DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 533–546

ON THE GEOMETRY OF NON–HOLONOMIC

MECHANICAL SYSTEMS ^{∗)}

OLGA KRUPKOVÁ

Abstract. A geometric approach to the theory of non-holonomic systems is presented. General first order mechanical systems (not necessarily regular or Lagrangian) are studied. A constraint structure on J^{1}Y is given by a submanifold fibered over Y, naturally endowed with a distribution (of a constant rank, possibly nonintegrable). Constraint force is studied and geometric arguments for obtaining the Chetaev formula are presented. Constrained mechanical systems are geometrically described, their motion equations in intrinsic form are found, and dynamics are characterized by means of distributions on the constraint submanifold.

1. Non-holonomic constraints in classical mechanics

In classical mechanics one naturally encounters different kinds of constraints on the motion, mostly holonomic and linear non-holonomic constraints (integrable or non-integrable). While mechanics of holonomic systems is well-established both from the physical and geometric point of view, in non-holonomic mechanics still many questions remain open. The main problems studied concern namely geometric structures arising due to presence of constraints, a direct geometric characterization of constrained systems (without help of Lagrange multipliers), variationality and regularity of constrained systems, the concept of constraint force, a generalization of the concept of virtual displacements, Principle of virtual work, etc., all for the case of general non-holonomic constraints (which are not supposed linear in the first derivatives, as it is done within classical mechanics). Following classical works (e.g. [4]), recently there have appeared many papers dealing with geometric aspects of the problem (see [3], [5], [8]–[21] and references therein).

1991 Mathematics Subject Classification. 58F05, 70F25, 70H35. Key words and phrases. Mechanical system, Lagrangian system, non-holonomic constraints,

holonomic constraints, constraint distribution, Chetaev bundle, virtual displacements, constraint force, constrained equations of motion, regularity.

∗) Work supported by Grant No. VS 96003 “Global Analysis” of the Czech Ministry of Education, Youth and Sports, and Grant No. 201/98/0853 of the Czech Grant Agency.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Let us turn to a brief characterization of systems subject to non-holonomic constraints. To this purpose consider mechanical systems modeled on a fibered manifold π : Y → X where dim X = 1 and dim Y = m + 1. Denote by (t, q^{1}, …, q^{m}) fiber coordinates on Y, and by dotted resp. doubledotted q’s the associated coordinates on the first and second jet prolongation of π. Within this scheme, dynamics is usually represented by a semispray distribution on J^{1}Y. It should be stressed that such a geometric description is suitable only for regular mechanical systems; generalization to non-regular cases requires to consider distributions on J^{1}Y of rank greater than one [7].

A system of k non-holonomic constraints is a submanifold Q of codimension k in J^{1}Y, where 1 ≤ k < m, fibered over Y; locally it is given by a system of k equations

(1.1)  f^{i}(t, q^{1}, …, q^{m}, q̇^{1}, …, q̇^{m}) = 0, 1 ≤ i ≤ k ,

where

rank(\frac{∂f^{i}}{∂q̇^{σ}} )= k .

Consider a mechanical system on π locally represented by motion equations

\frac{∂L}{∂q^{σ}} − \frac{d}{dt} \frac{∂L}{∂q̇^{σ}} = F_{σ} ,

where L is the Lagrange function of the system (in classical mechanics L = T − V, the kinetic minus potential energy), and F_{σ} have the meaning of components of the sum of non-conservative forces. The motion of the corresponding constrained system is then described by the equations

(1.2)  \frac{∂L}{∂q^{σ}} − \frac{d}{dt} \frac{∂L}{∂q̇^{σ}} = F_{σ} + μ_{i} Φ_{σ}^{i} ,

where μ_{i}, 1 ≤ i ≤ k, are Lagrange multipliers and Φ_{σ} = μ_{i} Φ_{σ}^{i} are the components of the so called non-holonomic constraint force, usually considered to be given by the formula

(1.3)  Φ_{σ}^{i} = \frac{∂f^{i}}{∂q̇^{σ}} ,

proposed by Chetaev in 1932 [4]. Equations (1.2) together with the equations of constraints (1.1) represent a system of m + k mixed first and second order ordinary differential equations for m + k unknown functions q^{σ}(t), μ_{i}(t). Solving them one finds both the explicit form of the Lagrange multipliers (as functions of the parameter t) and the trajectories of the constrained system. The classical procedure of excluding Lagrange multipliers from the motion equations with help of the equations of constraints means in fact a procedure to express the constrained system as a mechanical system on the fibered manifold Q → R.

Let us recall briefly at least a few points which have been of interest in nonholonomic mechanics.

– First of all, the equations (1.2) are not the only possibility for motion equations of non-holonomic systems proposed so far. Different equations (leading to different description of the motion) have been studied within vakonomic dynamics ([1], [2]). However, it seems that the vakonomic equations do not reflect the non-holonomic motions correctly.

– Also Chetaev’s approach, though widely accepted, carries some open problems, especially when non-linear non-holonomic constraints are considered. This concerns namely the constraint force: the formula (1.3) proposed by Chetaev is heuristic, and, at least at the heuristic level, not the only possibility which could be taken into account. A closely related question is a generalization of the Principle of virtual work. One needs more arguments, both physical and mathematical, to support the theory.

– Recently, studies of a geometric framework for non-holonomic mechanics have been a “hot topic” of research. This concerns geometric models for constraints and constrained systems, intrinsic forms of the constrained equations of motion, geometric characterization of properties of constrained systems (as e.g. regularity), etc. In particular, a geometric background for the procedure of excluding Lagrange multipliers has been found, and constrained systems have been modeled as mechanical systems on constraint manifolds. Apparently, geometric studies are helpful in clarifying problems of not only mathematical but also of physical nature.

– Many questions which remain open concern Lagrangian systems subject to non-holonomic constraints. Contrary to the case of holonomic constraints where a constrained system arising from a Lagrangian system remains Lagrangian with the Cartan form pulled-back to the constraint submanifold, non-holonomic constraints do not possess this lovely property. (Consult e.g. [3], [8], [19], [21]).

– Another remarkable property which can be violated under constraints is the property of regularity. More precisely, a regular mechanical system (i.e., defined by a regular Lagrangian, or, more generally, described by a semispray vector field) can become singular (i.e., not representable by means of a semispray on the constraint submanifold), and vice versa, a singular mechanical system can possibly become regular under constraints (the procedure of regularization via constraints).

– Last but not least, one should notice that equivalent motion equations (i.e., whose solutions are the paths of the same semispray) subject to the same constraints may result in essentially different constrained systems (with different dynamics on the constraint submanifold).

The aim of this paper is to explain and further develop the approach to the theory of non-holonomic systems presented in [8]. The theory is proposed for general (not only regular or Lagrangian) mechanical systems on fibered manifolds, and for general non-holonomic constraints. The key-points are the following:

– the concept of a mechanical system as an equivalence class of 2-forms on J^{1}Y, called Lepagean 2-forms, which, roughly speaking, represents a geometric model generalizing e.g. the symplectic model of regular Lagrangian mechanics on tangent spaces,

– the concept of constraint structure as a submanifold Q of J^{1}Y, fibered over Y, endowed with a distribution C (canonical distribution or Chetaev bundle), naturally generated by the constraint (similarly as in [8] and [15]). C is a distribution on Q of minimal possible rank such that the motion of every mechanical system constrained to Q is described by a subdistribution of C. In this sense, the canonical distribution has the meaning of an “universal covering distribution” for all possible dynamical distributions on Q. (In the unconstrained case the role of such a distribution is played simply by the tangent distribution to J^{1}Y. In the holonomic case, where Q = J^{1}Q_{0} of a fibered submanifold Q_{0} of Y, C is the tangent distribution T J^{1}Q_{0}).

A constrained system then can be modeled either by means of a local deformation of the original mechanical system in a neighborhood of the constraint submanifold (“physical description”) or as a mechanical system on the constraint structure (“geometrical description”). In both the cases we present equations of motion in intrinsic form. Within this scheme we also study constraint force and give geometric arguments for the Chetaev formula (1.3). We also deal with regularity and variationality of constrained systems.

2. Mechanical systems on fibered manifolds

Throughout the paper, manifolds and mappings are smooth, and the summation convention is used. We denote by d the exterior derivative of differential forms, by T and J^{r}, r = 1, 2, the tangent and the prolongation functor, respectively.

We consider a fibered manifold π : Y → X with dim X = 1, dim Y = m + 1, and the jet prolongations π_{1} : J^{1}Y → X and π_{2} : J^{2}Y → X of π. (Local) fiber coordinates on Y are denoted by (t, q^{σ}) where 1 ≤ σ ≤ m. The associated coordinates on J^{1}Y and J^{2}Y are denoted by (t, q^{σ}, q̇^{σ}) and (t, q^{σ}, q̇^{σ}, q̈^{σ}), respectively. We also use the projections π_{1,0} : J^{1}Y → Y, or π_{2,1} : J^{2}Y → J^{1}Y, etc.

A section δ of π_{1} is called holonomic if there exists a section γ of π such that δ = J^{1}γ.

By a distribution on J^{1}Y we shall mean a mapping D assigning to every point z ∈ J^{1}Y a vector subspace D(z) of the vector space T_{z}J^{1}Y. A distribution is said to be of a constant rank if dim D(z) does not depend on z. If D is a distribution, we denote by D^{0} its annihilator, i.e., the set of all 1-forms η_{κ} on J^{1}Y such that i_{ξ_{ι}}η_{κ} = 0 for every vector field ξ_{ι} belonging to D. Recall that a section δ of π_{1} is called an integral section of D if δ^{∗}η = 0 for every 1-form η belonging to D^{0}.

A differential p-form η on 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.

We denote by h the horizontalization mapping, assigning to every 1-form η on Y (resp. on J^{1}Y) a π-horizontal (resp. π_{1}-horizontal) 1-form hρ on J^{1}Y (resp. on J^{2}Y).

A form η on J^{1}Y is called contact if J^{1}γ^{∗}η = 0 for every section γ of π. In particular, a 2-form η is called 1-contact if for every π_{1}-vertical vector field ξ the 1-form i_{ξ}η is horizontal, and it is called 2-contact if i_{ξ}η is contact.

In what follows, we use the following contact 1-forms

ω^{σ} = dq^{σ} − q̇^{σ} dt, ω̇^{σ} = dq̇^{σ} − q̈^{σ} dt, 1 ≤ σ ≤ m ,

and the bases (dt, ω^{σ}, dq̇^{σ}), or (dt, ω^{σ}, ω̇^{σ}, dq̈^{σ}) of 1-forms on J^{1}Y (resp. on J^{2}Y).

Every 1-form η on J^{1}Y admits a unique decomposition into a sum of a horizontal and contact form. In fiber coordinates, if η = f dt + f_{σ} dq^{σ} + f̃_{σ} dq̇^{σ}, this decomposition reads

π_{2,1}^{∗}η = (f + f_{σ}q̇^{σ} + f̃_{σ}q̈^{σ})dt + f_{σ} ω^{σ} + f̃_{σ} ω̇^{σ} .

Similarly, every 2-form η on J^{1}Y admits a unique decomposition into a sum of a 1-contact and 2-contact form.We denote by pη (resp. p_{1}η) the contact (resp. 1-contact) part of η.

A 2-form E on J^{2}Y is called a dynamical form if it is 1-contact and π_{2,0}- horizontal. In this paper we restrict to dynamical forms on J^{2}Y with the components affine in the second derivatives, i.e., we suppose that in every fiber chart on J^{2}Y,

E = E_{σ}(t, q^{ν}, q̇^{ν}, q̈^{ν}) dq^{σ} ∧ dt ,

where

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

A section γ of π is called a path of E if

E ∘ J^{2}γ = 0 .

In fiber coordinates this equation represents a system of m second-order ordinary differential equations

A_{σ}(t, γ^{ν}, \frac{dγ^{ν}}{dt} ) + B_{σρ}(t, γ^{ν}, \frac{dγ^{ν}}{dt} )\frac{d^{2}γ^{ρ}}{dt^{2}} = 0

for the components γ^{ν}(t), 1 ≤ ν ≤ m, of γ. Notice that these equations need not be equations for paths of a semispray connection. However, they can be represented in terms of a distribution on J^{1}Y as follows.

Let α be a 2-form on J^{1}Y. Following [8] we say that α is a (generalized) Lepagean 2-form associated to a dynamical form E if p_{1}α = E. Immediately one gets for E the following equivalence class of associated Lepagean 2-forms:

(2.2)  [α] = A_{σ}ω^{σ} ∧ dt + B_{σν}ω^{σ} ∧ dq̇^{ν} mod F

where F runs over the 2-contact 2-forms of order one, F = F_{σν}(t, q^{ρ}, q̇^{ρ})ω^{σ} ∧ ω^{ν}. This equivalence class is then called a first-order mechanical system on π. Notice that 2-forms equivalent in the above sense are equivalent also in the sense of the Krupka variational sequence [6].

For every Lepagean 2-form α there is a distribution Δ_{α} on J^{1}Y defined by (2.3)

Δ_{α}^{0} = span{i_{ξ}α | ξ runs over the set of all π_{1}-vertical vector fields on J^{1}Y }
= span{A_{σ}dt + B_{σν}dq̇^{ν} + 2F_{σν}ω^{ν}, B_{σν}ω^{ν}} ,

and called the dynamical distribution of α. At each point of J^{1}Y, rank Δ_{α} ≥ 1, however, the rank need not be constant. Hence, we get the following:

Proposition 1.

(1) For any two equivalent Lepagean 2-forms the holonomic integral sections of the corresponding dynamical distributions coincide.

(2) The set of holonomic integral sections of any dynamical distribution of E coincides with the set of paths of E.

(3) A section γ of π is a path of a dynamical form E if and only if for every π_{1}-vertical vector field ξ on J^{1}Y

J^{1}γ^{∗}i_{ξ}α = 0 ,

where α is a Lepagean 2-form associated to E.

The class of dynamical distributions related to a mechanical system [α] will be denoted by [Δ_{α}].

Now, we get in a natural way a geometric concept of regularity of a mechanical system (cf. [7] and [8]): A mechanical system [α] is called regular if there exists a dynamical distribution Δ_{α} ∈ [Δ_{α}] such that rank Δ_{α} = 1.

We obtain the following useful characterization of regular mechanical systems:

Theorem 1 [8]. Let [α] be the mechanical system related to E, and let [Δ_{α}] be the corresponding class of dynamical distributions. Denote E = (A_{σ} +B_{σν}q̈^{ν})dq^{σ} ∧dt. The following conditions are equivalent:

(1) The mechanical system [α] is regular.

(2) The matrix (B_{σν}) is everywhere regular.

(3) Each of the dynamical distributions belonging to [Δ_{α}] has rank one.

(4) All the dynamical distributions belonging to [Δ_{α}] coincide, and

Δ_{α} = span\{\frac{∂}{∂t} + q̇^{σ} \frac{∂}{∂q^{σ}} − B^{σρ}A_{ρ} \frac{∂}{∂q̇^{σ}} \} ,
Δ_{α}^{0} = span{A_{σ}dt + B_{σν}dq̇^{ν} , ω^{σ}, 1 ≤ σ ≤ m} ,

where (B^{σν}) is the inverse matrix to (B_{σν}) (i.e., Δ_{α} is a semispray distribution).

(5) The equations for paths of E have an equivalent form

q̈^{σ} = −B^{σρ}A_{ρ} along J^{2}γ, 1 ≤ σ ≤ m .

For every regular mechanical system [α] the dynamical distribution Δ_{α} is the characteristic distribution of any of 2-forms α ∈ [α]. This means that for Γ ∈ Δ_{α} and all α ∈ [α],

i_{Γ}α = 0 .

Another particular but highly important case concerns mechanical systems arising from Lagrangians.

Theorem 2 [7]. Suppose that the dynamical form E is locally variational, i.e., that in a neighborhood of every point of J^{1}Y there exists a Lagrange function L(t, q^{σ}, q̇^{σ}) such that

E_{σ} = \frac{∂L}{∂q^{σ}} − \frac{d}{dt} \frac{∂L}{∂q̇^{σ}} .

Then the equivalence class [α] of Lepagean 2-forms of E contains a unique closed 2-form.

Conversely, if the equivalence class [α] of Lepagean 2-forms of E contains a closed 2-form then this is unique, and E is locally variational.

The unique closed representative of a locally variational form E is called the Lepagean equivalent of E, and the corresponding mechanical system is called a Lagrangian system (cf. [7] for more details).

3. The constraint structure

In what follows, we denote by Q a fibered submanifold π_{1,0}|_{Q} : Q → Y of the fibered manifold π_{1,0} : J^{1}Y → Y of codimension k, where 1 ≤ k ≤ m − 1. Locally, Q is defined by the equations

(3.1)  f^{i} = 0, 1 ≤ i ≤ k ,

where

(3.2)  rank(\frac{∂f^{i}}{∂q̇^{σ}} ) = k .

Q is called a constraint submanifold, and the equations (3.1) are called a a system of k (local) non-holonomic constraints.

Let U be a fiber chart at a point x ∈ Q with fiber coordinates (t, q^{σ}, q̇^{σ}). The equations (3.1) can be “locally solved” with respect to k of the functions q̇^{σ}. Without loss of generality one can suppose that it is possible to express the q̇^{m−k+1}, …, q̇^{m} as functions of the (t, q^{σ}, q̇^{1}, …, q̇^{m−k}). This means that we have an adapted chart (U, χ) to the submanifold Q at x of the form (t, q^{σ}, q̇^{1}, …, q̇^{m−k}, f^{1}, …, f^{k}). Such coordinates will be called adapted fiber coordinates. In particular, adapted fiber coordinates such that Q is described by the equations

(3.3)  q̇^{m−k+i} − g^{i}(t, q^{σ}, q̇^{1}, …, q̇^{m−k}) = 0

are called normal coordinates.

A section γ of π defined on an open set I ⊂ X is called a holonomic path in Q if for every x ∈ I, J^{1}γ(x) ∈ Q.

Consider a cover of Q by an atlas A of adapted fiber charts. For every (U, χ) ∈ A put

C_{U,φ}^{0} = span{φ^{i}, df^{i}, 1 ≤ i ≤ k} ,

where φ^{i} are linearly independent 1-forms such that

(3.4)  hφ^{i} = f^{i} dt, 1 ≤ i ≤ k .

Then C_{U,φ} is a distribution of rank 2m + 1 − 2k on U, in general not completely integrable. At each point x ∈ Q ∩ U it is a subdistribution of the tangent distribution to Q ∩ U (generated by means of the 1-forms df^{i}, 1 ≤ i ≤ k). Therefore, C_{U,φ} along the submanifold Q ∩ U is a distribution of corank k on Q ∩ U.

Holonomic integral sections of the distribution C_{U,φ} obviously coincide with the holonomic paths in Q ∩ U. Indeed, they are solutions of the equations J^{1}γ^{∗}φ^{i} = J^{1}γ^{∗}hφ^{i} = 0, J^{1}γ^{∗}df^{i} = 0, 1 ≤ i ≤ k, i.e., of the “constraint equations” f^{i} ∘J^{1}γ = 0 and their prolongations (df^{i}/dt) ∘ J^{2}γ = 0.

The above construction provides us, for a fixed adapted fiber chart (U, χ), with a family of distributions C_{U,φ} with the same holonomic integral sections. Indeed, the condition (3.4) means that the contact parts of the φ^{i}’s remain undetermined (unless all the forms φ^{i} are π_{1,0}-projectable). Namely, we have

(3.5)  φ^{i} = f^{i} dt + φ_{σ}^{i} ω^{σ} ,

where φ_{σ}^{i} are arbitrary functions on U such that the matrix (f^{i}, φ_{σ}^{i}) with k rows and m+1 columns has rank k on U. However, there is a natural choice for the φ^{i}’s, picking up a unique distribution from the family C_{U,φ}, giving a geometric sense to the physical concept of “possible generalized displacements”. To this end denote by V_{U}(π) (resp. V_{U}(π_{1,0})) the module of π-vertical vector fields on π_{1,0}(U) (resp. of π_{1,0}-vertical vector fields over the ring of π_{1,0}-projectable functions on U).

Definition. 1-forms φ^{i}, 1 ≤ i ≤ k, on U are called constraint 1-forms if

hφ^{i} = f^{i} dt, 1 ≤ i ≤ k ,

and for 1 ≤ i ≤ k, and every π-vertical vector field ξ on π_{1,0}(U),

(3.6)  i_{J^{1}ξ}φ^{i} = i_{F (ξ)}df^{i} ,

where F (ξ) is the image of ξ under the canonical morphism V_{U}(π) → V_{U}(π_{1,0}), defined by F (∂/∂q^{σ}) = ∂/∂q̇^{σ}, F (gξ) = (g ∘ π_{1,0})F (ξ).

The condition (3.6) gives a unique solution

(3.7)  φ_{σ}^{i} = \frac{∂f^{i}}{∂q̇^{σ}} , 1 ≤ i ≤ k

– the Chetaev expressions (1.3). Consequently, by (3.2), the obtained 1-forms are linearly independent. The corresponding distribution C_{U,φ} on U is called the constraint distribution, and denoted by C_{U}. It has the meaning of “possible generalized displacements”, its π_{1}-vertical subdistribution represents “virtual generalized displacements”, and its π_{1,0}-vertical subdistribution could be called a distribution of “virtual velocities”.

Summarizing the results, in a neighborhood of the constraint submanifold Q there exists a canonical, unique system of local constraint distributions, subordinate to a cover of Q by adapted fiber charts. The holonomic paths in Q (piecewise) coincide with holonomic integral sections of these distributions.

It is important to notice that if in a neighborhood U of x ∈ Q, f̄^{i} = 0 are other equations of Q, we get a different constraint distribution C̄_{U} on U. However, the holonomic integral sections of the distributions C_{U} and C̄_{U} coincide. In this sense, the description of holonomic paths in Q by means of a system of local constraint distributions does not depend on a choice of an atlas of adapted fiber charts to Q.

In what follows we denote by I(φ^{i}) the ideal generated by the constraint 1-forms φ^{i}, 1 ≤ i ≤ k, on U.

Physical constraint structure. A dynamical form Φ_{U} on U will be called a local constraint force or Chetaev force if Φ_{U} ∈ I(φ^{i}). We have

(3.8)  Φ_{U} = φ^{i} ∧ μ_{i} = μ_{0i} \frac{∂f^{i}}{∂q̇^{σ}} dq^{σ} ∧ dt .

The functions μ_{0i} on U are called Lagrange multipliers. The Chetaev force satisfies the Principle of virtual work

i_{ξ}Φ_{U} = 0

for every π_{1}-vertical vector field belonging to the constraint distribution C_{U}.

It can be shown using partition of unity techniques that local constraint forces define a constraint force Φ in a neighborhood of the constraint submanifold Q. This constraint force depends on the choice of an atlas A of adapted fiber charts. However, this dependence is unessential, since it is realized via Lagrange multipliers (see [8] for details).

A pair (Q, Φ) where Q ⊂ J^{1}Y is a constraint submanifold and Φ is a constraint force related to Q will be called a physical constraint structure on J^{1}Y.

Geometrical constraint structure. Consider on Q an atlas A of adapted fiber charts, and the subordinate system of local constraint distributions; recall that for a chart (U, χ) ∈ A we have

(3.9)  C_{U}^{0} = span\{φ^{i}, df^{i}, 1 ≤ i ≤ k\}, φ^{i} = f^{i}dt + \frac{∂f^{i}}{∂q̇^{σ}} ω^{σ} .

Denote ι : Q → J^{1}Y the canonical embedding.

Theorem 3 [8]. For each (U, χ) ∈ A consider the distribution C_{U} restricted to Q ∩ U, i.e., the distribution on Q ∩ U whose annihilator is spanned by the 1forms ι^{∗}φ^{i}, 1 ≤ i ≤ k. If φ̄^{i}, 1 ≤ i ≤ k, are the constraint 1-forms related to a chart (Ū, χ̄), Ū ∩ U ≠ ∅, then there is a regular matrix (a_{j}^{i}) on Ū ∩ U such that ι^{∗}φ̄^{i} = a_{j}^{i} ι^{∗}φ^{j}.

By the above theorem, putting

(3.10)  C^{0} = span{ι^{∗}φ^{i}, 1 ≤ i ≤ k} ,

where φ^{i} runs over all constraint 1-forms subordinate to the cover A of Q by adapted fiber charts, we get a distribution on Q of corank k (with respect to Q). This distribution does not depend on the choice of the atlas A. It is denoted by C and called the canonical distribution [8] or Chetaev bundle [15] of the constraint Q. Obviously, the holonomic integral sections of C coincide with the holonomic paths in Q.

Notice that the holonomic paths in Q are integral sections of (local) vector fields belonging to C which in normal coordinates, i.e., where Q is locally described by the equations q̇^{m−k+i} − g^{i}(t, q^{σ}, q̇^{1}, …, q̇^{m−k}) = 0, are given by

(3.11)  Γ_{Q} = \frac{∂}{∂t} + ∑_{l=1}^{m−k} q̇^{l} \frac{∂}{∂q^{l}} + ∑_{i=1}^{k} g^{i} \frac{∂}{∂q^{m−k+i}} + ∑_{l=1}^{m−k} Γ^{l} \frac{∂}{∂q̇^{l}} ,

where Γ^{l} are some functions. Such vector fields are called constraint semisprays.

A pair (Q, C) where Q ⊂ J^{1}Y is a constraint submanifold and C is the canonical distribution on Q will be called a geometrical constraint structure on J^{1}Y. The ideal on Q generated by the 1-forms annihilating the canonical distribution will be called the constraint ideal and will be denoted by I(C^{0}).

4. Geometrical models for classical constraints

Let us show relations of the above model of constraints with some frequently used geometric representations of classical (i.e., holonomic and linear non-holono-mic) constraints.

Submanifold of Y. Constraints which are realized as a fibered submanifold Q_{0} of Y of codimension k < m are called holonomic. Locally they are given by equations u^{i}(t, q^{σ}) = 0, 1 ≤ i ≤ k, such that rank(∂u^{i}/∂q^{σ}) = k. Within our scheme this means that f^{i} = du^{i}/dt, and

φ^{i} = \frac{du^{i}}{dt} dt + \frac{∂u^{i}}{∂q^{σ}} ω^{σ} = π_{1,0}^{∗}du^{i} ,

i.e., all the constraint 1-forms are π_{1,0}-projectable and closed. Consequently, I(C^{0}) is a differential ideal, the canonical distribution C is completely integrable, and along J^{1}Q_{0} it is nothing but the tangent distribution T J^{1}Q_{0}. Its projection is the tangent distribution to Q_{0}, giving sense to the concepts of degrees of freedom, possible displacements, and virtual displacements defined to be the dimension of the manifold Q_{0}, the tangent distribution to Q_{0}, and its π-vertical subdistribution, respectively. The geometrical and the physical constraint structure for holonomic systems becomes (J^{1}Q_{0}, T J^{1}Q_{0}), and (J^{1}Q_{0}, Φ), respectively, where

(4.1)  Φ = φ^{i} ∧ μ_{i} = μ_{0i} \frac{∂u^{i}}{∂q^{σ}} ω^{σ} ∧ dt .

Completely integrable distribution on Y. Consider a completely integrable distribution D of corank k < m on Y. This means that in a neighborhood of every point in Y there are k linearly independent 1-forms du^{i} spanning the annihilator D^{0}. Such a distribution represents constraints called semiholonomic or linear integrable (referring to the equations of the constraints which are affine in velocities).

In our notations, f^{i} = du^{i}/dt, φ^{i} = π_{1,0}^{∗}du^{i}. Thus, the constraint 1-forms are π_{1,0}-projectable and closed, i.e., I(C^{0}) is a differential ideal, and the corresponding constraint distribution is completely integrable and projects onto D. The geometrical constraint structure on J^{1}Y is (Q, C), where Q is locally defined by

\frac{du^{i}}{dt} = \frac{∂u^{i}}{∂t} + \frac{∂u^{i}}{∂q^{σ}} q̇^{σ} = 0, 1 ≤ i ≤ k ,

and C^{0} = span{ι^{∗}π_{1,0}^{∗}du^{i}, 1 ≤ i ≤ k}. The physical constraint structure is (Q, Φ), where Φ is given by (4.1).

Distribution on Y. In this case the constraints are given by a distribution D of corank k < m on Y which is not supposed to be completely integrable. They are called simple non-holonomic [8] or linear nonintegrable.

Denote D^{0} = span{η^{i} = a^{i} dt + b_{σ}^{i}dq^{σ}, 1 ≤ i ≤ k}. Then in our notations,

φ^{i} = π_{1,0}^{∗}η^{i} = (a^{i} + b_{σ}^{i}q̇^{σ})dt + b_{σ}^{i}ω^{σ} , f^{i} = a^{i} + b_{σ}^{i}q̇^{σ} ,

i.e., the constraint 1-forms are π_{1,0}-projectable (the corresponding constraint distribution projects onto D). The geometrical constraint structure on J^{1}Y is (Q, C), where Q is locally defined by

a^{i} + b_{σ}^{i}q̇^{σ} = 0, 1 ≤ i ≤ k ,

and C^{0} = span{ι^{∗}π_{1,0}^{∗}η^{i}, 1 ≤ i ≤ k}. The physical constraint structure is (Q, Φ), where

Φ = φ^{i} ∧ μ_{i} = μ_{0i} b_{σ}^{i} ω^{σ} ∧ dt .

5. Constrained systems

The point of view of physics: systems deformed by constraints. Consider on J^{1}Y a constraint structure (Q, Φ), where Q is a (fibered over Y) submanifold of codimension k < m and Φ is a corresponding constraint force defined in a neighborhood of Q. Let [α] be a mechanical system on J^{1}Y, E its dynamical form. Putting

α_{Φ} = α − Φ ,

we get a new mechanical system [α_{Φ}] defined in a neighborhood of Q, called a deformation of [α] by Φ [8]. If E = (A_{σ} + B_{σν}q̈^{ν})dq^{σ} ∧ dt, the f^{i} = 0 are local equations of the submanifold Q, and Φ is a Chetaev force (3.8) then

α_{Φ} = (A_{σ} − μ_{i0} \frac{∂f^{i}}{∂q̇^{σ}} )ω^{σ} ∧ dt + F_{σν}ω^{σ} ∧ ω^{ν} + B_{σν}ω^{σ} ∧ dq̇^{ν} .

The deformed mechanical system is regular if and only if the original mechanical system is regular.

Deformed dynamical distributions are defined by means of the 1-forms

(A_{σ} − μ_{i0} \frac{∂f^{i}}{∂q̇^{σ}} )dt + 2F_{σν}ω^{ν} + B_{σν}dq̇^{ν} , B_{σν}ω^{ν} .

The deformed equations of motion are J^{1}γ^{∗}i_{ξ}α_{Φ} = 0 for every π_{1}-vertical vector field ξ on J^{1}Y. In fiber coordinates,

f^{i} ∘ J^{1}γ = 0, A_{σ} + B_{σν}q̈^{ν} = μ_{i0} \frac{∂f^{i}}{∂q̇^{σ}} along J^{2}γ .

The point of view of geometry: constrained systems as mechanical systems on constraint submanifolds. With help of the canonical distribution C one can obtain an intrinsic description of mechanical systems constrained to a submanifold Q of J^{1}Y.

Let [α] be a mechanical system, and let (Q, C) be a constraint structure on J^{1}Y. Denote by ι the embedding of Q into J^{1}Y, and by I(C^{0}) the constraint ideal on Q. In keeping with [8], for α ∈ [α] put

α_{Q} = ι^{∗}α mod I(C^{0}) .

Denote by [α_{Q}] the mechanical system generated by the class α_{Q}. It is easy to see that if α_{1}, α_{2} ∈ [α] then [(α_{1})_{Q}] = [(α_{2})_{Q}]. The class [α_{Q}] is called the constrained system related to the mechanical system [α] and the constraint structure (Q, C).

Computing the coordinate expression in normal coordinates we get a representative

α_{Q} = ∑_{l=1}^{m−k} A′_{l}ω^{l} ∧ dt + ∑_{l,s=1}^{m−k} B′_{ls}ω^{l} ∧ dq̇^{s} ,

where

A′_{l} = (A_{l} + ∑_{p=1}^{k} A_{m−k+p} \frac{∂g^{p}}{∂q̇^{l}}

+ ∑_{s=1}^{k}(B_{l,m−k+s} + ∑_{j=1}^{k} B_{m−k+j,m−k+s} \frac{∂g^{j}}{∂q̇^{l}} )(\frac{∂g^{s}}{∂t} + \frac{∂g^{s}}{∂q^{σ}} q̇^{σ})) ∘ ι ,

B′_{ls} = (B_{ls} + ∑_{r=1}^{k}(B_{l,m−k+r} \frac{∂g^{r}}{∂q̇^{s}} + B_{m−k+r,s} \frac{∂g^{r}}{∂q̇^{l}} )

+ ∑_{r,j=1}^{k} B_{m−k+j,m−k+r} \frac{∂g^{j}}{∂q̇^{l}} \frac{∂g^{r}}{∂q̇^{s}} ) ∘ ι

(see also [8]). Now, the equations of motion have the following form:

Proposition 2. A section γ of π is a path of the constrained system [α_{Q}] if and only if J^{1}γ is an integral section of the canonical distribution C, and for every π_{1}-vertical vector field ξ ∈ C it satisfies the equation

J^{1}γ^{∗}i_{ξ}α_{Q} = 0 ,

where α_{Q} is (any) 2-form belonging to [α_{Q}].

In fiber coordinates this gives the following system of m − k second order ODE, and k first order ODE for the components γ^{1}, … γ^{m} of γ:

f^{i} ∘ J^{1}γ = 0 , (A′_{l} + ∑_{s=1}^{m−k} B′_{ls}q̈^{s}) ∘ J^{2}γ = 0 .

Now, a constrained dynamical distribution Δ_{α_{Q}} is a subdistribution of the canonical distribution C, generated by means of the 1-forms i_{ξ}α_{Q}, where ξ runs over all π_{1}-vertical vector fields on Q belonging to C.

In analogy with the unconstrained case, the constrained system is called regular if the equivalence class [Δ_{α_{Q}}] contains a representative of rank 1. In this case, by Theorem 1, the class [Δ_{α_{Q}}] contains only one element—a constraint semispray distribution, and regularity is equivalent with the condition that the (m − k) × (m − k)-matrix (B′_{sl}) is regular. Denote by (B^{′sl}) the inverse matrix. Apparently, Δ_{α_{Q}} = span Γ_{Q}, where Γ_{Q} is given by (3.11) with Γ^{l} = B^{′ls}A′_{s}, and we can see that

i_{Γ_{Q}}α_{Q} ∈ C^{0} .

Notice that a constrained system of a regular mechanical system need not be regular.

Let [α] be a Lagrangian system. By Theorem 2 this means that there exists a (unique) 2-form α_{E} ∈ [α] such that dα_{E} = 0. If λ is a Lagrangian and θ_{λ} its Cartan form, we have (at least locally) α_{E} = dθ_{λ}. Unfortunately, a corresponding constraint 2-form (α_{E})_{Q} = (dθ_{λ})_{Q} need not be closed. However, by Theorem 2, we have

Theorem 4. A constrained system [α] is Lagrangian if and only if the class [α_{Q}] contains a closed representative.

Corollary. Let (Q, C) be a non-holonomic constraint structure on J^{1}Y. A constrained system arising from a Lagrangian system on J^{1}Y is Lagrangian if and only if there exists a 2-form β ∈ I(C^{0}) and a 2-contact 2-form F on Q such that d(β + F) = 0.

In particular, if the ideal I(C^{0}) is closed then every constrained system arising from a Lagrangian system on J^{1}Y is Lagrangian.

Thus, trivially, Lagrangian systems subject to holonomic or semiholonomic (=affine integrable non-holonomic) constraints remain Lagrangian.

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Silesian University at Opava, Department of Mathematics Bezručovo nám. 13, 746 01 Opava, CZECH REPUBLIC E-mail: Olga.Krupkova@fpf.slu.cz