (γ_{j}^{a}) is a regular matrix, and df̄^{a} = γ_{j}^{a}df^{j}. In particular, at each point x ∈ Q ∩ U,

\frac{∂f̄^{a}}{∂q̇^{σ}} = γ_{j}^{a} \frac{∂f^{j}}{∂q̇^{σ}} .  (3.35)

Proposition 3.6. C_{U} is a subdistribution of both C̃_{U} and D_{U}. At the points of Q ∩ U, the distributions C_{U} and C̃_{U} ∩ D coincide, and define a distribution of corank k on Q ∩ U.

Theorem 3.3. Let Q be a non-holonomic constraint in J^{1}Y, let ι : Q → J^{1}Y be the canonical embedding of the submanifold Q into J^{1}Y. Put

φ̄^{a} = ι^{∗}ϕ^{a}, 1 ≤ a ≤ k.  (3.36)

Then

C = annih {φ̄^{a}, 1 ≤ a ≤ k}  (3.37)

is a distribution of corank k on Q.

Note that in fibered coordinates

φ̄^{a} = (\frac{∂f^{a}}{∂q̇^{σ}} ∘ ι)ω̄^{σ} = ω̄^{m−k+a} − \frac{∂g^{a}}{∂q̇^{l}} ω̄^{l}  (3.38)

where we have denoted

ω̄^{σ} = ι^{∗}ω^{σ} = dq^{σ} − (q̇^{σ} ∘ ι)dt,  (3.39)

i.e.,

\begin{matrix}ω̄^{l} = dq^{l} − q̇^{l}dt, 1 ≤ l ≤ m − k, \\ ω̄^{m−k+a} = dq^{m−k+a} − g^{a}dt, 1 ≤ a ≤ k. \end{matrix}  (3.40)

Definition 3.8. [15] The distribution C (3.37) on Q is called canonical distribution. 1-forms belonging to the annihilator C^{0} of C, are called canonical constraint 1-forms. The ideal in the exterior algebra of differential forms on Q generated by C^{0} is called canonical constraint ideal, and denoted by I(C^{0}); its homogeneous component of degree p is denoted by I^{p}(C^{0}). Elements of the ideal I(C^{0}) are called canonical constraint forms.

One can show by a direct computation that the canonical distribution can be equivalently locally spanned by the following system of vector fields:

\begin{matrix}\frac{∂_{c}}{∂t} ≡ \frac{∂}{∂t} + ∑_{a=1}^{k}(g^{a} − \frac{∂g^{a}}{∂q̇^{l}} q̇^{l})\frac{∂}{∂q^{m−k+a}} , \\ \frac{∂_{c}}{∂q^{s}} ≡ \frac{∂}{∂q^{s}} + ∑_{a=1}^{k} \frac{∂g^{a}}{∂q̇^{s}} \frac{∂}{∂q^{m−k+a}} , 1 ≤ s ≤ m − k, \\ \frac{∂}{∂q̇^{s}} , 1 ≤ s ≤ m − k. \end{matrix}  (3.41)

For the sake of simplicity we shall also use the following notations:

\begin{matrix}\frac{d_{c}}{dt} = \frac{∂}{∂t} + q̇^{l} \frac{∂}{∂q^{l}} + g^{a} \frac{∂}{∂q^{m−k+a}} + q̈^{l} \frac{∂}{∂q̇^{l}} = \frac{∂_{c}}{∂t} + q̇^{l} \frac{∂_{c}}{∂q^{l}} + q̈^{l} \frac{∂}{∂q̇^{l}} , \\ \frac{d′_{c}}{dt} = \frac{∂}{∂t} + q̇^{l} \frac{∂}{∂q^{l}} + g^{a} \frac{∂}{∂q^{m−k+a}} = \frac{∂_{c}}{∂t} + q̇^{l} \frac{∂_{c}}{∂q^{l}} \\ = \frac{d_{c}}{dt} − q̈^{l} \frac{∂}{∂q̇^{l}} = \frac{d′}{dt} ∘ ι. \end{matrix} (3.42)

In general, the canonical distribution is not completely integrable. There are two interesting particular cases of non-holonomic constraints as follows:

Definition 3.9. [15] A non-holonomic constraint Q is called

(1) simple if the canonical distribution C is projectable onto a distribution on Y,

(2) semiholonomic if the canonical distribution C is completely integrable.

It can be proved [15] that every semiholonomic constraint is simple. Consequently it can be equivalently modeled as either

• a fibered submanifold Q ⊂ J^{1}Y with the canonical distribution C completely integrable (i.e., the canonical constraint ideal I(C^{0}) closed), or

• a completely integrable, nowhere vertical distribution on the total space Y.

Another result shows [15] that a non-holonomic constraint is simple if and only if it is locally defined by equations affine in velocities. Consequently, a simple non-holonomic constraint can be equivalently modeled as either

• a fibered submanifold Q ⊂ J^{1}Y with the canonical distribution C projectable onto a distribution on Y, or

• a nowhere vertical distribution on Y, which need not be completely integrable.

3.4. Constrained Lagrangian systems. Let us consider a Lagrangian system on J^{1}Y. Recall that it is defined by a locally variational form E on J^{2}Y, as the first-order Lepage class [α] of E (see (3.19))

[α] = [dΘ_{λ}] = dΘ_{λ} mod Ω_{Y}^{0,2}(J^{1}Y).  (3.43)

If ι : Q → J^{1}Y is a non-holonomic constraint and I(C^{0}) the corresponding canonical constraint ideal, we have another equivalence, denoted by ≈, on 2-forms on Q (with the same domain of definition):

η_{1} ≈ η_{2} iff η_{1} − η_{2} = F̄ + φ,  (3.44)

where F̄ is a (local) 2-contact 2-form on Q, and φ is a constraint 2form. We denote by [[η]] the class of η. If [α] is a Lepage class on J^{1}Y associated with a locally variational form E ∈ Ω^{2}(J^{2}Y) then for any of its two elements defined on the same subset of J^{1}Y,

α_{1} ∼ α_{2} ⇒ ι^{∗}α_{1} ≈ ι^{∗}α_{2}.  (3.45)

Definition 3.10. [15, 20] Let [α] = [dΘ_{λ}] be a Lagrangian system on J^{1}Y. By the associated constrained Lagrangian system we mean the class [[ι^{∗}α]] = [[ι^{∗}dΘ_{λ}]]. Each form ι^{∗}dΘ_{λ} + φ, where φ ∈ I^{2}(C^{0}), is called constrained Cartan 2-form of λ.

Note that every element of [[ι^{∗}dΘ_{λ}]] is of the form

ᾱ = ι^{∗}dΘ_{λ} + F̄ + φ,  (3.46)

where F̄ ∈ Ω_{Y}^{0,2}(Q) and φ ∈ I^{2}(C^{0}).

In fibered coordinates, where Q is given by (3.32) and the Euler– Lagrange form of λ is represented by (3.23), (3.24), we have [15, 20]

ᾱ = Ā_{l}ω̄^{l} ∧ dt + B̄_{ls}ω̄^{l} ∧ dq̇^{s} + F̄_{ls} ω̄^{l} ∧ ω̄^{s} + φ,  (3.47)

where ω̄^{l} = ι^{∗}ω^{l}, φ ∈ I^{2}(C^{0}), F̄_{ls} are arbitrary, and

\begin{matrix}Ā_{l}=(A_{l}+A_{m−k+i} \frac{∂g^{i}}{∂q̇^{l}}+(B_{l,m−k+i}+B_{m−k+j,m−k+i} \frac{∂g^{j}}{∂q̇^{l}} ) \frac{d′g^{i}}{dt} ) ∘ ι, \\ B̄_{ls}=(B_{ls}+B_{l,m−k+i} \frac{∂g^{i}}{∂q̇^{s}}+B_{s,m−k+i} \frac{∂g^{i}}{∂q̇^{l}}+B_{m−k+i,m−k+j} \frac{∂g^{i}}{∂q̇^{l}} \frac{∂g^{j}}{∂q̇^{s}} )∘ ι, \end{matrix} (3.48)

and summations run over l, s = 1, 2, …, m − k and i, j = 1, 2, …, k. Since (B_{σν}) is a symmetric matrix, the above formula gives us that the matrix (B̄_{ls}) is symmetric.

Definition 3.11. [15] The constraint dynamical distribution related with a 2-form ᾱ, denoted by Δᾱ, is defined to be the subdistribution of the canonical distribution C, annihilated by the 1-forms i_{ξ}ᾱ, where ξ runs over all π_{1}-vertical vector fields on Q belonging to C. This means that

Δᾱ = annih {φ̄^{a}, 1 ≤ a ≤ k, i_{ξ}ᾱ, ∀ξ ∈ V(J^{1}Y) ∩ C}.  (3.49)

In particular, the constraint dynamical distribution related with a constrained Cartan 2-form is called constraint Euler–Lagrange distribution.

Definition 3.12. [15, 20] Let [[ι^{∗}dΘ_{λ}]] be a constrained Lagrangian system. Then for any representative ᾱ of the class [[ι^{∗}dΘ_{λ}]], equations for holonomic integral sections of the constraint dynamical distribution Δᾱ, i.e., the equations

J^{1}γ^{∗}i_{ξ}ᾱ = 0 for every π_{1}-vertical vector field ξ ∈ C,  (3.50)

where γ : W → Y, J^{1}γ(W) ⊂ Q, are called constrained Euler–Lagrange equations. Solutions of constrained Euler–Lagrange equations are called constrained extremals.

We note that (locally) constrained Euler–Lagrange equations do not depend upon the choice of a representative ᾱ of the class [[ι^{∗}dΘ_{λ}]]. This means that with help of a (local, possibly higher-order) Lagrangian λ for E we can write the constrained Euler–Lagrange equations in the form

J^{1}γ^{∗}i_{ξ}(ι^{∗}dΘ_{λ}) = 0 for every π_{1}-vertical vector field ξ ∈ C, (3.51)

where γ : W → Y, J^{1}γ(W) ⊂ Q.

For λ = L dt denote

L̄ = L ∘ ι, L̄_{a} = \frac{∂L}{∂q̇^{m−k+a}} ∘ ι, 1 ≤ a ≤ k,  (3.52)

and

Θ_{ι^{∗}λ} = L̄ dt + ∑_{s=1}^{m−k} \frac{∂L̄}{∂q̇^{s}} ω̄^{s}.  (3.53)

We get the following relation between the forms ι^{∗}Θ_{λ} and Θ_{ι^{∗}λ}:

Proposition 3.7. [20]

ι^{∗}Θ_{λ} = Θ_{ι^{∗}λ} + L̄_{a} φ̄^{a}.  (3.54)

For convenience we shall use the notation

μ_{s} = \frac{∂_{c}}{∂q^{s}} − \frac{d_{c}}{dt} \frac{∂}{∂q̇^{s}} , μ′_{s} = \frac{∂_{c}}{∂q^{s}} − \frac{d′_{c}}{dt} \frac{∂}{∂q̇^{s}}  (3.55)

for the so called C-modified Euler–Lagrange operator and cut C-modified Euler–Lagrange operator, respectively.

In fibered coordinates constrained Euler–Lagrange equations take the form a mixed system of m − k second-order and k first-order ODE’s for sections γ of π as follows:

Theorem 3.4. [15, 20] Let [dΘ_{λ}] be a Lagrangian system on J^{1}Y, Q ⊂ J^{1}Y a non-holonomic constraint. A section γ : W → Y of π is a constrained extremal if and only if it satisfies J^{1}γ(W) ⊂ Q, i.e.

f^{a} ∘ J^{1}γ = 0, 1 ≤ a ≤ k,  (3.56)

and the constrained Euler–Lagrange equations (3.51) . The latter take one of the following equivalent coordinate forms:

(1) By means of L,

(Ā_{l} + B̄_{ls}q̈^{s}) ∘ J^{2}γ = 0, 1 ≤ l ≤ m − k,  (3.57)

where Ā_{l}, B̄_{ls} are given by (3.24), (3.48).

(2) By means of L̄ and L̄_{a},

(μ_{l}(L̄) − L̄_{a} μ_{l}(g^{a})) ∘ J^{2}γ = 0, 1 ≤ l ≤ m − k.  (3.58)

Consequently, the functions Ā_{l}, B̄_{ls} are equivalently expressed as follows:

Ā_{l} = μ′_{l}(L̄) − L̄_{a} μ′_{l}(g^{a}), B̄_{ls} = −\frac{∂^{2}L̄}{∂q̇^{l}∂q̇^{s}} + L̄_{a} \frac{∂^{2}g^{a}}{∂q̇^{l}∂q̇^{s}} .  (3.59)

Definition 3.13. The operator

E_{s}^{C}(L̄, L̄_{a}) = μ_{s}(L̄) − L̄_{a} μ_{s}(g^{a}), 1 ≤ s ≤ m − k  (3.60)

is called the constraint Euler–Lagrange operator.

The definition of a regular constrained system is quite similar to the unconstrained case.

Definition 3.14. [15] A constrained Lagrangian system [[ι^{∗}dΘ_{λ}]] is called regular if around each point of Q there exists a constraint dynamical distribution Δᾱ such that rank Δᾱ = 1.

Regular constrained systems are characterized as follows (see [15, 38]):

Theorem 3.5. Let [[ι^{∗}dΘ_{λ}]] be a constrained Lagrangian system. The following conditions are equivalent:

(1) [[ι^{∗}dΘ_{λ}]] is regular.

(2) The (m − k) × (m − k)-matrix (B̄_{sl}) is regular, i.e.,

det(B̄_{sl}) ≠ 0.  (3.61)

(3) Every first-order Lagrangian λ = L dt satisfies the regularity condition

\begin{aligned}det ((\frac{∂^{2}L}{∂q̇^{l}∂q̇^{s}} & + & \frac{∂^{2}L}{∂q̇^{m−k+a} ∂q̇^{s}} \frac{∂g^{a}}{∂q̇^{l}} + \frac{∂^{2}L}{∂q̇^{m−k+a} ∂q̇^{l}} \frac{∂g^{a}}{∂q̇^{s}} \\ & + & \frac{∂^{2}L}{∂q̇^{m−k+a}∂q̇^{m−k+b}} \frac{∂g^{a}}{∂q̇^{l}} \frac{∂g^{b}}{∂q̇^{s}} ) ∘ ι) ≠ 0. \end{aligned} (3.62)

(4) Every first-order Lagrangian λ = L dt satisfies the regularity condition

det (\frac{∂^{2}L̄}{∂q̇^{l}∂q̇^{s}} − (\frac{∂L}{∂q̇^{m−k+a}} ∘ ι) \frac{∂^{2}g^{a}}{∂q̇^{l}∂q̇^{s}} ) ≠ 0,  (3.63)

where L̄ = L ∘ ι.

(5) Every constraint dynamical distribution is locally spanned by the following constraint semispray:

ζ = \frac{∂}{∂t} + ∑_{l=1}^{m−k} q̇^{l} \frac{∂}{∂q^{l}} + ∑_{a=1}^{k} g^{a} \frac{∂}{∂q^{m−k+a}} − ∑_{l,s=1}^{m−k} B̄^{ls}Ā_{s} \frac{∂}{∂q̇^{l}} ,  (3.64)

where (B̄^{ls}) is the inverse matrix to (B̄_{ls}).

(6) The constrained Euler–Lagrange equations have an equivalent form

\begin{aligned}q̇^{m−k+a} & = & g^{a}(t, q^{σ}, q̇^{1}, …, q̇^{m−k}),  1 ≤ a ≤ k, \\ q̈^{l} & = & −B̄^{ls}Ā_{s},  1 ≤ l ≤ m − k. \end{aligned}  (3.65)

We stress that, as one can see from any of the above equivalent regularity conditions, a constrained system arising from a regular Lagrangian system need not be regular.

Corollary 3.2. If Q ⊂ J^{1}Y is a simple non-holonomic constraint then the regularity condition reads

det (\frac{∂^{2}L̄}{∂q̇^{l}∂q̇^{s}} ) ≠ 0,  (3.66)

where L̄ = L ∘ ι.

3.5. Constrained Hamilton equations. Constrained Hamiltonian systems were studied in detail in [3, 9, 38].

Let [α] = [dΘ_{λ}] be a Lagrangian system on J^{1}Y, ι : Q → J^{1}Y a nonholonomic constraint, [[ι^{∗}α]] = [[ι^{∗}dΘ_{λ}]] the corresponding constrained system. For every ᾱ = ι^{∗}α ∈ [[ι^{∗}dΘ_{λ}]] we have the constraint dynamical distribution Δᾱ defined on the domain of definition of ᾱ, say U ⊂ Q.

Directly from the definition of constraint dynamical distribution we can see that if ᾱ_{1} and ᾱ_{2} differ by a constraint form, their constraint dynamical distributions Δᾱ_{1} and Δᾱ_{2} coincide.

Definition 3.15. [38] Let [dΘ_{λ}] be a Lagrangian system on J^{1}Y. For every α ∈ [dΘ_{λ}] the equivalence class

ᾱ_{I} = ι^{∗}α mod I^{2}(C^{0})  (3.67)

is called constrained Hamiltonian system related with α and the constraint Q.

Equations for integral sections of the corresponding constraint dynamical distribution Δᾱ, i.e.

δ^{∗}φ̄^{a} = 0, 1 ≤ a ≤ k, δ^{∗}i_{ξ}ᾱ = 0 ∀ξ ∈ V(Q) ∩ C,  (3.68)

where ᾱ ∈ ᾱ_{I} and δ is a section of π_{1}|_{Q} : Q → X, are called constrained Hamilton equations.

Note that for every ᾱ ∈ [[ι^{∗}dΘ_{λ}]] on U, holonomic integral sections of Δᾱ coincide with prolongations of constrained extremals in U.

Theorem 3.6. [38] Let [[ι^{∗}dΘ_{λ}]] be a regular constrained Lagrangian system. Then for any two its Hamiltonian systems ᾱ_{1I}, ᾱ_{2I} on an open subset U their constrained dynamical distributions coincide, i.e. Δᾱ_{1} = Δᾱ_{2}.

Corollary 3.3. If the regularity condition (3.61) is satisfied then constrained Euler–Lagrange equations are equivalent with (any) constrained Hamilton equations.

For regular constrained systems we can introduce a constraint Legendre transformation.

Theorem 3.7. [38] Let ι : Q → J^{1}Y be a non-holonomic constraint, [[ι^{∗}dΘ_{λ}]] a constrained Lagrangian system. Let x ∈ Q be a point. Suppose that in a neighborhood of x,

\frac{∂B̄_{ls}}{∂q̇^{r}} = \frac{∂B̄_{lr}}{∂q̇^{s}} , 1 ≤ l, r, s ≤ m − k.  (3.69)

Then there exists a neighborhood U ⊂ Q of x, and, on U, functions P_{l}, 1 ≤ l ≤ m − k, and a 1-form η, such that the class [[ι^{∗}dΘ_{λ}]] has a representative of the form

ᾱ′ = η ∧ dt + dP_{l} ∧ dq^{l}.  (3.70)

If, moreover, the constrained system [[ι^{∗}dΘ_{λ}]] is regular, then (t, q^{σ}, q̇^{l}) → (t, q^{σ}, P_{l}) is a coordinate transformation on U.

Proof. In a neighborhood of x, let us consider the elements of the equivalence class [[ι^{∗}dΘ_{λ}]] in the form (3.47). By assumption, from the Poincaré Lemma we get a neighborhood U ⊂ Q of x and functions P_{l}, 1 ≤ l ≤ m − k, on U such that

B̄_{ls} = −\frac{∂P_{l}}{∂q̇^{s}} .  (3.71)

Hence, in the class [[ι^{∗}dΘ_{λ}]] there is a local representative of the form

\begin{matrix}ᾱ = Ā_{l} ω̄^{l} ∧ dt + \frac{∂P_{l}}{∂q̇^{s}} dq̇^{s} ∧ ω̄^{l} \\ = Ā_{l}ω̄^{l} ∧ dt + dP_{l} ∧ ω̄^{l} − \frac{∂P_{l}}{∂t} dt ∧ ω̄^{l} − \frac{∂P_{l}}{∂q^{s}} dq^{s} ∧ ω̄^{l} \\ −\frac{∂P_{l}}{∂q^{m−k+a}} dq^{m−k+a} ∧ ω̄^{l} \\ = (Ā_{l} + \frac{∂P_{l}}{∂t} ) dq^{l} ∧ dt + dP_{l} ∧ dq^{l} − q̇^{l} dP_{l} ∧ dt \\ −\frac{∂P_{l}}{∂q^{s}} (ω̄^{s} + q̇^{s}dt) ∧ ω̄^{l} − \frac{∂P_{l}}{∂q^{m−k+a}} (ω̄^{m−k+a} + g^{a}dt) ∧ ω̄^{l} \\ = (Ā_{l} + \frac{∂P_{l}}{∂t} + \frac{∂P_{l}}{∂q^{s}} q̇^{s} + \frac{∂P_{l}}{∂q^{m−k+a}} g^{a}) dq^{l} ∧ dt − q̇^{l}dP_{l} ∧ dt \\ + dP_{l} ∧ dq^{l} + \frac{∂P_{l}}{∂q^{s}} ω̄^{l} ∧ ω̄^{s} − \frac{∂P_{l}}{∂q^{m−k+a}} ω̄^{m−k+a} ∧ ω̄^{l}. \end{matrix}  (3.72)

This means that we also have a representative

\begin{matrix}ᾱ′ = (Ā_{l} + \frac{∂P_{l}}{∂t} + \frac{∂P_{l}}{∂q^{s}} q̇^{s} + \frac{∂P_{l}}{∂q^{m−k+a}} g^{a}) dq^{l} ∧ dt − q̇^{s}dP_{s} ∧ dt \\ + dP_{l} ∧ dq^{l} \\ = (Ā_{l} + \frac{∂P_{l}}{∂t} + (\frac{∂P_{l}}{∂q^{s}} − \frac{∂P_{s}}{∂q^{l}} ) q̇^{s} + \frac{∂P_{l}}{∂q^{m−k+a}} g^{a}) dq^{l} ∧ dt \\ −\frac{∂P_{s}}{∂q^{m−k+a}} q̇^{s}dq^{m−k+a} ∧ dt − \frac{∂P_{s}}{∂q̇^{l}} q̇^{s}dq̇^{l} ∧ dt + dP_{l} ∧ dq^{l}. \end{matrix} (3.73)

We can write it in the form ᾱ′ = η ∧ dt + dP_{l} ∧ dq^{l} with

η = η̄_{0}dt + η̄_{l}dq^{l} + η̄_{m−k+a}dq^{m−k+a} + η̃_{l}dq̇^{l},  (3.74)

where η̄_{0} is an arbitrary function on U, and

\begin{aligned}η̄_{l} & = & Ā_{l} + \frac{∂P_{l}}{∂t} + (\frac{∂P_{l}}{∂q^{s}} − \frac{∂P_{s}}{∂q^{l}} ) q̇^{s} + \frac{∂P_{l}}{∂q^{m−k+a}} g^{a}, \\ η̃_{l} & = & −\frac{∂P_{s}}{∂q̇^{l}} q̇^{s},  η̄_{m−k+a} = −\frac{∂P_{l}}{∂q^{m−k+a}} q̇^{l}. \end{aligned}  (3.75)

Finally, the regularity condition for the transformation (t, q^{σ}, q̇^{l}) → (t, q^{σ}, P_{l}) coincides with (3.61). □

Remark 3.2. Condition (3.69) rewritten in terms of a first-order Lagrangian reads

(\frac{∂}{∂q̇^{r}} (\frac{∂L}{∂q̇^{m−k+a}} ∘ ι)) \frac{∂^{2}g^{a}}{∂q̇^{l}∂q̇^{s}} =(\frac{∂}{∂q̇^{s}} (\frac{∂L}{∂q̇^{m−k+a}} ∘ ι)) \frac{∂^{2}g^{a}}{∂q̇^{l}∂q̇^{r}} . (3.76)

The integrability condition for the B̄_{sl}’s ((3.69), (3.76)) ensures that one can express functions P_{l} explicitly. To this purpose we consider a mapping χ : [0, 1] × W → W defined by (u, t, q^{σ}, q̇^{l}) → (t, q^{σ}, uq̇^{l}), where W ⊂ Q is an appropriate open set. Then Poincaré Lemma gives us a solution [38]

\begin{matrix}P_{l} = −q̇^{s} ∫_{0}^{1}(B̄_{ls} ∘ χ) du \\ = \frac{∂L̄}{∂q̇^{l}} − q̇^{s} ∫_{0}^{1} ((\frac{∂L}{∂q̇^{m−k+a}} ∘ ι) \frac{∂^{2}g^{a}}{∂q̇^{l}∂q̇^{s}} ) ∘ χ du. \end{matrix}  (3.77)

Definition 3.16. [38] We call the above functions P_{l}, 1 ≤ l ≤ k, constraint momenta, and the corresponding coordinate transformation constraint Legendre transformation. The 1-form η in (3.70) is called a constraint energy 1-form.

The 1-form η is determined up to a constraint 1-form, and need not be closed. In constraint Legendre coordinates we can write

η_{I} = η_{0} dt + η_{l} dq^{l} + η^{l} dP_{l} mod I^{1}(C^{0}).  (3.78)

Corollary 3.4. If the Lagrangian system [[ι^{∗}dΘ_{λ}]] is regular, then the constraint Euler–Lagrange equations are equivalent with constraint Hamilton equations. In constraint Legendre coordinates Hamilton equations take the following canonical form

\frac{d}{dt}(P_{l} ∘ δ) = η_{l}, \frac{d}{dt}(q^{l} ∘ δ) = −η^{l}, \frac{d}{dt}(q^{m−k+a} ∘ δ) = g^{a}, (3.79)

where 1 ≤ l ≤ m − k, 1 ≤ a ≤ k. Constraint Hamilton equations depend upon the choice of a representative α ∈ [dΘ_{λ}], rather than on a particular Lagrangian λ.

For simple non-holonomic constraints, which, as we have seen in Sec. 3.3, can be modeled by a distribution on Y, and are given by (3.30) (resp. (3.32)) where the functions f^{a} (resp. g^{a}) are affine in the velocities) the situation essentially simplifies:

Theorem 3.8. Assume that Q is a simple non-holonomic constraint. Then (3.69) is fulfilled identically and the constraint momenta are defined by

P_{l} = \frac{∂L̄}{∂q̇^{l}} , 1 ≤ l ≤ m − k.  (3.80)

Regularity condition takes the form

det (\frac{∂^{2}L̄}{∂q̇^{l}∂q̇^{s}} ) ≠ 0.  (3.81)

Moreover, if the constraint Q is semiholonomic then the family of energy 1-forms (3.78) contains a closed 1-form equal to −dH̄, where

H̄ = −L̄ + P_{l}q̇^{l}.  (3.82)

3.6. Holonomic Lagrangian systems. The case of Lagrangian systems subjected to holonomic constraints can be considered as a special case of the nonholonomic theory (see [15]).

Definition 3.17. Let π : Y → X be a fibered manifold, dim X = 1, dim Y = m + 1, m > 1. By a holonomic constraint in Y we mean a fibered submanifold π_{0} : Q_{0} → X of π.

If X = R and Y = R × M where M is a manifold of dimension m, we also speak about a rheonomic constraint. If a rheonomic constraint is of the form Q_{0} = R × N where N is a submanifold of M, it is called skleronomic.

We denote by ι_{0} : Q_{0} → Y the canonical inclusion of Q_{0} into Y, and assume codim Q_{0} = k, where 1 ≤ k < m. The constraint Q_{0} is locally defined by a system of algebraic equations

u^{a}(t, q^{σ}) = 0, 1 ≤ a ≤ k,  (3.83)

where the functions u^{a} satisfy the rank condition

rank (\frac{∂u^{a}}{∂q^{σ}} ) = k.  (3.84)

Hence, around every point x ∈ Q_{0} there is a fibered chart (U, χ) on Y, adapted to the submanifold Q_{0}, i.e. χ = (t, q^{1}, …, q^{m−k}, u^{1}, …, u^{k}).

The fibered submanifold Q_{0} can be prolonged to J^{1}Q_{0} ⊂ J^{1}Y, codim J^{1}Q_{0} = 2k, locally defined by the equations

u^{a} = 0, \frac{du^{a}}{dt} = 0, 1 ≤ a ≤ k.  (3.85)

J^{1}Q_{0} is a submanifold of the manifold Q ⊂ J^{1}Y, defined by the equations

f^{a} ≡ \frac{du^{a}}{dt} = 0, 1 ≤ a ≤ k.  (3.86)

We can see that Q is a semiholonomic constraint in J^{1}Y, codim Q = k.

The only admissible holonomic paths in Q are sections γ of the fibered manifold Q_{0} → X, i.e. such that J^{1}γ ∈ J^{1}Q_{0}. This means that for a Lagrangian system [dΘ_{λ}] on J^{1}Y, the corresponding constrained system [[ι^{∗}dΘ_{λ}]] can be restricted to J^{1}Q_{0}.

Proposition 3.8. Let Q_{0} ⊂ Y be a holonomic constraint, Q the semiholonomic constraint related with Q_{0}. Let C be the canonical distribution on Q. Then for every x ∈ J^{1}Q_{0}

C(x) = T_{x}J^{1}Q_{0}.  (3.87)

Equivalently, the annihilator C^{0} of the canonical distribution on J^{1}Q_{0} is trivial, C^{0}(x) = {0}.

Corollary 3.5. Let Q_{0} be a holonomic constraint in Y, Q ⊂ J^{1}Y the associated semiholonomic constraint. Then the canonical distribution C on Q is completely integrable and projects onto a (completely integrable) distribution on Y. Along J^{1}Q_{0} the canonical distribution coincides with the tangent bundle to J^{1}Q_{0} and projects onto the tangent bundle T Q_{0} → Q_{0}.

Proposition 3.9. Let ι_{0} : Q_{0} → Y be a holonomic constraint, [dΘ_{λ}] a Lagrangian system on J^{1}Y. Then for every α ∈ [dΘ_{λ}]

(J^{1}ι_{0}^{∗} α)_{I} = J^{1}ι_{0}^{∗} α.  (3.88)

Moreover, for every first-order Lagrangian λ

J^{1}ι_{0}^{∗} Θ_{λ} = Θ_{J^{1}ι_{0}^{∗} λ}.  (3.89)

This means that the corresponding constrained system on J^{1}Q_{0} satisfies

[[J^{1}ι_{0}^{∗} dΘ_{λ}]] = [J^{1}ι_{0}^{∗} dΘ_{λ}] = J^{1}ι_{0}^{∗} [dΘ_{λ}] = [dΘ_{J^{1}ι_{0}^{∗} λ}]

≡ dΘ_{J^{1}ι_{0}^{∗} λ} mod Ω_{Y}^{0,2}(J^{1}Q_{0}). (3.90)

For simplicity of notations we write

λ̄ = J^{1}ι_{0}^{∗} λ, L̄ = L ∘ J^{1}ι_{0}  (3.91)

for the restricted Lagrangian.

By the above propositions, contrary to the non-holonomic case, holonomic constraints represent no constraints in the tangent bundle to the constraint submanifold. Consequently, holonomic constrained systems are treated in the same way as unconstrained systems on fibered manifolds. Simply, instead of a Lagrangian λ and a constraint Q_{0} in Y one can consider the restricted Lagrangian λ̄ on J^{1}Q_{0}.

Let us summarize some of the main properties of holonomic systems:

Corollary 3.6. The holonomic constraint Euler–Lagrange form satisfies

E_{λ}^{C} = Eλ̄,  (3.92)

and constraint Euler–Lagrange equations become simply equations for sections γ of the fibered manifold π_{0} : Q_{0} → X as follows:

J^{1}γ^{∗}i_{ξ}dΘλ̄ = 0 ∀ξ ∈ V(J^{1}Q_{0}),  (3.93)

or, in adapted fibered coordinates,

\frac{∂L̄}{∂q^{l}} − \frac{d}{dt} \frac{∂L̄}{∂q̇^{l}} = 0, 1 ≤ l ≤ m − k.  (3.94)

Hamilton equations are then equations for sections of the prolonged manifold (π_{0})_{1} : J^{1}Q_{0} → X,

δ^{∗}i_{ξ}dΘλ̄ = 0 ∀ξ ∈ V(J^{1}Q_{0}).  (3.95)

The regularity condition reads

det(\frac{∂^{2}L̄}{∂q̇^{l}∂q̇^{s}} ) ≠ 0.

The class (3.90) has the canonical form

−dH̄ ∧ dt + dP_{l} ∧ dq^{l} mod Ω_{Y}^{0,2}(J^{1}Q_{0}),  (3.96)

where the Hamiltonian and momenta take the form

H̄ = −L̄ + P_{l}q̇^{l}, P_{l} = \frac{∂L̄}{∂q̇^{l}} , 1 ≤ l ≤ m − k.  (3.97)

The holonomic Hamilton equations then take the canonical form

\frac{d}{dt}(P_{l} ∘ δ) = −\frac{∂H̄}{∂q^{l}} , \frac{d}{dt}(q^{l} ∘ δ) = \frac{∂H̄}{∂P_{l}} .  (3.98)

3.7. Example: A sleigh on an inclined plane. Let us consider an example of a nonholonomic motion. The situation is presented on the following picture:

There is an object on the inclined plane and a cutting knife. The center of mass of the object lies on the straight line along the knife edge at a distance a from the point (x, y) of contact of the knife and the plane and β measures the angle between the straight line along the knife edge and x axis, see [29].

This mechanical system is modeled on the fibered manifold π : ℝ × S^{1} × ℝ^{2} → ℝ, (t, β, x, y) are coordinates on ℝ × S^{1} × ℝ^{2}.

We introduce “generalized coordinates”

q^{1} = β, q^{2} = x, q^{3} = y,  (3.99)

β ∈ (−\frac{π}{2}, \frac{π}{2}).

The Lagrange function L of the system consists of three parts, the first one represents the energy of rotation, the second one characterizes the kinetic energy of translation of the mechanical system and the third one is the potential energy of the system.

Because the center of mass C does not coincide with the point of contact, the energy of rotation has to be modified. At first we write down the formulas for angular velocities associated with Euler angles which represent rotational motion of this system,

\begin{aligned}ω′_{x} & = & 0, \\ ω′_{y} & = & 0, \\ ω′_{z} & = & β̇, \end{aligned}  (3.100)

for β̄ = β, Θ̄ = 0, ψ̄ = 0. Then

T^{rot} = \frac{1}{2}Jβ̇^{2} + mv_{tr}(ω′ × R′),  (3.101)

where ω′ is the angular velocity vector, R′ = (x′, y′, z′) is the position vector between point of contact and the center of mass and v_{tr} is the velocity vector of a translation. The following identities hold

\begin{aligned}ω′ × R′ & = & (ω′_{x}, ω′_{y}, ω′_{z}) × (x′, y′, z′) \\ & = & (ω′_{y}z′ − ω′_{z}y′, −ω′_{x}z′ + ω′_{z}x′, ω′_{x}y′ − ω′_{y}x′) \\ & = & (−β̇y′, β̇x′, 0) = (−β̇a sin β, β̇a cos β, 0). \end{aligned}  (3.102)

The second term of (3.101) now can be expressed as follows

mv_{tr}(ω′ × R′) = m(ẋ, ẏ, ż)(ω′ × R′) = maβ̇(ẏ cos β − ẋ sin β). (3.103)

The kinetic energy of translation and the potential energy take the form

T^{tr} = \frac{1}{2}m(ẋ^{2} + ẏ^{2}), V = −mgx sin α,  (3.104)

where angle α represents an inclination of the plane.

Finally the Lagrange function L of this mechanical system is expressed by

L = \frac{1}{2}m(ẋ^{2} + ẏ^{2}) + \frac{1}{2}Jβ̇^{2} + maβ̇(ẏ cos β − ẋ sin β) − mgx sin α, (3.105)

g is the gravitational acceleration. For the variations of dx and dy we obtain

dy = tan βdx.  (3.106)

This is equivalent to

ẏ = ẋ tan β.  (3.107)

Then constraint function f_{1} is expressed by

f_{1} = ẏ − g_{1} = ẏ − ẋ tan β.  (3.108)

With respect to (3.24) we have

A_{1} = 0,
A_{2} = −mg sin α + maβ̇^{2} cos β,
A_{3} = maβ̇^{2} sin β,

(3.109)

and

B_{σν} = ( \begin{matrix}−J & ma sin β & −ma cos β \\ ma sin β & −m & 0 \\ −ma cos β & 0 & −m \end{matrix} ) .  (3.110)

The matrix B_{σν} is regular, det(B_{σν}) ≠ 0. The Euler–Lagrange equations are then

β̈ = −\frac{amg sin α sin β}{J − ma^{2}} ,

ẍ = −\frac{Jg sin α−Jaβ̇^{2} cos β−ma^{2}g sin α cos^{2} β+ma^{3}β̇^{2} cos β}{J − ma^{2}} ,

ÿ = \frac{a sin β(mag sin α cos β + Jβ̇^{2} − ma^{2}β̇^{2})}{J − ma^{2}}.

(3.111)

Let us return to the constrained case. Now a = 1, k = 1. The rank condition (3.31) is satisfied. Indeed,

rank (\frac{∂f^{1}}{∂q̇^{σ}} ) = rank ( 0, − tan β, 1 ) = 1.  (3.112)

So, constraint (3.108) generates a constraint submanifold Q ⊂ J^{1}Y by

Q = {(t, β, x, y, β̇, ẋ, ẏ); ẏ = ẋ tan β} .  (3.113)

The constraint distribution on Q is annihilated by one 1-form φ̄^{1} = ι^{∗}ϕ^{1}, where

ϕ^{1} = dy − tan βdx.  (3.114)

The constraint distribution in not completely integrable, i.e. there does not exist any 1-form μ such that

dφ̄^{1} = μ ∧ φ̄^{1}.  (3.115)

Indeed,

dφ̄^{1} = d(dy − tan βdx) = −\frac{1}{cos^{2} β} dβ ∧ dx.  (3.116)

The constraint is affine in the velocities, i.e. the constraint is simple and it can be equivalently represented by a (non-integrable) distribution on Y, generated by ϕ^{1}.

From (3.48) we get

Ā_{1} = −\frac{maβ̇ẋ}{cos β},
Ā_{2} = −\frac{m(g sin α cos^{3} β − aβ̇^{2} cos^{2} β + β̇ẋ sin β)}{cos^{3} β},

(3.117)

and

B̄ = ( \begin{matrix}−J & 0 \\ 0 & −m − m tan^{2} β \end{matrix} ) .  (3.118)

The determinant of B̄ is non-zero, so the constrained system is regular.

Constrained Euler–Lagrange equations (3.65) consist of two equations of the second order

β̈ = −\frac{maβ̇ẋ}{J cos β} ,
ẍ = −g sin α cos^{2} β + aβ̇^{2} cos β − β̇ẋ tan β,

(3.119)

and of one equation of the first order

ẏ = ẋ tan β.  (3.120)

The “constraint Lagrangian” L̄ = L ∘ ι is expressed by

L̄ = \frac{1}{2} \frac{mẋ^{2}}{cos^{2} β} + \frac{1}{2}Jβ̇^{2} − mgx sin α,  (3.121)

and we can check by a direct computation that the same equations are obtained from the functions L̄ and

L̄_{1} = mẋ tan β + maβ̇ cos β  (3.122)

using Euler–Lagrange equations in the form (3.58).

The constraint Legendre transformation is given by

(t, β, x, y, β̇, ẋ) → (t, β, x, y, P_{β}, P_{x}),  (3.123)

where constraint momenta (3.77) take the form

\begin{aligned}P_{β} & = & Jβ̇, \\ P_{x} & = & \frac{mẋ}{cos^{2} β} . \end{aligned}  (3.124)

The class of constraint energy 1-forms is then expressed in constraint Legendre coordinates by

η_{I} = η̄_{0}dt

−\frac{aP_{β}P_{x} cos β}{J} dβ+(−mg sin α + \frac{maP_{β}^{2}}{J^{2} cos β} + \frac{P_{β}P_{x} sin β}{J cos β} ) dx

−\frac{P_{β}}{J} dP_{β} − \frac{P_{x} cos^{2} β}{m} dP_{x} mod I^{1}(C^{0}).

(3.125)

Hence, constrained Hamilton equations consist of four first-order equations which, in simplified notation, can be written as follows:

Ṗ_{β} = −\frac{aP_{β}P_{x} cos β}{J} ,

Ṗ_{x} = −mg sin α + \frac{maP_{β}^{2}}{J^{2} cos β} + \frac{P_{β}P_{x} sin β}{J cos β} ,

β̇ = \frac{P_{β}}{J} ,

ẋ = \frac{P_{x} cos^{2} φ}{m},

(3.126)

and of the equation of the constraint ẏ = ẋ tan β.

4. Fields with differential constraints

In the sequel we consider a fibered manifold π : Y → X where dim X = n, and its jet prolongations. As above, m denotes the fiber dimension (i.e. dim Y = m + n), and we assume m > 1.

First, we summarize main concepts from the theory of unconstrained Lagrangian and Hamiltonian systems, then we turn to the constraint structure in J^{1}Y, and finally we are interested in Lagrangian and Hamiltonian constrained field equations. Main sources for this section are [7, 10, 12, 13, 19, 21] for the unconstrained theory, [22] for the nonholonomic constraint structure, and [2, 22, 37, 38], for the nonholonomic constrained systems. In this section we also present new results concerning constrained Hamilton–De Donder systems. We study constrained Hamilton–De Donder equations, regularity, and existence of constraint Legendre transformation.

4.1. Dynamical forms and locally variational forms. Let E be a dynamical form on J^{2}Y. In fibered coordinates,

E = E_{σ}ω^{σ} ∧ ω_{0},  (4.1)

where E_{σ} are functions of (x^{i}, y^{ν}, y_{i}^{ν}, y_{ij}^{ν}), and ω_{0} denotes the local volume element (2.2). The coordinate form of the equation for paths of E (3.1) is a system of m second-order partial differential equations for the components (γ^{ν}) of sections γ of π as follows:

E_{σ}(x^{i}, γ^{ν}(x^{i}), \frac{∂γ^{ν}}{∂x^{i}} , \frac{∂^{2}γ^{ν}}{∂x^{i}∂x^{j}} ) = 0, 1 ≤ σ ≤ m.  (4.2)

Again, equations for paths of dynamical forms can be represented by means of exterior differential systems; now, however, locally generated by n-forms.

Proposition 4.1. [19] Let E be a dynamical form on J^{2}Y. A section γ of π is a path of E if and only if

J^{2}γ^{∗}i_{ξ}α = 0 ∀ξ ∈ V(J^{2}Y),  (4.3)

where α is any (n + 1)-form such that p_{1}α = E.

Proof of this statement is the same as that of Proposition 3.1.

In the “PDE situation” we can proceed in full analogy with the case of mechanics, and consider Lepage classes and corresponding Hamiltonian systems:

Definition 4.1. [22] Let E ∈ Ω_{Y}^{n+1}(J^{2}Y) be a dynamical form. The equivalence class of (n + 1)-forms (on an open subset U ⊂ J^{2}Y) defined by

α_{1} ∼ α_{2} iff p_{1}α_{1} = p_{1}α_{2} = E|_{U}  (4.4)

is called Lepage class of E on U. The family of all local Lepage classes of E is referred to as Lepage class of E and is denoted by [α]_{E}, or simply [α].

By the above proposition, the equation for paths of E (on U) coincides with equations for holonomic integral sections of the exterior differential system H_{α}, generated by the following system of n-forms

i_{ξ}α ∀ξ ∈ V(J^{2}Y),  (4.5)

where α is any representative of the Lepage class of E (on U).

Definition 4.2. [19, 22] Let [α] be a Lepage class of E. Every representative α ∈ [α] is called a Hamiltonian system associated with E. The exterior differential system H_{α} is called a Hamiltonian EDS related to E. Equations for (all) integral sections of H_{α} are called Hamilton equations associated with E.

Let us turn to locally variational dynamical forms. By definition this means that around each point E = E_{λ}, for a local Lagrangian λ (recall that a Lagrangian of order r is defined to be a horizontal n-form on J^{r}Y).

In fibered coordinates, the components E_{σ} of E take the form of Euler– Lagrange expressions of λ = L ω_{0}. Necessary and sufficient conditions for a (second-order) dynamical form to be locally variational read [1, 11]

\begin{aligned}\frac{∂E_{σ}}{∂y_{ij}^{ν}} − \frac{∂E_{ν}}{∂y_{ij}^{σ}} & = & 0, \\ \frac{∂E_{σ}}{∂y_{i}^{ν}} + \frac{∂E_{ν}}{∂y_{i}^{σ}} − 2\frac{d}{dx^{j}} \frac{∂E_{ν}}{∂y_{ij}^{σ}} & = & 0, \\ \frac{∂E_{σ}}{∂y^{ν}} − \frac{∂E_{ν}}{∂y^{σ}} + \frac{d}{dx^{i}} \frac{∂E_{ν}}{∂y_{i}^{σ}} − \frac{d}{dx^{i}} \frac{d}{dx^{j}} \frac{∂E_{ν}}{∂y_{ij}^{σ}} & = & 0, \end{aligned}  (4.6)

and the corresponding Tonti Lagrangian for E is

L = y^{σ} ∫_{0}^{1} E_{σ}(x^{i}, uy^{ν}, uy_{i}^{ν}, uy_{ij}^{ν})du.  (4.7)

Notice that, contrary to the case of ordinary differential equations, variationality conditions (4.6) do not imply that the locally variational form E should be affine in the second derivatives. Moreover, second-order locally variational forms need not come from Lagrangians of the first order. This means that Tonti Lagrangian need not be reducible to a firstorder Lagrangian. (From the formula for Euler–Lagrange expressions we can see immediately that a necessary condition for reducibility is that E_{σ} should be affine in the second derivatives y_{ij}^{ν}).

Let us summarize basic definitions:

Definition 4.3. The Lepage class [α] of a locally variational form E is called Lagrangian system. Every element α ∈ [α] is called a Hamiltonian system associated with E. Paths of a locally variational form E are called extremals. Equations for paths of a locally variational form (respectively, equations for holonomic integral sections of associated Hamiltonian EDS) are called Euler–Lagrange equations. Equations for integral sections of the Hamiltonian EDS are called Hamilton equations, their integral sections are then called Hamilton extremals.

4.2. Euler–Lagrange and Hamilton–De Donder equations in firstorder field theory. In what follows we shall be concerned merely with locally variational forms that arise from local first-order Lagrangians. Such dynamical forms, among others, have the following properties:

• Around each point in J^{2}Y it holds E = E_{λ}, where λ is a Lagrangian defined on an open subset of J^{1}Y. In fibered coordinates where λ = Lω_{0},

E_{λ} = E_{σ}ω^{σ} ∧ ω_{0}, E_{σ} = \frac{∂L}{∂y^{σ}} − \frac{d}{dx^{j}} \frac{∂L}{∂y_{j}^{σ}} .  (4.8)

This means that the Euler–Lagrange equations in fibered coordinates take the familiar form

(\frac{∂L}{∂y^{σ}} − \frac{d}{dx^{j}} \frac{∂L}{∂y_{j}^{σ}} ) ∘ J^{2}γ = 0, 1 ≤ σ ≤ m.  (4.9)

• In every fibered chart, components E_{σ} of E are affine in the second derivatives, i.e.

E_{σ} = A_{σ} + B_{σν}^{ij} y_{ij}^{ν} = A_{σ} + B_{σν}^{(ij)} y_{ij}^{ν},  (4.10)

where A_{σ} and B_{σν}^{ij} are functions of (x^{i}, y^{ρ}, y_{l}^{ρ}) that in terms of a first-order Lagrangian for E take the form

\begin{matrix}B_{σν}^{ij} = −\frac{∂^{2}L}{∂y_{i}^{ν}∂y_{j}^{σ}} , \\ A_{σ} = \frac{∂L}{∂y^{σ}} − \frac{d′}{dx^{j}} \frac{∂L}{∂y_{j}^{σ}} = \frac{∂L}{∂y^{σ}} − \frac{∂^{2}L}{∂x^{j} ∂y_{j}^{σ}} − \frac{∂^{2}L}{∂y^{ν}∂y_{j}^{σ}} y_{j}^{ν}. \end{matrix}  (4.11)

Note that the B_{σν}^{ij} need not be symmetric in the upper indices. B_{σν}^{(ij)} denotes the symmetric part in the i, j.

• Every first-order Lagrangian for E has a first-order Lepage equivalent that is not unique. Lepage equivalents of λ take the form

ρ = Θ_{λ} + dν + μ,  (4.12)

where

Θ_{λ} = L ω_{0} + \frac{∂L}{∂y_{j}^{σ}} ω^{σ} ∧ ω_{j},  (4.13)

μ is an arbitrary at least 2-contact n-form, and ν is an arbitrary contact (n − 1)-form. Θ_{λ} is called the Poincaré–Cartan form associated with λ. It is the unique at most 1-contact n-form such that hΘ_{λ} = λ and p_{1}dΘ_{λ} is π_{1,0}-horizontal. This means that the Euler–Lagrange form E_{λ} of λ is also unique, since by (4.12) it does not depend upon the choice of a Lepage equivalent ρ of λ:

E_{λ} = p_{1}dρ = p_{1}dΘ_{λ}.  (4.14)

Note that contrary to mechanics, one generally has dΘ_{λ_{1}} ≠ dΘ_{λ_{2}} for equivalent Lagrangians λ_{1}, λ_{2}.

• E can be locally represented by a first-order Lepage class that is called Lagrangian system associated with the locally variational form E. With a similar inaccuracy as in Sec. 3, in order to simplify notations, we write

[α] = [dΘ_{λ}] = dΘ_{λ} mod Ω^{n+1,(≥2)}(J^{1}Y).  (4.15)

Consequently, extremals and Hamilton extremals are described by Hamiltonian exterior differential systems defined on (open subsets of) J^{1}Y.

Remark 4.1. In the sequel we shall again assume that E, defined on J^{2}Y, is everywhere nontrivially of order 2. This means that (B_{σν}^{ij}) in (4.10) is everywhere a non-zero matrix, or, equivalently, for every firstorder Lagrangian λ the Poincaré–Cartan (n + 1)-form dΘ_{λ} is everywhere nontrivially of order one.

As we can see, the Euler–Lagrange and Hamilton equations in firstorder field theory now have the following EDS formulation:

Proposition 4.2. Let [α] be a Lagrangian system on J^{1}Y, E the corresponding locally variational form.

A section γ of π is an extremal of E (on an open set W ⊂ X in dom γ) if and only if

J^{1}γ^{∗}i_{ξ}α = 0 ∀ξ ∈ V(J^{1}Y),  (4.16)

where α is any (n+1)-form belonging to the class [α] (defined on π_{1}^{−1}(W)).

A section δ of π_{1} is a Hamilton extremal of E, related with the Hamiltonian system α ∈ [α] (defined in π_{1}^{−1}(dom δ)) if and only if

δ^{∗}i_{ξ}α = 0 ∀ξ ∈ V(J^{1}Y).  (4.17)

The concept of regularity of a Lagrangian system is, similarly as in mechanics, related with the properties of the associated Hamiltonian exterior differential systems. The situation in field theory is, however, much more rich and interesting than that in mechanics: the reason is the non-uniqueness of the Poincaré–Cartan form Θ_{λ} of a Lagrangian λ. For more details we refer to [19, 23, 24]. In this paper we shall study the most simple case related just to the properties of the Hamiltonian differential systems H_{dΘ_{λ}} related with the forms dΘ_{λ} (Hamilton–De Donder equations [5, 7]).

To this end, let us recall the following definition [25].

Definition 4.4. Let [α] (on J^{1}Y) be a Lagrangian system related with a locally variational form E. An element α of the class [α] is called Hamilton–De Donder system related with E if

α = dΘ_{λ} + F,  (4.18)

where λ is a Lagrangian for E and F ∈ Ω_{Y}^{n+1,(≥2)}(J^{1}Y). The corresponding Hamilton equations, i.e.

δ^{∗}i_{ξ}(dΘ_{λ} + F) = 0 ∀ξ ∈ V(J^{1}Y)  (4.19)

are called Hamilton–De Donder equations.

It is easy to see that Hamilton–De Donder systems can be locally expressed in the so-called canonical form as follows:

Proposition 4.3. Let α = dΘ_{λ} + F be a Hamilton–De Donder system on an open set U ⊂ J^{1}Y; we may assume that U is endowed with fibered coordinates (x^{i}, y^{σ}, y_{j}^{σ}). Then there exist functions H and p_{σ}^{i}, 1 ≤ σ ≤ m, 1 ≤ i ≤ n, such that

α = −dH ∧ ω_{0} + dp_{σ}^{i} ∧ dy^{σ} ∧ ω_{i} + F.  (4.20)

H and p_{σ}^{i} are defined by

p_{σ}^{i} = \frac{∂L}{∂y_{i}^{σ}} , H = −L + p_{ν}^{j}y_{j}^{ν},  (4.21)

where Lω_{0} is a first-order Lagrangian whose Poincaré–Cartan (n + 1)- form coincides with dΘ_{λ}.

Definition 4.5. Functions H and p_{σ}^{i}, 1 ≤ σ ≤ m, 1 ≤ i ≤ n, defined by formula (4.20) are called a Hamiltonian and momenta of the Hamilton– De Donder system α.

Note that the family of a Hamiltonian and momenta (4.21) of a Hamilton–De Donder system is non-unique and depends upon the choice of a Lagrangian for the form dΘ_{λ}. On the other hand, in the following subclass of the equivalence class (4.15),

[α]_{Y} = [dΘ_{λ}]_{Y} ≡ dΘ_{λ} mod Ω_{Y}^{n+1,(≥2)}(U),  (4.22)

all elements posses the same families of momenta&Hamiltonian. This is due to the fact that if α_{1}, α_{2} ∈ [α]|_{U} are such that α_{1} = dΘ_{λ_{1}} + F_{1} and α_{2} = dΘ_{λ_{2}} + F_{2} where dΘ_{λ_{1}} ≠ dΘ_{λ_{2}} and F_{1}, F_{2} ∈ Ω_{Y}^{n+1,(≥2)}(U), then α_{1} and α_{2} are not equivalent in the sense of (4.22), and vice versa (see [25], Proposition 3.1 and its proof).

Let us turn to the concept of regularity of a Hamilton–De Donder system ([19]).

Definition 4.6. A Hamilton–De Donder system α is called regular if H_{α} contains all the canonical contact n-forms

ω^{σ} ∧ ω_{i}, 1 ≤ σ ≤ m, 1 ≤ i ≤ n.  (4.23)

A Lagrangian system [α] is called De Donder regular if around each point in J^{1}Y there exists a related regular Hamilton–De Donder system.

Proposition 4.4. Let α be a regular Hamilton–De Donder system. Then every integral section of H_{α} is holonomic. Consequently, Hamilton–De Donder equations of α are equivalent with the Euler–Lagrange equations of [α].

Proof. If a Hamilton–De Donder system α is regular then for every integral section δ of H_{α},

\begin{matrix}δ^{∗}(ω^{σ} ∧ ω_{i}) = (\frac{∂(y^{σ}∘δ)}{∂x^{i}} − (y_{i}^{σ} ∘ δ))ω_{0} = 0, \\ 1 ≤ σ ≤ m, 1 ≤ i ≤ n, \end{matrix}  (4.24)

meaning that δ = J^{1}γ for a section γ of π. □

Theorem 4.1. Let α = dΘ_{λ} +F be a Hamilton–De Donder system. The following conditions are equivalent:

(1) The Hamilton–De Donder system α is regular.

(2) A system of generators of H_{α} has maximal rank (i.e. equal to m + mn).

(3) Every Lagrangian λ for dΘ_{λ} satisfies the regularity condition

det (\frac{∂^{2}L}{∂y_{i}^{σ}∂y_{j}^{ν}} ) ≠ 0.  (4.25)

Proof. Computing explicitly generators of H_{α} we obtain the following system of m + mn differential n-forms:

\begin{matrix}A_{σ}ω_{0}+(2F_{σν}^{j} + \frac{∂^{2}L}{∂y^{σ}∂y_{j}^{ν}} − \frac{∂^{2}L}{∂y^{ν}∂y_{j}^{σ}} )ω^{ν}∧ ω_{j} +B_{σν}^{ij} dy_{j}^{ν} ∧ ω_{i}+μ_{σ}, \\ B_{σν}^{ij} ω^{ν} ∧ ω_{j}, \end{matrix} (4.26)

where 1 ≤ σ ≤ m, 1 ≤ i ≤ n, A_{σ} and B_{σν}^{ij} are given by (4.11), and μ_{σ} are at least 2-contact (precisely, μ_{σ} is the at least 2-contact part of i_{∂/∂y^{σ}}F). This means that the matrix of generators of H_{α} is the following matrix with m + mn rows (and 1 + mn + mn^{2} + ⋯ columns):

( \begin{matrix}A_{σ} & 2F_{σν}^{j} + \frac{∂^{2}L}{∂y^{σ}∂y_{j}^{ν}} − \frac{∂^{2}L}{∂y^{ν}∂y_{j}^{σ}} & B_{σν}^{ij} & ⋯ \\ 0 & B_{σν}^{ij} & 0 & 0 \end{matrix} ) .  (4.27)

First, we prove the equivalence of (1) and (2).

If α is regular then all the generators B_{σν}^{ij} ω^{ν} ∧ ω_{j} are independent, meaning that the matrix (B_{σν}^{ij}) is regular. Consequently, all rows of (B_{σν}^{ij}) (labelled by (σ, i)) are linearly independent, for every fixed i. Then, however, the matrix (B_{σν}^{ij}) with m rows labelled by σ, and mn columns labelled by (ν, j), has the maximal rank, m, and the rank of the matrix (4.27) is equal to m + mn, as desired.

Conversely, if the rank of the matrix (4.27) is maximal then its square submatrix (B_{σν}^{ij}) (with rows labelled by (σ, i)) is regular. This means that all the forms ω^{σ} ∧ ω_{i} are independent. Hence α is regular.

The equivalence of (3) and (2) is now clear: by the above, H_{α} has maximal rank iff the matrix (B_{σν}^{ij}) is regular, However, in terms of the Lagrangian λ for α,

B_{σν}^{ij} = −\frac{∂^{2}L}{∂y_{i}^{σ}∂y_{j}^{ν}} .  (4.28)

This completes the proof. □

Theorem 4.2. If dΘ_{λ} +F is a regular Hamilton–De Donder system then every form α ∈ [dΘ_{λ}]_{Y} is regular. Consequently,

(1) For every α ∈ [dΘ_{λ}]_{Y}, all Hamilton extremals are holonomic.

(2) For every α ∈ [dΘ_{λ}]_{Y}, the Hamilton equations are equivalent with the Euler–Lagrange equations of λ.

(3) For every α ∈ [dΘ_{λ}]_{Y}, every Hamilton extremal of α is a prolongation of an extremal of λ.

(4) Hamilton equations of all elements in the class [dΘ_{λ}]_{Y} are equivalent.

Proof. Looking at the generators (4.26) we can see immediately that regularity does not depend upon the choice of functions F_{σν}^{j}, i.e., upon the choice of α ∈ [dΘ_{λ}]_{Y}.

The rest of the proof is elementary. □

Corollary 4.1. Let α = dΘ_{λ} + F be a regular Hamilton–De Donder system. Then momenta p_{σ}^{j}, 1 ≤ σ ≤ m, 1 ≤ j ≤ n, of α are independent, and (x^{i}, y^{σ}, p_{σ}^{j}) are local coordinates on J^{1}Y, called Legendre coordinates. The Hamiltonian differential system H_{α} has generators that in Legendre coordinates take the form (since ω^{ν} ∧ ω_{i} ∈ H_{α}, ∀ν, i),

\frac{∂H}{∂y^{σ}} ω_{0} + dp_{σ}^{j} ∧ ω_{j} + μ_{σ}, \frac{∂H}{∂p_{σ}^{j}} ω_{0} − dy^{σ} ∧ ω_{j},  (4.29)

where μ_{σ} = i_{∂/∂y^{σ}F} ∈ Ω_{Y}^{n(≥2)}(J^{1}Y). Hamilton equations of α in Legendre coordinates then read

\frac{∂p_{σ}^{j}}{∂x^{j}} = −\frac{∂H}{∂y^{σ}} ,  \frac{∂y^{σ}}{∂x^{j}} = \frac{∂H}{∂p_{σ}^{j}} ,  (4.30)

where the appearing functions are considered along sections δ of π_{1}.

4.3. Non-holonomic constraints in field theory. The aim of the section is to present the concept of the non-holonomic constraint structure [22].

Non-holonomic constraints in the case dim X > 1 are defined in the same way as for one independent variable:

Definition 4.7. By a constraint submanifold or a non-holonomic constraint in J^{1}Y we shall understand a submanifold Q ⊂ J^{1}Y, fibered over Y, precisely speaking, a surjective submersion π_{1,0}|_{Q} : Q → Y.

Put codim Q = κ and assume 1 ≤ κ ≤ mn − 1. Locally Q can be expressed by a system of first-order partial differential equations

f^{α}(x^{i}, y^{σ}, y_{j}^{σ}) = 0, 1 ≤ α ≤ κ,  (4.31)

such that

rank (\frac{∂f^{α}}{∂y_{j}^{σ}} ) = κ, where α labels rows and σ, j columns. (4.32)

Definition 4.8. Let Q be a non-holonomic constraint in J^{1}Y, codim Q = κ, 1 ≤ κ ≤ mn − 1. If

rank (\frac{∂f^{α}}{∂y_{j}^{σ}} ) = k, where α, j label rows and σ columns, (4.33)

for some k, 1 ≤ k ≤ m − 1, we say that Q is a regular non-holonomic constraint of corank (κ, k).

It can be shown that the above definition is correct (coordinate independent) [22].

Given a regular non-holonomic constraint Q in J^{1}Y there naturally arise the following local distributions, defined on the domain U of definition of the functions f^{α}:

(1) D_{U} = annih {df^{α}, 1 ≤ α ≤ κ}; rank D_{U} is constant on U due to (4.32) and equal to m + n + mn − κ.

(2) C̃_{U} = annih {ϕ^{αj}}, where

ϕ^{αj} = f^{α}dx^{j} + \frac{1}{n} \frac{∂f^{α}}{∂y_{j}^{σ}} ω^{σ}, 1 ≤ α ≤ κ, 1 ≤ j ≤ n.  (4.34)

The forms (4.34) are not linearly independent, however, due to rank condition (4.33), there exist functions c_{αj}^{a}, 1 ≤ a ≤ k, 1 ≤ α ≤ κ, 1 ≤ j ≤ n, on U, such that the (k × m)-matrix

M = (M_{σ}^{a}), where M_{σ}^{a} = \frac{1}{n}c_{αj}^{a} \frac{∂f^{α}}{∂y_{j}^{σ}} ,  (4.35)

has maximal rank equal to k. Thus,

\begin{matrix}ϕ^{a} = c_{αj}^{a}ϕ^{αj} = c_{αj}^{a}f^{α}dx^{j} + \frac{1}{n}c_{αj}^{a} \frac{∂f^{α}}{∂y_{j}^{σ}} ω^{σ} = c_{αj}^{a}f^{α}dx^{j} + M_{σ}^{a}ω^{σ}, \\ 1 ≤ a ≤ k, \end{matrix} (4.36)

are linearly independent at each point in U. Hence, the distribution C̃_{U} has constant corank equal to k, i.e. rank C̃_{U} = m + n + mn − k.

(3) C_{U} = annih {ϕ^{a}, df^{α}, 1 ≤ a ≤ k, 1 ≤ α ≤ κ}.

The following results have been obtained in [22]:

Proposition 4.5. Q ∩ U is an integral submanifold of D_{U}. Hence, for every x ∈ Q, the forms df^{α}(x), 1 ≤ α ≤ κ, annihilate the tangent space T_{x}Q to the manifold Q at x, i.e., along Q, D = annih {df^{α}, 1 ≤ α ≤ κ} = T Q.

Corollary 4.2. Let Q be a constraint of codimension κ in J^{1}Y, and let f^{α} = 0 and f′^{α} = 0, where 1 ≤ α ≤ κ, be two sets of equations of Q on an open set U ⊂ V_{1} ⊂ J^{1}Y. Then there are functions γ_{β}^{α} on U such that at each point of U, (γ_{β}^{α}) is a regular matrix, and df′^{α} = γ_{β}^{α}df^{β}. In particular, at each point x ∈ Q ∩ U,

\frac{∂f′^{α}}{∂y_{j}^{σ}} = γ_{β}^{α} \frac{∂f^{β}}{∂y_{j}^{σ}} .  (4.37)

Proposition 4.6. C_{U} is a subdistribution of both C̃_{U} and D_{U}. At the points of Q ∩ U, the distributions C_{U} and C̃_{U} ∩ D coincide, and define a distribution of corank k on Q ∩ U.

The local distributions on Q mentioned above unite into a (global) distribution on Q:

Theorem 4.3. Let Q be a regular non-holonomic constraint in J^{1}Y of corank (κ, k), let ι : Q → J^{1}Y be the canonical embedding of the submanifold Q into J^{1}Y. If ϕ^{a}, 1 ≤ a ≤ k, are independent 1-forms (4.36), put

φ^{a} = ι^{∗}ϕ^{a} = (M_{σ}^{a} ∘ ι) ι^{∗}ω^{σ}, 1 ≤ a ≤ k.  (4.38)

Then

C = annih {φ^{a}, 1 ≤ a ≤ k}  (4.39)

is a distribution of corank k on Q.

The proof of the theorem can be found in [22].

Definition 4.9. The distribution C (4.39) on Q is called canonical distribution. 1-forms belonging to the annihilator, C^{0}, of C, are called canonical constraint 1-forms. The ideal in the exterior algebra of differential forms on Q generated by C^{0} is called canonical constraint ideal, and denoted by I(C^{0}); its homogeneous component of degree p is denoted by I^{p}(C^{0}). Elements of the ideal I(C^{0}) are called canonical constraint forms.

Theorem 4.4. The canonical distribution C on Q is locally spanned by the following vector fields:

\begin{matrix}\frac{∂_{c}}{∂x^{i}} ≡ \frac{∂}{∂x^{i}} + ∑_{a=1}^{k}(F_{i}^{a} ∘ ι)\frac{∂}{∂y^{m−k+a}} , 1 ≤ i ≤ n, \\ \frac{∂_{c}}{∂y^{s}} ≡ \frac{∂}{∂y^{s}} + ∑_{a=1}^{k}(G_{s}^{a} ∘ ι)\frac{∂}{∂y^{m−k+a}} , 1 ≤ s ≤ m − k, \\ \frac{∂}{∂z^{J}} , 1 ≤ J ≤ nm − κ, \end{matrix}  (4.40)

where (x^{i}, y^{σ}, z^{J}, f^{α}), 1 ≤ i ≤ n, 1 ≤ σ ≤ m, 1 ≤ J ≤ nm − κ, 1 ≤ α ≤ κ, denote fibered coordinates adapted to the submanifold ι : Q → J^{1}Y, the functions G_{s}^{a} represent (at each point) a fundamental system of solutions of the system of independent homogeneous algebraic equations for m unknowns Ξ^{σ}, 1 ≤ σ ≤ m,

M_{σ}^{a} Ξ^{σ} = 0, 1 ≤ a ≤ k,  (4.41)

and, for every i = 1, 2, …, n, the F_{i}^{a} are solutions of the equations

M_{σ}^{a} F_{i}^{σ} = M_{σ}^{a}y_{i}^{σ} − f^{α}c_{αi}^{a}, 1 ≤ a ≤ k,  (4.42)

(where y_{j}^{σ} are considered as functions of z^{J}, f^{β}) corresponding to the choice of all the parameters equal to zero.

A section γ of π defined on an open set W ⊂ X is called a holonomic path in Q if for every x ∈ W

J^{1}γ(x) ∈ Q.  (4.43)

Remark 4.2. We shall use the following notations and objects, adapted to the constraint structure, introduced in [22].

(i) Conventions concerning notation of indices:

\begin{matrix}1 ≤ i, j, l ≤ n,  1 ≤ α, β, γ ≤ κ,  1 ≤ J ≤ nm − κ, \\ 1 ≤ σ, ν, ρ ≤ m, 1 ≤ a, b, c ≤ k,  1 ≤ p, r, s ≤ m − k. \end{matrix} (4.44)

(ii) Taking into account that the matrix (4.35) in (4.36) has maximal rank, k, one can express k of the contact 1-forms ω^{σ} by means of the constraint forms ϕ^{a}, 1 ≤ a ≤ m, and the remaining ω^{ν}’s. Without loss of generality we may suppose that this concerns the forms ω^{m−k+a}, where 1 ≤ a ≤ k. In an adapted basis (x^{i}, y^{σ}, z^{J}, f^{α}), and in the notations of the above theorem it holds

\begin{matrix}ω^{m−k+a} = μ_{b}^{a}(ϕ^{b} − M_{s}^{b} ω^{s} − c_{αj}^{b}f^{α}dx^{j}) \\ = μ_{b}^{a}ϕ^{b} + G_{s}^{a} ω^{s} + (F_{j}^{a} + G_{s}^{a}y_{j}^{s} − y_{j}^{m−k+a})dx^{j}, \end{matrix}  (4.45)

where (μ_{b}^{a}) is an appropriate regular matrix. Here and in what follows, y_{j}^{σ} are considered as functions of the coordinates (x^{i}, y^{σ}, z^{J}, f^{α}). Similarly, the rank condition (4.32) guarantees that one can express the forms dz^{α} by means of (df^{β}, dx^{i}, dy^{σ}, dz^{J}). Thus, we have on J^{1}Y the following bases of 1-forms, adapted to the constraint structure:

(dx^{i}, dy^{s}, ϕ^{a}, dz^{J}, df^{α}), or (dx^{i}, ω^{s}, ϕ^{a}, dz^{J}, df^{α});  (4.46)

Consequently, with obvious notations we may write

ω̄^{m−k+a} ≡ ι^{∗}ω^{m−k+a} = φ̄^{a} + Ḡ_{s}^{a} ω̄^{s},  (4.47)

where ω̄^{s} = ι^{∗}ω^{s}, and φ̄^{a} = ι^{∗}(μ_{b}^{a}ϕ^{b}) = (μ_{b}^{a} ∘ ι)φ^{b}. We can see that, on Q, instead of a canonical basis (dx^{i}, dy^{σ}, dz^{J}), or a basis (dx^{i}, ω̄^{σ}, dz^{J}) adapted to the induced contact structure, it is worth to work with bases adapted to the constraint structure, where the canonical constraint 1forms appear:

(dx^{i}, dy^{s}, φ̄^{a}, dz^{J}),  (dx^{i}, ω̄^{s}, φ̄^{a}, dz^{J}).  (4.48)

(iii) Keeping the above notations we can express the functions G_{s}^{a} and F_{j}^{a} appearing in (4.40) as follows:

G_{s}^{a} = μ_{b}^{a}M_{s}^{b},  F_{j}^{a} = y_{j}^{m−k+a} − G_{s}^{a}y_{j}^{s} − μ_{b}^{a}c_{αj}^{b}f^{α}.  (4.49)

We also put

y_{j}^{σ} ∘ ι = g_{j}^{σ}.  (4.50)

With this notation,

Ḡ_{s}^{a} = (μ_{b}^{a}M_{s}^{b}) ∘ ι,  F̄_{j}^{a} = g_{j}^{m−k+a} − Ḡ_{s}^{a}g_{j}^{s},  (4.51)

i.e.

g_{j}^{m−k+a} = F̄_{j}^{a} + Ḡ_{s}^{a} g_{j}^{s}.  (4.52)

(iv) The vector fields ∂_{c}/∂x^{i} and ∂_{c}/∂y^{s} on Q defined by (4.40) are called constraint partial derivative operators. We put

\begin{matrix}\frac{d′_{c}}{dx^{i}} = \frac{∂_{c}}{∂x^{i}} + g_{i}^{s} \frac{∂_{c}}{∂y^{s}} , \\ \frac{d_{c}}{dx^{i}} = \frac{∂_{c}}{∂x^{i}} + g_{i}^{s} \frac{∂_{c}}{∂y^{s}} + z_{i}^{J} \frac{∂}{∂z^{J}} = \frac{d′_{c}}{dx^{i}} + z_{i}^{J} \frac{∂}{∂z^{J}} , \end{matrix}  (4.53)

and call the above operators the i-th cut constraint total derivative operator and i-th constraint total derivative operator, respectively.

(v) The exterior derivative of a function f on Q is expressed as follows:

df = \frac{d′_{c}f}{dx^{j}} dx^{j} + \frac{∂_{c}f}{∂y^{s}} ω̄^{s} + \frac{∂f}{∂y^{m−k+a}} φ̄^{a} + \frac{∂f}{∂z^{J}} dz^{J}.  (4.54)

(vi) Next, denote

C_{Jj}^{a} = Ḡ_{s}^{a} \frac{∂g_{j}^{s}}{∂z^{J}} − \frac{∂g_{j}^{m−k+a}}{∂z^{J}} = −\frac{∂F̄_{j}^{a}}{∂z^{J}} − \frac{∂Ḡ_{s}^{a}}{∂z^{J}} g_{j}^{s},  C_{js}^{ai} = C_{Jj}^{a} \frac{∂z^{J}}{∂y_{i}^{s}} . (4.55)

(vii) For dφ̄^{a} we have

\begin{matrix}dφ̄^{a} = (Ḡ_{s}^{a} \frac{d′_{c}g_{j}^{s}}{dx^{i}} − \frac{d′_{c}g_{j}^{m−k+a}}{dx^{i}} ) dx^{i} ∧ dx^{j} \\ + (Ḡ_{s}^{a} \frac{∂_{c}g_{j}^{s}}{∂y^{r}} + \frac{d′_{c}Ḡ_{r}^{a}}{dx^{j}} − \frac{∂_{c}g_{j}^{m−k+a}}{∂y^{r}} ) ω̄^{r} ∧ dx^{j} \\ +C_{Jj}^{a}dz^{J} ∧ dx^{j} − \frac{∂_{c}Ḡ_{s}^{a}}{∂y^{r}} ω̄^{r} ∧ ω̄^{s} − \frac{∂Ḡ_{s}^{a}}{∂z^{J}} dz^{J} ∧ ω̄^{s} \\ + (Ḡ_{s}^{a} \frac{∂g_{j}^{s}}{∂y^{m−k+b}} − \frac{∂g_{j}^{m−k+a}}{∂y^{m−k+b}} ) φ̄^{b} ∧ dx^{j} − \frac{∂Ḡ_{s}^{a}}{∂y^{m−k+b}} φ̄^{b} ∧ ω̄^{s}. \end{matrix} (4.56)

There are several interesting particular cases of regular non-holonomic constraints in field theory. We wish to mention here very briefly the following ones (precise definitions and further properties can be found in [22]:

• Constraints whose canonical distribution is projectable onto a distribution on Y, i.e. constraints that can be modeled by a distribution or codistribution on Y.

• Constraints whose canonical distribution is completely integrable; these constraints are called semiholonomic, and can be equivalently modeled by a completely integrable, nowhere vertical distribution on Y.

• Lagrangian constraints: these are characterized by the property that the codistributions C̃_{U}^{0} can be generated by a system of (independent) Lepage 1-forms; for Lagrangian constraints it holds

C_{Jj}^{a} = 0, C_{js}^{ai} = 0  (4.57)

for all values of indices. In this context it is interesting to note that for dim X = 1 (mechanics) all non-holonomic constraints are Lagrangian.

• π-adapted constraints: can be locally represented by equations “in normal form”,

y_{j}^{m−k+a} = g_{j}^{a}(x^{i}, y^{σ}, y_{l}^{s}), 1 ≤ j ≤ n, 1 ≤ a ≤ k.  (4.58)

These constraints are Lagrangian.

Lagrangian and Hamiltonian systems subjected to π-adapted constraints are studied in detail in [25].

• Holonomic constraints, defined as fibered submanifolds of π, can again be easily treated in terms of the theory of regular non-holonomic constraints as a (very) particular case. The situation is completely analogous to that in mechanics (for details see [22]).

4.4. Constrained Lagrangian systems. The aim of the section is to introduce the concept of the constrained Lagrangian systems. For more details and the proofs of the assertions we refer to [22].

Let us consider a Lagrangian system on J^{1}Y. Recall from Sec. 4.2 that it is defined to be a first-order Lepage class. We write it in the form

[α] = [dΘ_{λ}] = dΘ_{λ} mod Ω^{n+1,(≥2)}(J^{1}Y).  (4.59)

If ι : Q → J^{1}Y is a regular non-holonomic constraint and I(C^{0}) the corresponding canonical constraint ideal, we have another equivalence, denoted by ≈, on (n+1)-forms on Q (with the same domain of definition):

η_{1} ≈ η_{2} iff η_{1} − η_{2} = F̄ + φ,  (4.60)

where F̄ is a (local) at least 2-contact (n + 1)-form on Q, and φ is a constraint (n + 1)-form. We denote by [[η]] the class of η. If [α] is a Lepage class on J^{1}Y then for any of its two elements defined on the same subset of J^{1}Y,

α_{1} ∼ α_{2} ⇒ ι^{∗}α_{1} ≈ ι^{∗}α_{2}.  (4.61)

Definition 4.10. Let [α] = [dΘ_{λ}] be a Lagrangian system on J^{1}Y and ι : Q → J^{1}Y a regular non-holonomic constraint. By the associated constrained Lagrangian system we mean the class [[ι^{∗}α]] = [[ι^{∗}dΘ_{λ}]]. Each form ι^{∗}dΘ_{λ} + φ, where φ ∈ I^{n+1}(C^{0}), is called constrained Poincaré-Cartan (n + 1)-form of λ.

Note that every element of [[ι^{∗}dΘ_{λ}]] is of the form

ᾱ = ι^{∗}dΘ_{λ} + F̄ + φ,  (4.62)

where F̄ ∈ Ω^{n+1,(≥2)}(Q) and φ ∈ I^{n+1}(C^{0}).

Definition 4.11. Consider the following system of forms on dom ᾱ ⊂ Q:

i_{ξ}ᾱ ∀ξ ∈ V(Q) ∩ C, φ ∈ C^{0}.  (4.63)

The exterior differential system generated by (4.63) is called constraint Hamiltonian EDS related with ᾱ, and is denoted by Hᾱ.

Definition 4.12. Let [[ι^{∗}dΘ_{λ}]] be a constrained Lagrangian system. Then for any representative ᾱ of the class [[ι^{∗}dΘ_{λ}]], equations for holonomic integral sections of the constraint Hamiltonian exterior differential system Hᾱ, i.e., the equations

J^{1}γ^{∗}i_{ξ}ᾱ = 0 for every π_{1}-vertical vector field ξ ∈ C,  (4.64)

where γ : W → Y, J^{1}γ(W) ⊂ Q, are called constrained Euler–Lagrange equations. Solutions of constrained Euler–Lagrange equations are called constrained extremals.

We note that (on an open subset of Q) constrained Euler–Lagrange equations do not depend upon the choice of a representative ᾱ of the class [[ι^{∗}dΘ_{λ}]]. This means that with help of a local Lagrangian λ for E we can write the constrained Euler–Lagrange equations in the form

J^{1}γ^{∗}i_{ξ}(ι^{∗}dΘ_{λ}) = 0 for every π_{1}-vertical vector field ξ ∈ C, (4.65)

where γ : W → Y, J^{1}γ(W) ⊂ Q.

For λ = Lω_{0} denote

L̄ = L ∘ ι, L̄_{a}^{j} = \frac{∂L}{∂y_{j}^{m−k+a}} ∘ ι,  (4.66)

considered as functions of adapted fibered coordinates (x^{i}, y^{σ}, z^{J}) on Q, and put

Θ_{ι^{∗}λ} = L̄ ω_{0} + \frac{∂L̄}{∂y_{j}^{s}} ω̄^{s} ∧ ω_{j} = L̄ ω_{0} + \frac{∂L̄}{∂z^{J}} \frac{∂z^{J}}{∂y_{j}^{s}} ω̄^{s} ∧ ω_{j}.  (4.67)

In keeping with notations in Remark 4.2 we can easily find the following relation:

Proposition 4.7.

ι^{∗}Θ_{λ} = Θ_{ι^{∗}λ} + L̄_{a}^{i}C_{is}^{aj} ω̄^{s} ∧ ω_{j} + L̄_{a}^{j} φ̄^{a} ∧ ω_{j}.  (4.68)

For convenience of notations let us introduce the C-modified Euler– Lagrange operator and cut C-modified Euler–Lagrange operator, respectively:

\begin{aligned}μ_{s} & = \frac{∂_{c}}{∂y^{s}} − \frac{d_{c}}{dx^{i}} (\frac{∂}{∂y_{i}^{s}} ) − \frac{∂_{c}g_{j}^{r}}{∂y^{s}} \frac{∂}{∂y_{j}^{r}} \\ & = \frac{∂_{c}}{∂y^{s}} − \frac{d_{c}}{dx^{i}} (\frac{∂z^{J}}{∂y_{i}^{s}} \frac{∂}{∂z^{J}} ) − \frac{∂_{c}g_{j}^{r}}{∂y^{s}} \frac{∂z^{J}}{∂y_{j}^{r}} \frac{∂}{∂z^{J}} , \\ μ′_{s} & = \frac{∂_{c}}{∂y^{s}} − \frac{d′_{c}}{dx^{i}} (\frac{∂}{∂y_{i}^{s}} ) − \frac{∂_{c}g_{j}^{r}}{∂y^{s}} \frac{∂}{∂y_{j}^{r}} \\ & = \frac{∂_{c}}{∂y^{s}} − \frac{d′_{c}}{dx^{i}} (\frac{∂z^{J}}{∂y_{i}^{s}} \frac{∂}{∂z^{J}} ) − \frac{∂_{c}g_{j}^{r}}{∂y^{s}} \frac{∂z^{J}}{∂y_{j}^{r}} \frac{∂}{∂z^{J}} . \end{aligned}  (4.69)

Theorem 4.5. Let λ be a Lagrangian in J^{1}Y, Q ⊂ J^{1}Y a regular nonholonomic constraint. Denote by γ local sections of the fibered manifold π : Y → X such that J^{1}γ(W) ⊂ Q. In adapted fibered coordinates, the constrained Euler–Lagrange equations take one of the following equivalent forms:

(1) By means of L,

(A_{s} + B_{sJ}^{i} z_{i}^{J}) ∘ J^{2}γ = 0,  (4.70)

where A_{s}, B_{sJ}^{i} are given by

\begin{matrix}A_{s} = Ā_{s} + Ā_{m−k+a}Ḡ_{s}^{a} + (B̄_{sν}^{ji} + B̄_{m−k+a ν}^{ji}Ḡ_{s}^{a})\frac{d′_{c}g_{j}^{ν}}{dx^{i}} , \\ B_{sJ}^{i} = (B̄_{sν}^{ji} + B̄_{m−k+a ν}^{ji}Ḡ_{s}^{a})\frac{∂g_{j}^{ν}}{∂z^{J}} , \end{matrix}  (4.71)

where (cf. (4.11))

Ā_{σ} = ε′_{σ}(L) ∘ ι, B̄_{σν}^{ij} = − (\frac{∂^{2}L}{∂y_{i}^{σ}∂y_{j}^{ν}} ) ∘ ι  (4.72)

(2) By means of L̄ and L̄_{a}^{j},

(μ_{s}(L̄) − L̄_{a}^{j} μ_{s}(g_{j}^{m−k+a}) − C_{js}^{ai} \frac{d_{c}L̄_{a}^{j}}{dx^{i}} ) ∘ J^{2}γ = 0,  (4.73)

meaning that the functions A_{s}, B_{sJ}^{i} are equivalently expressed as follows:

\begin{matrix}A_{s} = μ′_{s}(L̄) − L̄_{a}^{j} μ′_{s}(g_{j}^{m−k+a}) − C_{js}^{ai} \frac{d′_{c}L̄_{a}^{j}}{dx^{i}} , \\ B_{sJ}^{i} = −\frac{∂}{∂z^{J}} (\frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ) + L̄_{a}^{i} \frac{∂}{∂z^{J}} (Ḡ_{s}^{a}δ_{i}^{j} − C_{is}^{aj}) − C_{js}^{ai} \frac{∂L̄_{a}^{j}}{∂z^{J}} \\ = −\frac{∂}{∂z^{J}} (\frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ) + L̄_{a}^{j} \frac{∂}{∂z^{J}} (\frac{∂g_{j}^{m−k+a}}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ) − C_{js}^{ai} \frac{∂L̄_{a}^{j}}{∂z^{J}} . \end{matrix} (4.74)

Definition 4.13. The operator defined by (4.73), i.e.

E_{s}^{C}(L̄, L̄_{a}^{j}) = μ_{s}(L̄) − L̄_{a}^{j} μ_{s}(g_{j}^{m−k+a}) − C_{js}^{ai} \frac{d_{c}L̄_{a}^{j}}{dx^{i}}  (4.75)

is called the constraint Euler–Lagrange operator.

Remark 4.3. Lagrangian and semiholonomic constraints. Recall that if the constraint Q in J^{1}Y is Lagrangian, then

C_{js}^{ai} = 0, C_{Jj}^{a} = 0  (4.76)

for all values of indices. Consequently, formulas become much simpler. In particular,

ι^{∗}Θ_{λ} = Θ_{ι^{∗}λ} + L̄_{a}^{j} φ̄^{a} ∧ ω_{j},  (4.77)

and the constraint Euler–Lagrange operator reads

E_{s}^{C}(L̄, L̄_{a}^{j}) = μ_{s}(L̄) − L̄_{a}^{j} μ_{s}(g_{j}^{m−k+a}).  (4.78)

If, moreover, Q is a semiholonomic constraint, i.e., if dφ̄^{a} ∈ I(C^{0}) for all a, we get

ι^{∗}dΘ_{λ} = dΘ_{ι^{∗}λ} + a constraint form.  (4.79)

This means that dΘ_{ι^{∗}λ} ≈ ι^{∗}dΘ_{λ}, and even that dΘ_{ι^{∗}λ} is a constrained Poincaré–Cartan (n + 1)-form of λ. Then, of course, the constrained Euler–Lagrange equations (4.65) have the equivalent form

J^{1}γ^{∗}i_{ξ}dΘ_{ι^{∗}λ} = 0 for every π_{1}-vertical vector field ξ ∈ C.  (4.80)

Since in this case κ = kn, we have on Q local coordinates (x^{i}, y^{σ}, y_{j}^{s}), and all formulas take a much simpler form (cf. e.g. [22, 25]).

4.5. Constrained Hamilton–De Donder equations. Let [α] = [dΘ_{λ}] be a Lagrangian system on J^{1}Y, ι : Q → J^{1}Y a regular non-holonomic constraint of corank (κ, k), [[ι^{∗}α]] = [[ι^{∗}dΘ_{λ}]] the corresponding constrained system on Q. For every ᾱ = ι^{∗}α ∈ [[ι^{∗}dΘ_{λ}]] we have the constraint Hamiltonian exterior differential system Hᾱ defined on the domain of definition of ᾱ, say U ⊂ Q, and generated by the system of n-forms and 1-forms (4.63).

Directly from the definition of constraint Hamiltonian EDS we can see that if ᾱ_{1} and ᾱ_{2} differ by a constraint form, then Hᾱ_{1} = Hᾱ_{2}.

Definition 4.14. Let [dΘ_{λ}] be a Lagrangian system on J^{1}Y. For every α ∈ [dΘ_{λ}] the equivalence class

ᾱ_{I} = ι^{∗}α mod I^{n+1}(C^{0})  (4.81)

is called constrained Hamiltonian system related with α and the constraint Q.

Equations for integral sections of Hᾱ, i.e.

δ^{∗}i_{ξ}ᾱ = 0 ∀ξ ∈ V(Q) ∩ C, δ^{∗}φ̄^{a} = 0, 1 ≤ a ≤ k,  (4.82)

where ᾱ ∈ ᾱ_{I} and δ is a section of π_{1}|_{Q} : Q → X, are called constrained Hamilton equations.

Integral sections of Hᾱ are called constrained Hamilton extremals of [[ι^{∗}dΘ_{λ}]].

Similarly as in the unconstrained case, in what follows, we will be interested in constrained Hamiltonian systems that can be completely characterized by constrained Poincaré–Cartan (n + 1)-forms:

Definition 4.15. Let [α] be a Lagrangian system on J^{1}Y, ι : Q → J^{1}Y a regular non-holonomic constraint of corank (κ, k). A constrained Hamiltonian system ᾱ_{I} defined on U ⊂ Q is called constrained Hamilton– De Donder system of [α] if there exists a Lagrangian λ for [α] such that for every ᾱ ∈ ᾱ_{I},

ᾱ = ι^{∗}dΘ_{λ} + F̄, where F̄ ∈ Ω_{Y}^{n+1,(≥2)}(U).  (4.83)

The corresponding constrained Hamilton equations, i.e.

δ^{∗}i_{ξ}(ι^{∗}dΘ_{λ} + F̄) = 0 ∀ξ ∈ V(Q) ∩ C, δ^{∗}φ̄^{a} = 0, 1 ≤ a ≤ k, (4.84)

are called constrained Hamilton–De Donder equations.

Definition 4.16. A constrained Hamilton–De Donder system ᾱ_{I} is called regular if Hᾱ contains all the canonical contact n-forms

ι^{∗}ω^{σ} ∧ ω_{i}, 1 ≤ σ ≤ m, 1 ≤ i ≤ n.  (4.85)

A constrained Lagrangian system is called De Donder regular if around each point in Q there exists an associated regular constrained Hamilton– De Donder system.

Proposition 4.8. Let [α] be a Lagrangian system on J^{1}Y, ι : Q → J^{1}Y a regular non-holonomic constraint of corank (κ, k), ᾱ_{I} an associated regular constrained Hamilton–De Donder system. Then (for all ᾱ ∈ ᾱ_{I}) every integral section of Hᾱ is holonomic. Consequently, constrained Hamilton–De Donder equations of ᾱ_{I} are equivalent with the constrained Euler–Lagrange equations.

Proof. Let δ be an integral section of Hᾱ. If ᾱ is regular then, by definition,

δ^{∗}(ω̄^{σ} ∧ ω_{i}) = 0, 1 ≤ σ ≤ m, 1 ≤ i ≤ n.  (4.86)

This implies, however, that for all σ, δ^{∗}ω̄^{σ} = 0, meaning that δ is a holonomic section in Q. □

Theorem 4.6. Let ᾱ_{I} = ι^{∗}dΘ_{λ}+F̄ mod I^{n+1}(C^{0}), where F̄ ∈ Ω_{Y}^{n+1,(≥2)}(Q), be a constrained Hamilton–De Donder system. The following conditions are equivalent:

(1) For every ᾱ ∈ ᾱ_{I}, ᾱ is regular.

(2) For every ᾱ ∈ ᾱ_{I}, a system of generators of Hᾱ has maximal rank (equal to (n + 1)(m − k) + k).

(3) Every first-order Lagrangian λ for dΘ_{λ} satisfies the constraint regularity condition

rank (B_{sJ}^{i}) = n(m − k),  (4.87)

where B_{sJ}^{i} are given in terms of λ = Lω_{0} by (4.71) or (4.74).

Proof. We can write

ᾱ = A_{s}ω̄^{s} ∧ ω_{0} + B_{sJ}^{i}ω̄^{s} ∧ dz^{J} ∧ ω_{i} + F + φ,  (4.88)

where A_{s} and B_{sJ}^{i} are given in terms of a first-order Lagrangian λ for dΘ_{λ} by (4.71) or (4.74), F is the sum of F̄ and the 2-contact part of ι^{∗}dΘ_{λ}, and φ ∈ I^{n+1}(C^{0}). Denote

F = F_{ls}^{i}ω̄^{l} ∧ ω̄^{s} ∧ ω_{i} + ν, where F_{ls}^{i} = −F_{sl}^{i}.  (4.89)

Computing generators of Hᾱ, we obtain a mixed system of k (linearly independent) 1-forms

φ̄^{a} = ω̄^{m−k+a} − ∑_{s=1}^{m−k} Ḡ_{s}^{a}ω̄^{s}, 1 ≤ a ≤ k,  (4.90)

and m − k + mn − κ n-forms (that, in general, need not be independent) as follows:

\begin{matrix}A_{s}ω_{0} + 2F_{sl}^{i}ω̄^{l} ∧ ω_{i} + B_{sJ}^{i}dz^{J} ∧ ω_{i} + μ_{s}, \\ B_{sJ}^{i}ω̄^{s} ∧ ω_{i}, \end{matrix}  (4.91)

1 ≤ s ≤ m − k, 1 ≤ J ≤ mn − κ, where μ_{s} = i_{∂/∂y^{s}}ν.

Suppose that ᾱ_{I} is regular. Then Hᾱ is generated by the forms φ̄^{a} and

A_{s}ω_{0} + B_{sJ}^{i}dz^{J} ∧ ω_{i} + μ_{s}, ω̄^{s} ∧ ω_{i},  (4.92)

which means that the matrix B_{1} = (B_{sJ}^{i}) with mn − κ rows labelled by J and n(m − k) columns labelled by i, s, has rank n(m − k). Now, the rank of the system of generators of Hᾱ is > n(m − k). Let us compute the rank of the matrix B_{2} = (B_{sJ}^{i}) with m − k rows labelled by s and n(nm−κ) columns labelled by i, J. By the above we can see that for every fixed i, the matrix B_{1} has m − k linearly independent columns (labelled by s). Consequently, B_{2} has for every fixed i the submatrix (B_{sJ}^{•}) with m − k independent rows labelled by s (equal to transposed submatrix of B_{2} with the corresponding values of indices), hence rank B_{2} = m − k. Summarizing, we have obtained that the forms (4.91) (resp. (4.92)) are independent, meaning that the rank of the system of generators of Hᾱ (for all ᾱ ∈ ᾱ_{I}) is maximal and equal to n(m − k) + m − k + k = (n + 1)(m − k) + k, as desired.

Next, suppose that the rank of Hᾱ is maximal. Then the forms (4.91) are independent, meaning that the constrained regularity condition (4.87) holds.

Finally, if (4.87) holds, then the forms B_{sJ}^{i}ω̄^{s} ∧ ω_{i} in (4.91) are independent, i.e. the forms ω̄^{s} ∧ ω_{i}, 1 ≤ i ≤ n, 1 ≤ s ≤ m − k, belong to Hᾱ.

By definition of Hᾱ also all the n-forms φ̄^{a} ∧ ω_{i} ∈ Hᾱ. Hence, for all a, i, we have ω^{m−k+a} ∧ ω_{i} = φ̄^{a} ∧ ω_{i} + Ḡ_{s}^{a}ω̄^{s} ∧ ω_{i} ∈ Hᾱ, and we are done. □

Corollary 4.3. Let

ᾱ_{I} = ι^{∗}dΘ_{λ} + F̄ mod I^{n+1}(C^{0}), where F̄ ∈ Ω_{Y}^{n+1,(≥2)}(Q),

be a regular constrained Hamilton–De Donder system. Then Hamilton extremals of ᾱ_{I} (i.e. integral sections of (4.90) and (4.92)) do not depend upon the choice of F̄. This means that Hamilton–De Donder equations of all elements in the class

ι^{∗}dΘ_{λ} mod Ω_{Y}^{n+1,(≥2)}(Q) + I^{n+1}(C^{0})  (4.93)

are equivalent.

From the proof of Theorem 4.6 we can conclude that if ᾱ_{I} is regular then the matrix B_{1} has nm−κ rows and n(m−k) columns where nm−κ ≥ n(m − k). This means that κ ≤ nk, and we get the following result:

Corollary 4.4. Let [α] be a Lagrangian system on J^{1}Y, ι : Q → J^{1}Y a regular non-holonomic constraint of corank (κ, k). A necessary condition for the constrained Lagrangian system [[ι^{∗}α]] be De Donder regular is

κ ≤ nk.  (4.94)

4.6. Constraint Legendre transformation.

Theorem 4.7. Let ι : Q → J^{1}Y be a regular non-holonomic constraint of corank (κ, k), [[ᾱ]] a constrained Lagrangian system. Let x ∈ Q be a point. Suppose that in a neighborhood of x,

\frac{∂B_{sJ}^{i}}{∂z^{K}} = \frac{∂B_{sK}^{i}}{∂z^{J}} , 1 ≤ i ≤ n, 1 ≤ s ≤ m−k, 1 ≤ J, K ≤ mn−κ. (4.95)

Then there exists a neighborhood U ⊂ Q of x, and, on U, functions P_{s}^{i}, 1 ≤ i ≤ n, 1 ≤ s ≤ m − k, and a n-form η, such that the class [[ᾱ]] is represented by the (n + 1)-form

ᾱ = η ∧ ω_{0} + dP_{s}^{i} ∧ dy^{s} ∧ ω_{i}.  (4.96)

Proof. As we have seen in Sec. 4.4, around each point in Q, the constrained Lagrangian system [[ᾱ]] has a representative

A_{s}ω̄^{s} ∧ ω_{0} + B_{sJ}^{i}ω̄^{s} ∧ dz^{J} ∧ ω_{i},  (4.97)

where A_{s} and B_{sJ}^{i} are defined by (4.71) or (4.74). Assume that the given Lagrangian system [ᾱ] has a Lagrangian λ = Lω_{0} defined around x such that for the corresponding functions B_{sJ}^{i} integrability conditions (4.95) are satisfied. Applying the Poincaré Lemma we get a neighborhood U ⊂ Q of x and functions P_{s}^{i}, 1 ≤ i ≤ n, 1 ≤ s ≤ m − k, on U that are given by

B_{sJ}^{i} = −\frac{∂P_{s}^{i}}{∂z^{J}} .  (4.98)

Hence, in the class [[ᾱ]] on U we can find the following representatives, equivalent with ι^{∗}dΘ_{λ}, where λ is the above mentioned Lagrangian:

\begin{matrix}A_{s}ω̄^{s} ∧ ω_{0} + \frac{∂P_{s}^{i}}{∂z^{J}} dz^{J} ∧ ω̄^{s} ∧ ω_{i} \\ ≈ A_{s}ω̄^{s} ∧ ω_{0} + dP_{s}^{i} ∧ ω̄^{s} ∧ ω_{i} − \frac{d′_{c}P_{s}^{i}}{dx^{j}} dx^{j} ∧ ω̄^{s} ∧ ω_{i} \\ = (A_{s} + \frac{d′_{c}P_{s}^{i}}{dx^{i}} ) dy^{s} ∧ ω_{0} − g_{i}^{s}dP_{s}^{i} ∧ ω_{0} + dP_{s}^{i} ∧ dy^{s} ∧ ω_{i} \\ ≈ (A_{s} + \frac{d′_{c}P_{s}^{i}}{dx^{i}} − g_{i}^{l} \frac{∂_{c}P_{l}^{i}}{∂y^{s}} ) dy^{s} ∧ ω_{0} \\ −g_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{J}} dz^{J} ∧ ω_{0} + dP_{s}^{i} ∧ dy^{s} ∧ ω_{i}. \end{matrix} (4.99)

In this way we have obtained a representative

\begin{matrix}ᾱ = (A_{s} + \frac{d′_{c}P_{s}^{i}}{dx^{i}} − g_{i}^{r} \frac{∂_{c}P_{r}^{i}}{∂y^{s}} ) dy^{s} ∧ ω_{0} \\ −g_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{J}} dz^{J} ∧ ω_{0} + dP_{s}^{i} ∧ dy^{s} ∧ ω_{i}. \end{matrix}  (4.100)

Let us denote

ᾱ = η ∧ ω_{0} + dP_{s}^{i} ∧ dy^{s} ∧ ω_{i},  (4.101)

with

η = η̃_{j}dx^{j} + η̄_{s}dy^{s} + η̄_{J}dz^{J},  (4.102)

where η̃_{j}, 1 ≤ j ≤ n, are arbitrary functions on U, and

\begin{aligned}η̄_{s} & = & A_{s} + \frac{d′_{c}P_{s}^{i}}{dx^{i}} − g_{i}^{r} \frac{∂_{c}P_{r}^{i}}{∂y^{s}} , 1 ≤ r, s ≤ m − k, \\ η̄_{J} & = & −g_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{J}} , 1 ≤ s ≤ m − k, 1 ≤ J ≤ mn − κ. \end{aligned}  (4.103)

This completes the proof. □ □

Corollary 4.5. The class

ᾱ_{I} = η ∧ ω_{0} + dP_{s}^{i} ∧ dy^{s} ∧ ω_{i} mod I^{n+1}(C^{0})  (4.104)

constructed in the above Theorem is a constrained Hamilton–De Donder system, corresponding to the Lagrangian λ = Lω_{0}.

Proof. It is sufficient to check the computations to see that that ᾱ−ι^{∗}dΘ_{λ} is (up to a constraint form) a 2-contact form, horizontal with respect to the projection onto Y. □

Note that η in (4.104) is determined up to a constraint form. In this way, for a constrained Hamilton–De Donder system we have to consider the class

η_{I} = η mod I^{n}(C^{0}).  (4.105)

It should be stressed that in the class η_{I} there need not exist a closed representative.

Remark 4.4. Integrability condition (4.95) can be rewritten in terms of a Lagrangian λ = Lω_{0} giving rise to the functions B_{sJ}^{i} using (4.71) or (4.74). For example, with help of (4.74) it takes the form

\begin{matrix}\frac{∂L̄_{a}^{j}}{∂z^{K}} \frac{∂}{∂z^{J}} (\frac{∂g_{j}^{m−k+a}}{∂z^{I}} \frac{∂z^{I}}{∂y_{i}^{s}} ) − \frac{∂C_{js}^{ai}}{∂z^{K}} \frac{∂L̄_{a}^{j}}{∂z^{J}} \\ = \frac{∂L̄_{a}^{j}}{∂z^{J}} \frac{∂}{∂z^{K}} (\frac{∂g_{j}^{m−k+a}}{∂z^{I}} \frac{∂z^{I}}{∂y_{i}^{s}} ) − \frac{∂C_{js}^{ai}}{∂z^{J}} \frac{∂L̄_{a}^{j}}{∂z^{K}} . \end{matrix}  (4.106)

Let us find an explicit formula for the functions P_{s}^{i}.

Proposition 4.9. Let x ∈ U, and consider a mapping χ : [0, 1] × W → W, defined by

(u, x^{i}, y^{σ}, z^{J}) → (x^{i}, y^{σ}, uz^{J}),  (4.107)

where W ⊂ U ⊂ Q is an appropriate neighborhood of x. Then for arbitrary functions ψ_{s}^{i}(x^{j}, y^{ν}) (resp. ψ̃_{s}^{i}(x^{j}, y^{ν}) ), 1 ≤ s ≤ m − k, 1 ≤ i ≤ n, the functions

\begin{matrix}P_{s}^{i} = −z^{J} ∫_{0}^{1} (B_{sJ}^{i} ∘ χ) du + ψ_{s}^{i}(x^{j}, y^{ν}) \\ = \frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} +z^{J}∫_{0}^{1}(C_{js}^{ai} \frac{∂L̄_{a}^{j}}{∂z^{J}}−L̄_{a}^{j} \frac{∂}{∂z^{J}} (\frac{∂g_{j}^{m−k+a}}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ))∘ χdu \\ + ψ̃_{s}^{i}(x^{j}, y^{ν}), \end{matrix} (4.108)

are solutions of (4.98).

Proof. Integrability condition (4.95) guarantees that in a neighborhood of any point of U one can find solutions of (4.98) using Poincaré Lemma. Put

P_{s}^{i} = −z^{J} ∫_{0}^{1} (B_{sJ}^{i} ∘ χ) du + ψ_{s}^{i}(x^{j}, y^{ν}).  (4.109)

Then with help of (4.95)

\begin{aligned}\frac{∂P_{s}^{i}}{∂z^{J}} & = − ∫_{0}^{1} (B_{sJ}^{i} ∘ χ) du − z^{K} ∫_{0}^{1} (\frac{∂B_{sK}^{i}}{∂z^{J}} ∘ χ) u du \\ & = − ∫_{0}^{1} d(u(B_{sJ}^{i} ∘ χ)) = −B_{sJ}^{i}, \end{aligned}  (4.110)

as desired.

Using formula (4.74) and (4.109) we get

\begin{matrix}P_{s}^{i} = z^{J} ∫_{0}^{1} (\frac{∂}{∂z^{J}} (\frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ) + C_{js}^{ai} \frac{∂L̄_{a}^{j}}{∂z^{J}} \\ −L̄_{a}^{j} \frac{∂}{∂z^{J}} (\frac{∂g_{j}^{m−k+a}}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} )) ∘ χdu + ψ_{s}^{i}(x^{j}, y^{ν}) \\ = \frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} +z^{J}∫_{0}^{1}(C_{js}^{ai} \frac{∂L̄_{a}^{j}}{∂z^{J}}−L̄_{a}^{j} \frac{∂}{∂z^{J}} (\frac{∂g_{j}^{m−k+a}}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ))∘ χdu \\ +ψ̃_{s}^{i}(x^{j}, y^{ν}), \end{matrix} (4.111)

since

\begin{matrix}∫_{0}^{1} d (\frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ∘ χ) = [\frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ∘ χ]_{u=0}^{u=1} = \frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} − h_{s}^{i}(x^{j}, y^{ν}) \\ = ∫_{0}^{1} \frac{d}{du} (\frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} ∘ χ) du = z^{J} ∫_{0}^{1} (\frac{∂}{∂z^{J}} (\frac{∂L̄}{∂z^{K}} \frac{∂z^{K}}{∂y_{i}^{s}} )) ∘ χdu. \end{matrix}

This completes the proof. □ □

Keeping the above notations we can introduce the following concepts:

Definition 4.17. The local representative ᾱ (4.96) of the constrained Lagrangian system is called a representative in canonical form. Functions P_{s}^{i} are called constraint momenta, and (any) 1-form η in (4.104) is called energy 1-form associated with the corresponding Hamilton–De Donder system ᾱ_{I}.

By the next theorem, constraint Legendre transformation, associated with a regular constrained Hamilton–De Donder system is defined. It is a local coordinate transformation on the constraint submanifold Q.

Theorem 4.8. Let [[ᾱ]] be a De Donder regular constrained Lagrangian system on Q, let ᾱ_{I} be a corresponding regular Hamilton–De Donder system on a coordinate neighborhood U ⊂ Q, P_{s}^{i}, 1 ≤ i ≤ n, 1 ≤ s ≤ m − k, its associated constraint momenta. Then the set of functions (x^{i}, y^{σ}, P_{s}^{i}) can be completed to coordinates on U. In particular, the set of nm−κ indices labelled by J has a subset labelled by B, n(m−k)+1 ≤ B ≤ nm − κ, such that

(x^{i}, y^{σ}, z^{J}) → (x^{i}, y^{σ}, P_{s}^{i}, z^{B})  (4.112)

is a coordinate transformation on U.

Proof. By assumption, the matrix (B_{sJ}^{i}) = (∂P_{s}^{i}/∂z^{J}) has maximal rank (equal to n(m − k)). This means that it has a regular submatrix with n(m − k) linearly independent rows: we can label them by A, where 1 ≤ A ≤ n(m − k). The remaining rows will be labelled by B, i.e. n(m − k) + 1 ≤ B ≤ nm − κ. Now, the map (x^{i}, y^{σ}, z^{J}) → (x^{i}, y^{σ}, P_{s}^{i}, z^{B}) is regular, i.e. is a coordinate transformation on U. □ □

Finally, let us find the expression of constrained Hamilton–De Donder equations in constraint Legendre coordinates. First, we have up to a constraint form,

η = η_{j}dx^{j} + η_{s}dy^{s} + η_{i}^{s} dP_{s}^{i} + η_{B}dz^{B},  (4.113)

hence,

\begin{matrix}η = η_{i}dx^{i}+ η_{s}dy^{s}+ η_{i}^{s} (\frac{d′_{c}P_{s}^{i}}{dx^{j}} dx^{j}+\frac{∂_{c}P_{s}^{i}}{∂y^{r}} ω̄^{r}+\frac{∂P_{s}^{i}}{∂z^{J}} dz^{J})+ η_{B}dz^{B} \\ = (η_{j} + η_{i}^{s} \frac{d′_{c}P_{s}^{i}}{dx^{j}} − η_{i}^{s} \frac{∂_{c}P_{s}^{i}}{∂y^{r}} g_{j}^{r}) dx^{j} + (η_{r} + η_{i}^{s} \frac{∂_{c}P_{s}^{i}}{∂y^{r}} ) dy^{r} \\ + η_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{A}} dz^{A} + (η_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{B}} + η_{B}) dz^{B}. \end{matrix} (4.114)

Comparing with (4.102), (4.103) we obtain

\begin{matrix}η̄_{r} = η_{r} + η_{i}^{s} \frac{∂_{c}P_{s}^{i}}{∂y^{r}} = A_{r} + \frac{d′_{c}P_{r}^{i}}{dx^{i}} − g_{i}^{s} \frac{∂_{c}P_{s}^{i}}{∂y^{r}} , \\ η̄_{A} = η_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{A}} = −g_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{A}} , \\ η̄_{B} = η_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{B}} + η_{B} = −g_{i}^{s} \frac{∂P_{s}^{i}}{∂z^{B}} . \end{matrix}  (4.115)

The second relation gives us

(η_{i}^{s} + g_{i}^{s})\frac{∂P_{s}^{i}}{∂z^{A}} = 0, i.e. η_{i}^{s} = −g_{i}^{s}(x^{j}, y^{ν}, P_{l}^{j}, z^{B}),  (4.116)

since the matrix (\frac{∂P_{s}^{i}}{∂z^{A}} ) is regular. The first and third relation above

then read

η_{r} = η̄_{r} + g_{i}^{s} \frac{∂_{c}P_{s}^{i}}{∂y^{r}} = A_{r} + \frac{d′_{c}P_{r}^{i}}{dx^{i}} , η_{B} = 0.  (4.117)

Next, we can see that the vertical subbundle of the canonical distribution is in Legendre coordinates generated by the vector fields ∂_{c}/∂y^{s}, ∂/∂P_{s}^{i}, and ∂/∂z^{B}. Computing contractions of a representative ᾱ of (4.104) by these vector fields, we get the constrained Hamilton–De Donder equations δ^{∗}i_{ξ}ᾱ = 0, δ^{∗}φ̄^{a} = 0 in the following “canonical form”:

Theorem 4.9. Constrained Hamilton–De Donder equations (4.84) in constraint Legendre coordinates take the form

\begin{matrix}\frac{∂(P_{s}^{i} ∘ δ)}{∂x^{i}} = η_{s} ∘ δ, \frac{∂(y^{s} ∘ δ)}{∂x^{i}} = −η_{i}^{s} ∘ δ, \\ 1 ≤ s ≤ m − k, 1 ≤ i ≤ n, \end{matrix}  (4.118)

\frac{∂(y^{m−k+a} ∘ δ)}{∂x^{i}} = g_{i}^{m−k+a} ∘ δ, 1 ≤ a ≤ k, 1 ≤ i ≤ n.  (4.119)

Remark 4.5. In view of Theorem 4.8 we can see that for a regular constrained Lagrangian system one has on Q local adapted coordinates (x^{i}, y^{σ}, y_{j}^{s}, z^{B}), where 1 ≤ i ≤ n, 1 ≤ σ ≤ m, 1 ≤ s ≤ k, n(m − k) + 1 ≤ B ≤ nm − κ, and corresponding adapted bases of 1-forms (dx^{i}, dy^{s}, φ̄^{a}, dy_{j}^{s}, dz^{B}), respectively, (dx^{i}, ω̄^{s}, φ̄^{a}, dy_{j}^{s}, dz^{B}). In these coordinates many formulas simplify, since z^{A} = y_{j}^{s} (hence ∂z^{B}/∂y_{j}^{s} = 0).

The results on constrained Hamilton–De Donder systems can be easily reformulated for special cases of constraints. Let us recall the properties of semiholonomic constraints, that are quite similar to the unconstrained case [25].

Theorem 4.10. Let Q be a semiholonomic constraint in J^{1}Y (i.e. κ = kn and dI(C^{0}) ⊂ I(C^{0})), ᾱ_{I} a constrained Hamilton–de Donder system. Then the integrability condition (4.95) is satisfied identically and the regularity condition reads

det (\frac{∂^{2}L̄}{∂y_{i}^{l}∂y_{j}^{s}} ) ≠ 0.  (4.120)

Constrained momenta are given by the formula

P_{s}^{i} = \frac{∂L̄}{∂y_{i}^{s}} ,  (4.121)

the class of energy 1-forms η_{I} contains a closed form dH̄, where

H̄ = −L̄ + P_{s}^{i}y_{i}^{s},  (4.122)

and constraint Legendre transformation is a local diffeomorphism

(x^{i}, y^{σ}, y_{j}^{s}) → (x^{i}, y^{σ}, P_{s}^{j})  (4.123)

on the submanifold Q ⊂ J^{1}Y.

148 O. KRUPKOVÁ AND P. VOLNÝ

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Department of Algebra and Geometry, Faculty of Science Palacky University, Tomkova 40, 779 00 Olomouc, Czech Republic; Department of Mathematics, La Trobe University, Melbourne, Victoria 3086, Australia E-mail address: krupkova@inf.upol.cz

Petr Volný Department of Mathematics and Descriptive Geometry, VSB - Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic E-mail address: Petr.Volny@vsb.cz

Received July 24, 2006