DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 539–548

ALMOST PRODUCT STRUCTURES IN MECHANICS

MANUEL DE LEÓN AND DAVID MARTÍN DE DIEGO

Abstract. There are two different meanings for constrained systems in Lagrangian mechanics: “internal” constraints imposed by the singularity of the Lagrangian function, or “external”constraints imposed by forces of constraint acting on the system. In the first case, it is usual to apply the Gotay-Nester algorithm in the framework of presymplectic geometry and, in the second case, the classical technique of Lagrange multipliers. The aim of this paper it is to give a new look at singular and non-holonomic Lagrangian systems in the framework of almost product structures.

1. Introduction

We use almost product structures to study, at a first glance, different dynamical systems: degenerate and non-holonomic Lagrangian systems. Roughly speaking, an almost product structure on a manifold M consists of two complementary distributions on M. Therefore if L : T Q ⟶ ℝ is a singular Lagrangian function, it is very natural to take an almost product structure on T Q such that one of the two complementary distributions is just the singular distribution ker ω_{L}. The dynamics may be obtained by projecting the system on the complementary distribution. The use of almost product structures in order to obtain the “true” dynamics of a singular system was proposed in several recent papers (see de León, Martín de Diego & Pitanga [8], de León & Martín de Diego [11], Pitanga [15], Dubrovin, Giordano, Marmo & Simone [5] and references therein).

Given a Lagrangian system with non-holonomic constraints we construct an almost product structure on the tangent bundle of the configuration manifold such that the projection of the Euler-Lagrange vector field for the free Lagrangian system gives the dynamics of the system. We go further in the direction of the geometric formulation of non-holonomic Lagrangian systems started by Cariñena & Rañada [3], Rañada [16], Sarlet, Cantrijn & Saunders [17], de León & Martín de Diego[9].

By using both constructions, we study the general case of degenerate Lagrangian systems with non-holonomic constraints. In this general setting, several almost product structures arise.

1991 Mathematics Subject Classification. 58F05, 70H33, 70H35.

Key words and phrases. Presymplectic manifolds, non-holonomic constraints.

This research was supported in part by DGICYT (Spain), Project PB94-0106.

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

2. Almost product structures

An almost product structure on a manifold M is a tensor field F of type (1, 1) on M such that F^{2} = id (see [13]). If we set

A = \frac{1}{2}(id + F) , B = \frac{1}{2}(id − F) ,

then A and B are complementary projectors. We denote by ImA and ImB the corresponding complementary distributions. We also denote by A^{∗} and B^{∗} the transpose operators and ImA^{∗} and ImB^{∗} will be their corresponding images.

Let (M, ω) be a presymplectic manifold with a presymplectic form ω. An almost product structure F on M is said to be adapted to ω if ker ω = ker A. Define the linear mapping ♭ : 𝔛(M) ⟶ ∧^{1}(M) by ♭(X) = i_{X}ω. If F is adapted to ω, the restriction of ♭ to the distribution A induces an isomorphism of C^{∞}-modules ♭ : ImA ⟶ ImA^{∗}. Then, for a function f on M, there exists a unique vector field X_{f,A} ∈ ImA such that

i_{X_{f,A}}ω = A^{∗}(df) . (1)

A bracket of functions can be defined as follows:

{f, g}_{A} = ω(X_{f,A}, X_{g,A}) , f, g ∈ C^{∞}(M) .

{ , }_{A} satisfies all the properties of a Poisson bracket except the Jacobi identity. The Jacobi identity is equivalent to the integrability of the almost product structure F (see Dubrovin et al [5]). Let us recall that an almost product structure F is said to be integrable if both distributions ImA and ImB are integrable. In our case, the distribution ImB = ker ω is always integrable, but ImA is not necessarily so.

3. Lagrangian systems

Let Q be an n-dimensional manifold, T Q its tangent bundle, and τ_{Q} : T Q ⟶ Q the canonical projection. Denote by {q^{A}, q̇^{A}; 1 ≤ A ≤ n} the fibered coordinates on T Q. Let J be the canonical almost tangent structure and C the Liouville vector field on T Q.

Consider a Lagrangian function L on T Q. Let E_{L} = CL − L be the energy associated with L. We denote by α_{L} = J^{∗}(dL) the Poincaré-Cartan 1-form and, by ω_{L} = −dα_{L} the Poincar�-Cartan 2-form. A Lagrangian L is said to be regular if its Hessian matrix

(W_{AB}) = (\frac{∂^{2}L}{∂q̇^{A}∂q̇^{B}} ) (2)

is regular. If L is regular, ω_{L} is symplectic and, in this case, the motion equation

i_{X}ω_{L} = dE_{L} (3) has a unique solution ξ_{L} (the Euler-Lagrange vector field). Moreover, ξ_{L} turns out to be a second order differential equation (hereafter shortened as SODE), that is, ξ_{L} verifies that Jξ_{L} = C. The induced Poisson bracket on C^{∞}(T Q) is defined by

{f, g}_{L} = ω_{L}(X_{f}, X_{g}) , ∀f, g ∈ C^{∞}(T Q) ,

where X_{f} denotes the Hamiltonian vector field with Hamiltonian energy f, that is, i_{X_{f}} ω_{L} = df. Hence ξ_{L} = X_{E_{L}} and {f, E_{L}}_{L} = ξ_{L}(f).

If L : T Q ⟶ ℝ is singular, i.e., the Hessian matrix (2) is not regular, Equation (3) has no solution, in general, and even if it exists it will not be unique nor a SODE. We suppose, for the sake of simplicity, that the system admits a global dynamics, that is, a solution ξ on T Q of Equation (3) exists. Then, if Z ∈ ker ω_{L}, ξ + Z is also a solution of Equation (3).

Let T^{∗}Q be the cotangent bundle of Q with canonical projection π : T^{∗}Q ⟶ Q. We will denote by Leg : T Q ⟶ T^{∗}Q the Legendre map defined by L

Leg : (q^{A}, q̇^{A}) ⟶ (q^{A}, p̃_{A}) ,

with p̃_{A} = ∂L/∂q̇^{A}.

L is regular iff Leg is a local diffeomorphism. For the sake of simplicity, we suppose that L is hyper-regular, that is, Leg is a global diffeomorphism. If λ_{Q} is the Liouville 1-form and ω_{Q} = −dλ_{Q} is the canonical symplectic form on T^{∗}Q, we have that

Leg^{∗}ω_{Q} = ω_{L} , Leg^{∗}λ_{Q} = α_{L} .

We define the Hamiltonian function H on T^{∗}Q by H = E_{L} ∘ Leg^{−1}. If X_{H} denotes the Hamiltonian vector field of H we know that ξ_{L} and X_{H} are Leg-related. Then, both dynamics are equivalent.

Suppose that L is singular but almost regular, i.e., M_{1} = Leg(T Q) is a submanifold of T^{∗}Q and Leg is a submersion onto M_{1} with connected fibers. We denote by Leg_{1} : T Q ⟶ M_{1} the restriction of Leg : T Q ⟶ T^{∗}Q to its image. The submanifold M_{1} will be called the primary constraint submanifold. The energy E_{L} is constant along the fibers of Leg and, therefore, E_{L} projects onto a function h_{1} on M_{1}, i.e., h_{1}(Leg(x)) = E_{L}(x), ∀x ∈ T Q. Let {F, G} = ω_{Q}(X_{F} , X_{G}) , ∀F, G ∈ C^{∞}(T^{∗}Q) be the Poisson bracket on T^{∗}Q induced by ω_{Q}. If we denote by i : M_{1} ⟶ T^{∗}Q the natural embedding of M_{1} into T^{∗}Q, then we obtain a presymplectic system (M_{1}, ω_{1}, dh_{1}), where ω_{1} = i^{∗}ω_{Q}.

4. Singular Lagrangian systems and almost product structures

Suppose that (T Q, ω_{L}, dE_{L}) admits a global dynamics and take an almost product structure (A, B) adapted to ω_{L}. Then, we can fix a unique solution of the dynamics by taking the vector field ξ given by

i_{ξ}ω_{L} = dE_{L} and ξ ∈ ImA .

The presymplectic system (M_{1}, ω_{1}, dh_{1}) also admits a global dynamics ([6]), that is, there exists a vector field X on M_{1} such that

i_{X}ω_{1} = dh_{1}, (4) and each vector field X + Z with Z ∈ ker ω_{1} is also a solution of this equation. If (A, B) is projectable onto an almost product structure on M_{1} then ξ is projectable onto a vector field X which is a solution of Equation (4).

Since M_{1} is a submanifold of T^{∗}Q, there appear n − k independent constraints, the primary constraints, ϕ^{α} which describe M_{1} (see [4]). If H is an arbitrary extension of h_{1} to T^{∗}Q, all the Hamiltonian functions of the form

H̃ = H + λ_{α}ϕ^{α} , (5)

where λ_{α} are Lagrange multipliers, are weakly equal, that is, H̃_{/_{M_{1}}} = H_{/_{M_{1}}} = h_{1}. This shows that there exists an ambiguity in the description of the dynamics. Since ω_{Q} is symplectic, a solution of the equation i_{X}ω_{Q} = dH always exists. The constraints must be preserved in the time or, equivalently, the solution X must be tangent to M_{1}. Then, we get

({ϕ^{β}, H̃} + λ_{α}{ϕ^{β}, ϕ^{α}})_{/_{M_{1}}} = 0 . (6)

The vanishing of the expressions (6) implies that some of the arbitrary functions λ_{α} may be determined and the rest remains completely undetermined.

Dirac gave a classification of the constraints generated by this algorithm in order to clarify the ambiguity of the dynamics ([4, 2, 18]). A constraint ϕ of M_{1} is said to be first class if {ϕ, ϕ^{α}}_{/_{M_{1}}} = 0 for each constraint ϕ^{α} of M_{1}, and second class otherwise. Then, the coefficients of the primary first class constraints on M_{1} in (5) are completely undetermined, while the coefficients of the primary second class constraints are completely fixed.

We will distinguish three particular cases: all the primary constraints are second class, all them are first class and there exist first and second class constraints.

4.1. All the primary constraints are second class

Denote by Φ^{α}, 1 ≤ α ≤ s, the constraints of M_{1}. The matrix with entries C^{αβ} = {Φ^{α}, Φ^{β}} is non-singular on M_{1} and, in the sequel, we assume for simplicity that this matrix is non-singular in the entire phase space T^{∗}Q. This matrix is skewsymmetric and, then, the number of second class constraints is even. Denote by (C_{αβ}) its inverse matrix.

Let D be the smooth distribution generated by the vector fields X_{Φ^{α}}. A direct computation shows that

D^{⊥}(x) = {v ∈ T_{x}T^{∗}Q / ω_{Q}(x)(v, w) = 0 , ∀w ∈ D(x)} = T_{x}M_{1} , ∀x ∈ M_{1} .

Let B̃ : D ⊕ D^{⊥} ⟶ D be the projection onto D along D^{⊥} and Ã = id − B̃. The projector B̃ is given by

B̃ = C_{αβ}X_{Φ^{α}} ⊗ dΦ^{β} .

Take the 2-form Ω_{D} = Ã^{∗}ω_{Q} (that is, Ã^{∗}ω_{Q}(X, Y) = ω_{Q}(ÃX, ÃY)). Ω_{D} is an almost presymplectic form with constant rank 2n − s and, moreover, the almost product structure (Ã, B̃) is adapted to Ω_{D}. Notice that, if (Ã, B̃) is integrable, then Ω_{D} is closed and, hence, presymplectic. Thus, we can define a bracket {F, G}_{D}, called Dirac bracket, on T^{∗}Q as follows:

{F, G}_{D} = Ω_{D}(X_{F} , X_{G}) = {F, G} − {F, Φ^{α}}C_{αβ}{Φ^{β}, G} .

Consider now the projected Hamiltonian function h_{1} : M_{1} ⟶ ℝ. We can extend h_{1} to a function H on T^{∗}Q and consider the Hamiltonian functions of the form:

H̃ = H + λ_{α}Φ^{α} .

Take the Hamiltonian vector field XH̃. The consistency of the theory demands that the constraints Φ_{α} be preserved by XH̃; geometrically this means that the vector field XH̃ must be tangent to M_{1}.

The vector field

ÃX_{H} = X_{H} − C_{αβ}{Φ^{β}, H}X_{Φ^{α}} .

is tangent to M_{1} and its restriction (ÃX_{H})_{/_{M_{1}}} is the unique solution of the motion equations, that is,

i_{(_{A}X_{H})_{/_{M_{1}}}} ω_{1} = dh_{1} ,

because ω_{1} is symplectic.

4.2. All the primary constraints are first class

We denote by ϕ^{i}, 1 ≤ i ≤ p, the first class constraints. Since {ϕ^{i}, ϕ^{j}}_{/_{M_{1}}} = 0, then X_{ϕ_{i}}, 1 ≤ i ≤ p, the Hamiltonian vector field of ϕ^{i}, is tangent to M_{1}. Notice that the submanifold M_{1} is coisotropic into T^{∗}Q.

Since ker ω_{1} is generated by the restrictions of the Hamiltonian vector fields X_{ϕ^{i}}, in order to fix the gauge, we take an almost product structure (A_{1}, B_{1}) on M_{1} adapted to ker ω_{1}. Moreover, if the almost product structure is integrable, we can define a Poisson bracket on M_{1} as follows:

{f, g}_{A_{1}} = ω_{1}(X_{f,A_{1}}, X_{g,A_{1}}) , ∀f, g ∈ C^{∞}(M_{1}) ,

where X_{f,A_{1}} and X_{g,A_{1}} are the unique vector fields on M_{1} which belong to ImA_{1} and verify that i_{X_{f,A_{1}}} ω_{1} = A_{1}^{∗}df and i_{X_{g,A_{1}}} ω_{1} = A_{1}^{∗}dg, respectively. Therefore, if ξ is a solution of the motion equation, that is, i_{ξ}ω_{1} = dh_{1}, we can select a unique solution A_{1}ξ such that A_{1}ξ ∈ ImA_{1}. Thus, we have fixed the gauge.

Now, consider an arbitrary extension H to T^{∗}Q of the Hamiltonian h_{1} : M_{1} ⟶ ℝ. Since we have a global dynamics, the Hamiltonian vector field X_{H} is tangent to M_{1}, i.e., (X_{H})_{/_{M_{1}}} ∈ 𝔛(M_{1}) and

i_{(X_{H})_{/_{M_{1}}}} ω_{1} = dh_{1} .

We fix the gauge by taking A_{1}((X_{H})_{/_{M_{1}}} ).

4.3. There exist first and second class constraints

We denote by Φ^{α}, 1 ≤ α ≤ s, the second class constraints and by ϕ^{i}, 1 ≤ i ≤ p, the first class constraints. As in the first case, we can construct an almost product structure (Ã, B̃) on T^{∗}Q with B̃ given by

B̃ = C_{αβ}X_{Φ^{α}} ⊗ dΦ^{β} .

Here, (C_{αβ}) is the inverse matrix of ({Φ^{α}, Φ^{β}}). We also have the almost presymplectic form Ω_{D} = Ã^{∗}ω_{Q} with constant rank 2n − s and the Dirac bracket

{F, G}_{D} = {F, G} − {F, Φ^{α}}C_{αβ}{Φ^{β}, G} .

If we consider an arbitrary extension H of the Hamiltonian h_{1}, since we have a global dynamics, then the vector field Ã(X_{H}) is tangent to M_{1}. Now, we fix the gauge taking the vector field A_{1}(Ã(X_{H})_{/_{M_{1}}} ), where (A_{1}, B_{1}) is some almost product structure adapted to ω_{1}.

Remark 4.1. For an arbitrary Lagrangian system (T Q, ω_{L}, dE_{L}), we must apply the Gotay-Nester algorithm in order to obtain a final submanifold where there exists consistent solutions of the dynamics. From the equivalence theorem ([7]), this is equivalent to apply the algorithm to the presymplectic system (M_{1}, ω_{1}, dh_{1}) and we also have a final constraint submanifold M_{f} of M_{1}. In a similar way, we can obtain a classification of the constraints of M_{f} and then construct suitable almost product structures that guarantees the consistency and unicity of the dynamics.

Remark 4.2. If the almost product structure (A, B) on T Q is projectable on T Q and ξ is not a SODE then it is possible to construct from the projection ξ̃ a submanifold S of T Q where there exists a solution ξ_{S} of the dynamics which verifies the SODE condition, that is, J(ξ_{S})(x) = C(x) , ∀x ∈ S. The submanifold S is diffeomorphic to M_{1} and the dynamics on both sides are equivalent.

5. Non-holonomic regular Lagrangian systems and almost product structures

We assume that a regular lagrangian L is subjected to a system of m independent non-holonomic constraints {ϕ_{i}; 1 ≤ i ≤ m, m ≤ n} which are affine in the velocities. This means that

ϕ_{i} = μ̂_{i} + h_{i}^{V} ,

where μ_{i} = (μ_{i})_{A}(q)dq^{A} is a 1-form on Q and h_{i}^{V} = h_{i} ∘ τ_{Q} with h_{i} ∈ C^{∞}(Q). Here μ̂ : T Q ⟶ ℝ denotes the function defined by

μ̂(X_{q}) = ⟨μ(q), X_{q}⟩ , ∀X_{q} ∈ T_{q}Q .

We have to restrict the dynamics to the submanifold P̄_{1} of T Q defined by the vanishing of the functions ϕ_{i}. The dynamics of the constrained system is obtained solving the following system of equations:

\{ \begin{matrix}i_{X}ω_{L} & = & dE_{L} + λ^{i}μ_{i}^{V} , \\ dϕ_{i}(X) & = & 0 , \end{matrix} (7)

where μ_{i}^{V} = τ_{Q}^{∗}μ_{i}. The functions λ^{i} are Lagrange multipliers.

For any m-tuple λ = (λ^{1}, …, λ^{m}), we have the vector field Y_{λ} given by

Y_{λ} = ξ_{L} + λ^{i}Z_{i} ,

where Z_{i} are the vertical vector fields (see [3]) defined by

i_{Z_{i}}ω_{L} = μ_{i}^{V} .

Notice that Y_{λ} verifies the first equation of the system (7). If we demand that Y_{λ} satisfies the second condition, we obtain that

0 = ({ϕ_{j}, E_{L}}_{L} + λ^{i}Z_{i}(ϕ_{j}))_{/P̄_{1}} .

Denote by C the matrix on T Q of order m whose entries are C_{ij} = Z_{i}(ϕ_{j}). If C is regular, then the Lagrange multipliers λ^{i} are uniquely determined on T Q. In fact, assume that the matrix C is regular on T Q. Consider the (1,1) tensor field Q given by

Q = C^{ij}Z_{j} ⊗ dϕ_{i} ,

where C^{ij} are the entries of the inverse matrix of C, that is, C^{ij}C_{jk} = δ_{k}^{i}. A direct computation shows that Q^{2} = Q. If we set P = id − Q, then (P, Q) is an almost product structure on the phase space T Q. Moreover, we have that the vector field ((P(ξ_{L}))_{/P̄_{1}} is the unique solution of equations (7). In fact,

P(ξ_{L}) = ξ_{L} − C^{ij}ξ_{L}(ϕ_{i})Z_{j} ,

which implies that (P(ξ_{L}))_{/P̄_{1}} verifies the equation (7) for λ^{i} = − (C^{ji}ξ_{L}(ϕ_{j}))_{/P̄_{1}} .

Remark 5.1. Suppose that a Lagrangian system is subjected to holonomic constraints h_{i} ∈ C^{∞}(Q), (1 ≤ i ≤ m), with m < n. That is, the velocities do not enter into the constraint equations. In geometrical terms the motion equations are

\{ \begin{matrix}i_{X}ω_{L} & = & dE_{L} + λ^{i}dh_{i}^{V} , \\ dh_{i}^{c}(X) & = & 0 , \\ dh_{i}^{V} (X) & = & 0 . \end{matrix} (8)

Since dh_{i}^{V} (X) = h_{i}^{c} because X is a SODE, we can first study the system as a Lagrangian system subjected to the non-holonomic constrains h_{i}^{c} and solve the motion equations

\{ \begin{matrix}i_{X}ω_{L} & = & dE_{L} + λ^{i}dh_{i}^{V} , \\ dh_{i}^{c}(X) & = & 0 , \end{matrix} (9)

and later, we impose the constraints h_{i}^{V} = 0. In this manner, a holonomic system is, in some sense, a special case of a non-holonomic system.

If we suppose that the Lagrangian is hyper-regular, that is the Legendre map is a diffeomorphism, the study of the Hamiltonian formalism is straightforward, because we can transport all the above constructions to the Hamiltonian side.

6. Non-holonomic degenerate Lagrangian systems

Consider now a singular Lagrangian system subjected to a set of non-holonomic constraints {ϕ_{i}}, (1 ≤ i ≤ m) as in the previous section.

Let (T Q, ω_{L}, dE_{L}) be a Lagrangian system which admits a global dynamics, i.e., ker ω_{L}(E_{L}) = 0. Assume that the constraints are Leg-projectable, i.e., V (ker ω_{L})(ϕ_{i}) = 0. As a consequence, we obtain that

μ_{i}^{V} (ker ω_{L}) = 0 , (1 ≤ i ≤ m) .

We now suppose that there exists an almost product structure (A, B) adapted to the presymplectic form ω_{L} (ker ω_{L} = ker A). Then, we can fix a unique solution ξ of the free Lagrangian system as follows:

i_{ξ}ω_{L} = dE_{L} and ξ ∈ ImA .

Also, for each i, 1 ≤ i ≤ m, we have a unique vector field Z_{i} such that

i_{Z_{i}}ω_{L} = μ_{i}^{V} and Z_{i} ∈ ImA .

As in Section 5, if the matrix C = (C_{ij}) = (Z_{i}(ϕ_{j})), (1 ≤ i, j ≤ m) is regular we can construct an almost product structure (P, Q) on T Q as follows:

Q = C^{ij}Z_{j} ⊗ dϕ_{i} and P = id − Q .

Let P̄_{1} be the submanifold of T Q defined by the vanishing of all the constraints. We deduce that (P(ξ))_{/P̄_{1}} is a solution of the dynamics of the constrained system.

In order to obtain the Hamiltonian formalism, consider the presymplectic system (M_{1}, ω_{1}, dh_{1}) subjected to the constraints ϕ̄_{i}, (1 ≤ i ≤ m) where ϕ̄_{i} are defined by ϕ̄_{i}(Leg(x)) = ϕ_{i}(x), ∀ x ∈ T Q. Denote by M̄_{1} the submanifold of M_{1} defined by the vanishing of all the constraints ϕ̄_{i}. It is clear that Leg(P̄_{1}) = M̄_{1}.

If the almost product structure (A, B) is projectable onto M_{1} (see [8]) we obtain an almost product structure (A_{1}, B_{1}) on M_{1} adapted to ω_{1}. Moreover, the vector field ξ ∈ ImA solution of the motion equation is projectable on M_{1} onto a vector field ξ̃ solution of the equation

iξ̃ω_{1} = dh_{1} .

From construction the almost product structure (P, Q) is also projectable onto an almost product structure (P̄, Q̄) on M_{1}.

As in Section 4 we can classify the constraints of M_{1} in first class and second class constraints. We consider three different cases:

6.1. All the primary constraints are second class

As in Section 4.1, we can construct an almost product structure (Ã, B̃) on T^{∗}Q. Moreover, we have the almost product structure (P̄, Q̄) on M_{1}. Take now an arbitrary extension H of the Hamiltonian h_{1} on M_{1}. We fix the unique solution (because (M_{1}, ω_{1}) is symplectic) of the constrained dynamics by taking the vector field on M̄_{1}

(P̄(Ã(X_{H})_{/_{M_{1}}} ))_{/M̄_{1}} .

In fact, Ã(X_{H})_{/_{M_{1}}} = ξ̃.

6.2. All the primary constraints are first class

As in Section 4.2, we take the almost product structure (A_{1}, B_{1}) on M_{1} adapted to the presymplectic form ω_{1}. Now, we fix a solution of the constrained dynamics by taking

(P̄(A_{1} (X_{H}_{/_{M_{1}}} )))_{/M̄_{1}} .

We have ξ̃ = A_{1} (X_{H}_{/_{M_{1}}} ).

6.3. There exist first class and second class constraints

In this case, we have three different almost product structures (Ã, B̃) on T^{∗}Q and (A_{1}, B_{1}) and (P̄, Q̄) on M_{1}. A solution of the dynamics is just

(P̄(A_{1}(Ã(X_{H})_{/_{M_{1}}} )))_{/M̄_{1}} .

We have ξ̃ = A_{1}(Ã(X_{H})_{/_{M_{1}}} ).

Remark 6.1. If the vector field P̄_{1}(ξ)_{/P̄_{1}} is not a SODE then we can construct a submanifold S̄ of P̄_{1} and a diffeomorphism α : M̄_{1} ⟶ S̄ such that ξS̄ = T α(P̄(ξ̃)_{/M̄_{1}} ) is a solution of the constrained dynamics which, moreover, verifies the SODE condition.

Remark 6.2. For an arbitrary degenerate Lagrangian system subjected to constraints it is possible to construct an algorithm similar to that constructed by Gotay and Nester (see [10] for details).

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Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 123, 28006 Madrid, SPAIN

E-mail address : mdeleon@pinar1.csic.es

Dep. de Economía Aplicada Cuantitativa Facultad de Ciencias Económicas y Empresariales, UNED 28040 Madrid, SPAIN

E-mail address : ceedd05@cc.csic.es