Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 497–508

Degenerate time-dependent Lagrangians of second order: the fourth order differential equation problem^{∗}

Manuel de León  Juan C. Marrero

Abstract. We develop a constraint algorithm for degenerate time-dependent Lagrangians of second order and solve the so-called fourth order differential equation problem. The study is inspired in the Gotay and Nester constraint algorithm and the notion of intermediate spaces introduced by Gràcia, Pons and Román-Roy. Keywords. Degenerate, time-dependent, higher order Lagrangians, constraint algorithm.

MS classification. 58F05, 53C15, 53C05, 70H35.

1 Introduction

Several physical theories use higher order Lagrangians. A classical example of a second order regular Lagrangian appears in the elastic theory of straight beams [1, 18]. Also, the study of classical and relativistic spinning particles involved second order Lagrangians [37, 38, 9, 2]. Degenerate higher order Lagrangians appear in the study of the Einstein-Hilbert’s action for gravitation (in fact, the Lagrangian of the Einstein theory of relativity does in fact contains second order derivatives of the metric field [10]), the Polyakov’s formulation of the string [36] by adding a term dependent on the extrinsic curvature (see also [31]), the Podolski’s generalized electrodynamics [35, 23] and its generalizations to Yang-Mills theories [11, 12], or the Lagrangian of two-points particles interacting gravitationally [39] (see also [19]). Also, the Hamiltonian formulation of the non-linear system described by the Korteweg-de Vries equations involved higher order Lagrangians [1, 32, 33]. We remit to [30] for a recent and interesting survey about the problems in the formulation of theories involving higher order derivatives.

The global geometric formulation of higher order Lagrangian systems was developed in the framework of higher order tangent bundles by using its underlying canonical structure, the almost tangent structure of higher order [22, 23, 24, 4]. An alternative procedure by using the higher order variational calculus on fibred manifolds is due to García and Muñoz [13]. A recent approach based on the notion of Lepagean forms introduced by Krupka is due to Krupková [20] (see also the references therein). Notice that the local study of these systems is due to Ostrogradskii and dates back to the past century [42].

∗Supported by DGICYT-SPAIN, Proyecto PB91-0142 and Consejería de Educación del Gobierno de Canarias-SPAIN.

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

Recently, Gràcia et al. [17] have developed a geometric constraint algorithm for degenerate Lagrangians of higher order by using the so called intermediate spaces (see [3]for the order 2). This technique permit us to decompose the Legendre-Ostrogradskii transformation in several steps and consider the constraint algorithm in each step. Further, on each step the corresponding higher order differential equation problem is considered. To do this, they used the notion of intermediate evolution operators. Another approach to this problem is due to Cariñena and López [6]. They have developed a constraint algorithm by extending the formulation of Skinner and Rusk [40, 41]. Also they introduce a time-evolution operator which passes Hamiltonian to Lagrangian constraints.

The aim of this paper is to extend the Gotay and Nester algorithm [14, 15, 16] to higher order degenerate time-dependent Lagrangians. We follow the ideas introduced in [7, 8] for degenerate time-dependent Lagrangians of order 1. Our approach is different to that of Gràcia, Pons and Román-Roy and we do not use the notion of evolution operators. However the notion of intermediate spaces will be widely used. In fact, our purpose is to follow the spirit of the Gotay and Nester algorithm. Since we deal with time-dependent Lagrangians, then the symplectic formalism is not adequate. Thus, we must use the cosymplectic formalism developed in [25] (see [26, 27, 5] for the order 1). We notice that only the second order case is considered, but the results may be easily extended to the general case. The paper is structured as follows. In section 2 we recall the cosymplectic formalism for regular time-dependent second order Lagrangians; a first constraint algorithm is considered. In section 3 we introduce the intermediate spaces for time-dependent Lagrangians and develop a constraint algorithm for each step. The fourth order differential equation problem is solved in section 4. We end the paper with an application to linear Lagrangians on the accelerations (see [29] for the autonomous case).

We thank P. Pitanga for several helpful conversations.

2 Time-dependent second order Lagrangians

The generalized evolution space of order k of an n-dimensional manifold Q is the (k+1)n+1 -dimensional manifold J^{k}(R, Q) of all k-jets j^{k}σ of differentiable curves in Q (see [28]). We may identify in a canonical way J^{k}(R, Q) with the product manifold R × T^{k}Q. If (q^{i}) are coordinates on Q, then we denote by (t, q_{0}^{i}, q_{1}^{i}, ⋯, q_{k}^{i}) the induced coordinates on J^{k}(R, Q). We denote by ρ̃_{r}^{k} : J^{k}(R, Q) ⟶ J^{r}(R, Q), r < k the canonical projection. From the natural identification J^{k}(R, Q) ≅ R×T^{k}Q we may transport all the geometric structures on T^{k}Q to J^{k}(R, Q). For instance, the canonical almost tangent structure J_{1} of order k on T^{k}Q can be extended to a tensor field of type (1, 1) also denoted by J_{1} and locally given by J_{1} = ∑_{r=0}^{k−1}(r +1)\frac{∂}{∂q_{r+1}^{i}} ⊗dq_{r}^{i}. For each r, 2 ≤ r ≤ k, we set J_{r} = J_{1}^{r},

the composition of J_{1} r-times. Also, let C_{1} be the Liouville vector field on T^{k}Q may be translated to J^{k}(R, Q). The translated vector field is also denoted by C_{1} and it is locally given by C_{1} = ∑_{r=1}^{k} rq_{r}^{i} \frac{∂}{∂q_{r}^{i}}. We define C_{r} = J_{r−1}C_{1}, 2 ≤ r ≤ k. We define the following (1, 1)-tensor field: J̄_{1} = J_{1} + \frac{∂}{∂t} ⊗ dt. J̄_{1} is the canonical almost stable tangent structure of order k on J^{k}(R, Q) (see [21]).

Recall that a vector field ξ on J^{k}(R, Q) is said to be a (k + 1)-th differential equation if J̄_{1}ξ = (∂/∂t) + C_{1}, or equivalently, if dt(ξ) = 1 and J_{1}ξ = C_{1}. A solution of ξ is a curve σ in Q such that j^{k}σ is an integral curve of ξ.

We deduce that a (k + 1)-th differential equation ξ on J^{k}(R, Q) is locally expressed by ξ = \frac{∂}{∂t} + q_{1}^{i} \frac{∂}{∂q_{0}^{i}} + q_{2}^{i} \frac{∂}{∂q_{1}^{i}} + ⋯ + q_{k}^{i} \frac{∂}{∂q_{k−1}^{i}} + ξ^{i} \frac{∂}{∂q_{k}^{i}}, where ξ^{i} = ξ^{i}(t, q_{0}^{j}, q_{1}^{j}, ⋯, q_{k}^{j}). Then a

curve σ(t) = (q^{i}(t)) in Q is a solution of ξ if and only if it satisfies the following system of non-autonomous differential equations of order k + 1:

\frac{d^{k+1}q^{i}}{dt^{k+1}} = ξ^{i}(t, q(t), \frac{dq}{dt} , ⋯, \frac{d^{k}q}{dt^{k}} ), 1 ≤ i ≤ n.

Let L : J^{2}(R, Q) ⟶ R be a Lagrangian of order 2 and define the 1-form α_{L} = d_{J_{1}}L − \frac{1}{2}d_{T} d_{J_{2}}L, the 2-form ω_{L} = −dα_{L} and the function E_{L} = C_{1}L − \frac{1}{2}d_{T} C_{2}L − L on J^{3}(R, Q). E_{L} is the energy associated to L and then the Poincaré-Cartan 2-form on J^{3}(R, Q) is given by Ω_{L} = −dα_{L} + dE_{L} ∧ dt. Let us recall that the generalized momenta are given by p̂_{0}^{i} = \frac{∂L}{∂q_{1}^{i}} −d_{T} (\frac{∂L}{∂q_{2}^{i}}), p̂_{1}^{i} = \frac{∂L}{∂q_{2}^{i}}. Then we have E_{L} = p̂_{0}^{i}q_{1}^{i}+p̂_{1}^{i}q_{2}^{i}−L(t, q_{0}^{i}, q_{1}^{i}, q_{2}^{i}) and Ω_{L} = dq_{0}^{i} ∧ dp̂_{0}^{i} + dq_{1}^{i} ∧ dp̂_{1}^{i} + dE_{L} ∧ dt.

The Legendre-Ostrogradskii transformation is the fibred mapping Leg : J^{3}(R, Q) ⟶ R × T^{∗}T Q over T Q locally given by Leg(t, q_{0}^{i}, q_{1}^{i}, q_{2}^{i}, q_{3}^{i}) = (t, q_{0}^{i}, q_{1}^{i}, p̂_{0}^{i}, p̂_{1}^{i}).

The Lagrangian L is said to be regular if the Hessian matrix (W_{ij} = \frac{∂^{2}L}{∂q_{2}^{i}∂q_{2}^{j}}) is non-singular. Otherwise L is said to be singular or degenerate. The following assertions are equivalent: (i) L is regular; (ii) (Ω_{L}, dt) is a cosymplectic structure; and (iii) the Legendre-Ostrogradskii transformation Leg : J^{3}(R, Q) ⟶ R × T^{∗}T Q is a local diffeomorphism. If Leg is a global diffeomorphism then L is said to be hyperregular.

Now consider the global equations of motion :

ι_{ξ}Ω_{L} = 0, i_{ξ}dt = 1.  (1)

If L is regular, then there exists a unique vector field ξ_{L} on J^{3}(R, Q) satisfying (1). Furthermore ξ_{L} is a fourth order differential equation such that its solutions are the Euler-Lagrange equations for L:

\frac{d^{2}}{dt^{2}} (\frac{∂L}{∂q_{2}^{i}} ) − \frac{d}{dt}(\frac{∂L}{∂q_{1}^{i}} ) + \frac{∂L}{∂q_{0}^{i}} = 0.

In fact, ξ_{L} is the Reeb vector field of the cosymplectic structure (Ω_{L}, dt).

For the sake of simplicity, we suppose that L is hyperregular. Then we have the Hamiltonian function H : R × T^{∗}T Q ⟶ R, H ∘ Leg = E_{L} (if L is not hyperregular, then H is only locally defined). Let λ_{T Q} = p_{0}^{i}dq_{0}^{i} + p_{1}^{i}dq_{1}^{i} be the Liouville 1-form on T^{∗}T Q (here considered on R × T^{∗}T Q) and ω_{T Q} = −dλ_{T Q} the canonical symplectic 2form on T^{∗}T Q (also considered on R × T^{∗}T Q). Then we have a cosymplectic structure (Ω_{H} = ω_{T Q} +dH ∧dt, dt). Let X_{H} be the Reeb vector field of the structure (Ω_{H}, dt). X_{H} is the Hamiltonian vector field for H, i.e., i_{X_{H}} Ω_{H} = 0, i_{X_{H}} dt = 1. Moreover T Leg(ξ_{L}) = X_{H} and the integral curves of X_{H} satisfying the Hamilton equations for H:

\frac{dq_{0}^{i}}{dt} = \frac{∂H}{∂p_{0}^{i}} , \frac{dp_{0}^{i}}{dt} = −\frac{∂H}{∂q_{0}^{i}} , \frac{dq_{1}^{i}}{dt} = \frac{∂H}{∂p_{1}^{i}} , \frac{dp_{1}^{i}}{dt} = −\frac{∂H}{∂q_{1}^{i}} .  (2)

From the local expression of X_{H} we can check that X_{H} satisfies a second order differential equation condition: J(T γ_{2}(X_{H})) = C, where J is the canonical tensor field on J^{1}(R, Q), γ_{2} : R × T^{∗}T Q ⟶ R × T Q the canonical projection and C the Liouville vector field on T Q, here considered in R × T Q. In fact (2) becomes:

\frac{dq_{0}^{i}}{dt} = \frac{∂H}{∂p_{0}^{i}} , \frac{dp_{0}^{i}}{dt} = −\frac{∂H}{∂q_{0}^{i}} , \frac{d^{2}q_{0}^{i}}{dt^{2}} = \frac{∂H}{∂p_{1}^{i}} , \frac{dp_{1}^{i}}{dt} = −\frac{∂H}{∂q_{1}^{i}} .

If the Lagrangian L is singular, then (Ω_{L}, dt) is not longer a cosymplectic structure and hence the equation (1) has no necessarily one solution; even, if there exists one solution, say ξ, then ξ is not necessarily unique nor a fourth order differential equation. Furthermore, in general there will not be a Hamiltonian counterpart.

A first attempt to solve this problem consists in to develop a constraint algorithm for the equations (1). In fact, we can extend the algorithm for time-dependent Lagrangians of order 1 [7, 8] and hence we obtain a sequence of submanifolds

⋯ S_{3} ↪^{g_{3}} S_{2} ↪^{g_{2}} S_{1} = J^{3}(R, Q).

One of three possibilities may occur: (i) there exists a positive integer r such that S_{r} = ∅; (ii) we obtain a submanifold S_{r} ≠ ∅ such that dim S_{r} = 0; (iii) there exists a positive integer r such that S_{r} = S_{r+1}, and dim S_{r} ≠ 0. In the first case S_{r} = ∅ implies that (1) have no solution. In the second case the final submanifold S_{r} consists of isolated points and the solution of (1) is ξ = 0. In the third case we have a final submanifold S_{F} = S_{r} and there exists a vector field ξ on S_{F} such that (i_{ξ}Ω_{L} = 0, i_{ξ}dt = 1)_{|_{S_{F}}}. In such a case the submanifold S_{F} will be called the final constraint submanifold.

Next, we consider the problem of the existence of a Hamiltonian formulation of the dynamics. We need an additional hypothesis on the Lagrangian. We suppose that M_{1} = Leg(R×T^{3}Q) is a submanifold of R×T^{∗}T Q and denote by j_{1} : M_{1} ⟶ R×T^{∗}T Q the corresponding inmersion. Hence we may consider the map Leg_{1} : R × T^{3}Q ⟶ M_{1}; in other words, the map Leg_{1} is the restriction of the map Leg onto its image. We say that L almost regular if Leg is a submersion onto M_{1}, and the fibers Leg^{−1}(Leg (z)) of Leg are connected for all z ∈ J^{3}(R, Q). The manifold M_{1} is called the primary constraint submanifold. Since Ker TLeg = Ker Ω_{L}∩Im J_{2} then the manifold M_{1} may be canonically identified with the leaf space of the foliation on J^{3}(R, Q) determined by the involutive distribution Ker T Leg, i. e. M_{1} ≅ R × T Q/Ker T Leg. Now we can check that the energy E_{L} is constant on the fibers of the submersion Leg and then we may define a Hamiltonian function H_{1} : M_{1} ⟶ R such that E_{L} = H_{1} ∘ Leg_{1}. We put ω_{1} = j_{1}^{∗}ω_{T Q}, η_{1} = j_{1}^{∗}(dt), Ω_{H_{1}} = ω_{1} + dH_{1} ∧ η_{1}. Then we have Leg_{1}^{∗} ω_{1} = ω_{L}, Leg_{1}^{∗} η_{1} = dt and Leg_{1}^{∗} Ω_{H_{1}} = Ω_{L}. Hence we may apply the constraint precosymplectic algorithm in both sides of the above diagram.

On the Hamiltonian side, the algorithm gives another sequence of submanifolds

⋯ M_{4} ↪^{j_{4}} M_{3} ↪^{j_{3}} M_{2} ↪^{j_{2}} M_{1},

in such a way that Leg_{1}(S_{l}) = M_{l}, for all l. Consequently, Leg_{1} induces a sequence of submersions Leg_{l} : S_{l} ⟶ M_{l} such that the following diagrams are commutative

[figure: S_{l−1}; M_{l−1}; Leg_{l−1}; S_{l}; M_{l}; g_{l}; j_{l}; Leg_{l}]

Therefore, the behaviour of the algorithm on both sides is the same. Thus, if we obtain a final constraint submanifold S_{F′} on the Lagrangian side then we have a final constraint submanifold M_{F′} on the Hamiltonian side and conversely (see [6] for the autonomous case).

Suppose now that X is a vector field on S_{F′} such that satisfies (1); then, if X is Leg_{F′}-projectable to a vector field T Leg_{F′}(X) on M_{F′} we have (i_{T Leg_{F′} (X)}Ω_{H_{1}} = 0, i_{T Leg_{F′} (X)}η_{1} = 1)_{|_{M_{F′}}}. Conversely, if Y is a vector field on M_{F′} such that (i_{Y} Ω_{H_{1}} = 0, i_{Y} η_{1} = 1)_{|_{M_{F′}}}, then every X ∈ (T Leg_{F′})^{−1}(Y) satisfies (i_{X}Ω_{L} = 0, i_{X}dt = 1)_{|_{S_{F′}}}. Thus, if L is almost regular then there exists a Hamiltonian formulation of the dynamics, and the Lagrangian and Hamiltonian formulations are equivalent. Notice that from the construction procedure the submanifolds S_{F′} and M_{F′} are maximal and that for all x ∈ M_{F′}, Leg^{−1}(x) ⊆ S_{F′}. Therefore the involutive distribution Ker T Leg restricts to the submanifold S_{F′}.

In order to solve the fourth order differential equation problem this formulation is not adequate. In fact, following Gotay and Nester [14, 15, 16] and Chinea, de León and Marrero [7, 8] we must construct a submanifold S of S_{F′} and a vector field X on S such that (i_{X}Ω_{L} = 0, i_{X}dt = 1, J_{1}X = C_{1})_{|_{S}}. For this purpose we must prove that if X is a Leg_{F′}-projectable vector field on S_{F′} satisfying (1) then J_{1}X − C_{1} ∈ Ker T Leg. But, in general, it is not true. Thus this procedure does not work. In fact, we must incorporate to the constraint algorithm the condition J_{3}X = C_{3} on the Lagrangian side and the condition of second order differential equation on the Hamiltonian side.

3 The constraint algorithms

Let Leg : J^{3}(R, Q) ⟶ R×T^{∗}T Q be the Legendre-Ostrogradskii transformation defined by a Lagrangian L of order 2. Following Gràcia, Pons and Román-Roy [17] we first define the intermediate space P_{1} that connects J^{3}(R, Q) and R × T^{∗}T Q, through a decomposition of Leg

P_{0} = J^{3}(R, Q) ⟶^{α_{0}} P_{1} ⟶^{α_{1}} P_{2} = R × T^{∗}T Q,

where α_{0}(t, q_{0}^{i}, q_{1}^{i}, q_{2}^{i}, q_{3}^{i}) = (t, q_{0}^{i}, q_{1}^{i}, q_{2}^{i}, p̂_{0}^{i}), and α_{1}(t, q_{0}^{i}, q_{1}^{i}, q_{2}^{i}, p_{0}^{i}) = (t, q_{0}^{i}, q_{1}^{i}, p_{0}^{i}, p̂_{1}^{i}). Then Leg = α_{1} ∘α_{0}. Let us briefly describe the intermediate space P_{1} (for an alternative description see [6]). Define P̄_{1} = (R × T^{2}Q)_{×_{R×T Q}}(R × T^{∗}T Q). Let λ_{T Q} be the Liouville 1-form on T^{∗}T Q and set ϕ = (^{t}J)^{∗}λ_{T Q}, where ^{t}J : T^{∗}T Q ⟶ T^{∗}T Q is the adjoint operator of the canonical almost tangent structure J on T Q and (^{t}J)^{∗}λ_{T Q} is the pullback of λ_{T Q} via the mapping ^{t}J. On the other hand, let ϕ̃ = J_{1}^{∗}(d_{J_{1}}L − \frac{1}{2}d_{T} d_{J_{2}}L); ϕ̃ is a 1-form on R × T^{3}Q which projects onto a 1-form ϕ̄ on R × T^{2}Q. Then P_{1} = {y = ((t, v), (t, α)) ∈ P̄_{1} | ϕ̄(y) = ϕ(y)}. Since ϕ = p_{1}^{i}dq_{0}^{i} and ϕ̄ = p̂_{1}^{i}dq_{0}^{i} we deduce that P_{1} = {y = ((t, q_{0}^{i}, q_{1}^{i}, q_{2}^{i}, p_{0}^{i}, p_{1}^{i} = p̂_{1}^{i}) ∈ P̄_{1}}. P_{1} is a (4n + 1)-dimensional submanifold of P̄_{1}. We know that P_{1} is fibred over J^{2}(R, Q) ≅ R × T^{2}Q, the canonical projection being denoted by γ_{1} : P_{1} ⟶ R × T^{2}Q. We also define the vector subbundle V_{1} of T P_{1} by V_{1}(y) = {Y ∈ T_{y}P_{1} | J̄_{1}(T γ_{1}Y) = 0}. On P_{1} we define the partial energy function E_{1} = p_{0}^{i}q_{1}^{i} + p̂_{1}^{i}q_{2}^{i} − L and the partial Poincaré-Cartan 2-form Ω_{E_{1}} = α_{1}^{∗}ω_{T Q} + dE_{1} ∧ τ_{1}, where τ_{1} = α_{1}^{∗}dt.

From a direct computation we prove that rank α_{0} = rank α_{1} = 3n + 1 + rank W_{ij}. Then L is regular iff any (and hence both) α_{l}, l = 0, 1 is a local diffeomorphism. Then the following assertions are equivalent: (i) L is regular; (ii) (Ω_{L}, dt) is cosymplectic; and (iii) (Ω_{E_{1}}, τ_{1}) is cosymplectic. In such a case we obtain three vector fields ξ_{0} = ξ_{L}, ξ_{1} and ξ_{2} = X_{H} determined by the three motion equations i_{ξ_{0}}Ω_{L} = 0, i_{ξ_{0}}dt = 1; i_{ξ_{1}}Ω_{E_{1}} = 0, i_{ξ_{1}}τ_{1} = 1; and i_{ξ_{2}}Ω_{H} = 0, i_{ξ_{2}}dt = 1, respectively. A direct computation shows that J_{1}ξ_{0} = C_{1}, J_{1}(T γ_{1}(ξ_{1})) = C_{1}, and J(T γ_{2}(ξ_{2})) = C. Notice that in the second equation J_{1} is the canonical tensor field on J^{2}(R, Q). In other words, ξ_{0} is a fourth order differential equation and ξ_{1} (resp. ξ_{2}) satisfies a third (resp. second) order differential equation condition.

We set P_{11} = α_{0}(P_{0}), P_{21} = α_{1}(P_{1}), and α_{01} : P_{0} ⟶ P_{11}, α_{11} : P_{1} ⟶ P_{21} the corresponding restrictions. Further we set P_{22} = α_{11}(P_{11}) and α_{12} : P_{11} ⟶ P_{22} the restriction of α_{11}. Fron now on, we are assuming the following hypotheses: (i) P_{11}, P_{21} and P_{22} are submanifolds; and (ii) α_{01}, α_{11} and Leg_{1} are submersions with connected fibres. Then we obtain that α_{12} is also a submersion. Moreover, we have the following identifications:

P_{11} ≅ \frac{J^{3}(R, Q)}{Ker T α_{0}}, P_{21} ≅ \frac{P_{1}}{Ker T α_{1}} , M_{1} = P_{22} ≅ \frac{J^{3}(R, Q)}{Ker T Leg} .

We remark that if the Legendre-Ostrogradskii transformation and the Hessian matrix (W_{ij} = \frac{∂^{2}L}{∂q_{2}^{i}∂q_{2}^{j}}) of L have constant rank, then P_{11}, P_{21} and P_{22} are, locally, submanifolds and α_{01}, α_{11} and Leg_{1} are submersions.

We can easily check that the partial energy function E_{1} is constant along the fibres of the submersion α_{11} : P_{1} ⟶ P_{21}. Hence it induces a function H̃_{1} : P_{21} ⟶ R such that H̃_{1} ∘ α_{11} = E_{1}. Furthermore we have H̃_{1} ∘ j_{22} = H_{1}, where j_{22} : P_{22} ⟶ P_{21} is the canonical inclusion. We set ΩH̃_{1} = j_{21}^{∗}ω_{T Q} + dH̃_{1} ∧ η̃_{1}, being j_{21} : P_{21} ⟶ P_{2} the inclusion and η̃_{1} = j_{21}^{∗}dt.

Next we apply the constraint algorithm to the three following equations:

i_{X}Ω_{L} = 0, i_{X}dt = 1, J_{3}X = C_{3},
i_{X}Ω_{E_{1}} = 0, i_{X}τ_{1} = 1, with X tangent to P_{11},
i_{X}ΩH̃_{1} = 0, i_{X} η̃_{1} = 1, with X tangent to P_{22}.

In the first case, we obtain a sequence of submanifolds

⋯ P_{03} ↪^{j_{03}} P_{02} ↪^{j_{02}} P_{01} ↪^{j_{01}} P_{0} = J^{3}(R, Q),

where

P_{01} = {x ∈ P_{0} | ∃X ∈ T_{x}P_{0} such that i_{X}Ω_{L} = 0, i_{X}dt = 1, J_{3}X = C_{3}}.

On P_{01} there exists a solution but in general it is not tangent to P_{01} and then we must consider a new submanifold P_{02} on which there exists a solution tangent to P_{01}, and so on.

In the second case, we obtain a sequence

⋯ P_{13} ↪^{j_{13}} P_{12} ↪^{j_{12}} P_{11} ↪^{j_{11}} P_{1},

where

P_{12} = {y ∈ P_{11} | ∃Y ∈ T_{y}P_{11} such that ⟨T_{y}P_{1}, i_{Y} Ω_{E_{1}}⟩ = 0, i_{Y} τ_{1} = 1}.

As above, on P_{12} there exists a solution but in general it is not tangent to P_{12} and then we must consider a new submanifold P_{13} on which there exists a solution tangent to P_{12}, and so on.

Finally, in the third case we obtain a sequence of submanifolds

⋯ P_{23} ↪^{j_{23}} P_{22} ↪^{j_{22}} P_{21} ↪^{j_{21}} P_{2} = R × T^{∗}T Q,

where

P_{23} = {z ∈ P_{22} | ∃Z ∈ T_{z}P_{22} such that ⟨T_{z}P_{21}, i_{Z}ΩH̃_{1}⟩ = 0, i_{Z} η̃_{1} = 1}.

By a similar device we obtain P_{24}, P_{25}, and so on.

A direct computation shows that α_{0}(P_{01}) = P_{12} and the induced mapping α_{02} : P_{01} ⟶ P_{12} is also a submersion. Proceeding by induction we obtain a sequence of submersions α_{0,r+1} : P_{0r} ⟶ P_{1,r+1}. Furthermore, we can check that α_{12}(P_{12}) = P_{23} and the induced mapping α_{13} : P_{12} ⟶ P_{23} is also a submersion. Then we obtain a sequence of submersions α_{1,r+1} : P_{1r} ⟶ P_{2,r+1}. Therefore the three algorithms are equivalent. If one of these stops in some level, then the other two stop in the same level. In such a case we obtain three final constraint submanifolds P_{0F} , P_{1,F +1} and P_{2,F +2} which are connected by the partial Legendre-Ostrogradskii transformations:

P_{0F} ⟶^{α_{0,F +1}} P_{1,F +1} ⟶^{α_{1,F +2}} P_{2,F +2}.

The following diagram summarizes the above results:

[figure: P_{0} = J^{3}(R, Q); j_{0F}; j_{02}; j_{01}; α_{0}; P_{1}; j_{11}; j_{12}; j_{13}; j_{1,F +1}; j_{21}; j_{22}; j_{23}; j_{24}; j_{2,F +2}; α_{1}; α_{01}; α_{02}; α_{03}; α_{0,F +1}; P_{11}; P_{12}; P_{13}; α_{11}; α_{12}; α_{13}; α_{14}; α_{1,F +2}; P_{2} = R × T^{∗}T Q; P_{21}; M_{1} = P_{22}; P_{23}; P_{24}; P_{01}; P_{02}; P_{0F}; P_{12}; P_{13}; P_{1,F +1}; P_{2,F +2}; Leg]

It is clear that P_{0F} ⊆ S_{F′} and P_{2,F +2} ⊆ M_{F′}.

4 The fourth order differential equation problem

A direct computation shows that dim Ker T α_{0} ≤ dim Ker T Leg ≤ 2dim Ker T α_{0}. In this section we shall assume that dim Ker T Leg = 2dim Ker T α_{0}. Thus, we have Ker T α_{1} = T α_{0}(Ker T Leg), which implies that the involutive distribution Ker T α_{1} restricts to the submanifold P_{11}. Therefore for all p_{1} ∈ P_{11} we deduce that α_{1}^{−1}(α_{1}(p_{1})) ⊆ P_{11}.

Suppose that we have obtained three final constraint submanifods P_{0F} , P_{1,F +1} and P_{2,F +2} which are connected by the partial Legendre–Ostrogradskii α_{0,F +1} and α_{1,F +2}.

Proceeding by induction we have that for all p_{1} ∈ P_{1,F +1}, α_{1}^{−1}(α_{1}(p_{1})) ⊆ P_{1,F +1} which implies that the involutive distribution Ker T α_{1} restricts to the submanifold P_{1,F +1}.

Let ξ_{2} be a vector field on P_{2,F +2} such that (i_{ξ_{2}}ΩH̃_{1} = 0, i_{ξ_{2}} η̃_{1} = 1)_{|_{P_{2,F +2}}} and (t, q_{0}, q_{1}, q_{2}, p_{0}) local coordinates on P_{1}. Since α_{11}^{∗}ΩH̃_{1} = Ω_{E_{1}}, (T α_{1})(\frac{∂}{∂p_{0}} ) = \frac{∂}{∂p_{0}} and i_{\frac{∂}{∂p_{0}}} Ω_{E_{1}} = q_{1}dt − dq_{0}, we obtain that ξ_{2} satisfies the second order differential equation condition, i.e., J(T γ_{2}(ξ_{2})) = C.

Next, let Y be a vector field on P_{1,F +1} such that (i_{Y} Ω_{E_{1}} = 0, i_{Y} τ_{1} = 1)_{|_{P_{1,F +1}}} . Then there exists a unique vector field Y^{∗} on P_{1,F +1} such that Y^{∗} ∈ (Ker T_{α_{1}})_{|_{P_{1,F +1}}} and (T γ_{1})(Y^{∗}) = J_{1}((T γ_{1})Y) − C_{1}. Suppose that Y is α_{1}-semiprojectable, i.e., Y = Z + U, where Z is α_{1}-projectable and U ∈ V_{1}. Denote by ỹ′ the leaf of the distribution Ker T α_{1} passing through y′. By a similar technique to that used in [7, 8] we can choose an unique point y on each leaf ỹ′ of (Ker T α_{1})_{|_{P_{1,F +1}}} such that Y^{∗}(y) = 0, or equivalently J_{1}(T γ_{1}(Y (y))) = C_{1}(γ_{1}(y)), i.e., Y satisfies the third order differential equation condition at y. In fact, if y′ = (z, β) ∈ (R × T^{2}Q) ×_{(R×T Q)} (R × T^{∗}T Q) then y = (T (ρ̃_{1}^{2} ∘ γ_{1})(Y (y′)), β). Notice that T (ρ̃_{1}^{2} ∘ γ_{1})(Y (y′)) ∈ R × T^{2}Q since Y satisfies the second order differential equation condition and i_{Y} τ_{1} = 1. Denote y = W_{Y}. If Y′ is another α_{1}-semiprojectable solution such that Y′−Y ∈ V_{1}, then W_{Y′} = W_{Y}. Put y = W_{[Y]}, where [Y] denotes the equivalence class of Y. Let S_{[Y]} the set of these points W_{[Y]} and define the canonical inclusion j_{1} : P_{2,F +2} ≅ _{\frac{P_{1,F +1}}{(Ker T α_{1})_{|_{P_{1,F +1}}}}} ⟶ P_{1,F +1}, ỹ ⟶ W_{[Y]}(ỹ),

where ỹ denotes the leaf through y and W_{[Y]}(ỹ) is the unique point on the leaf ỹ such that J_{1}(T γ_{1}(Y (y))) = C_{1}(γ_{1}(y)). Then S_{[Y]} = Im j_{1} and the inclusion j_{1} is a section of the submersion α_{1,F +2} : P_{1,F +1} ⟶ P_{2,F +2} in such a way that S_{[Y]} ≅ P_{2,F +2} ≅ \frac{P_{1,F +1}}{(Ker T α_{1})_{|_{P_{1,F +1}}}} . Furthermore, along S_{[Y]} we have i_{Y} Ω_{E_{1}} = 0, i_{Y} τ_{1} = 1, J_{1}(T γ_{1}(Y)) =

C_{1} ∘ γ_{1}, but Y is not necessarily tangent to S_{[Y]}. Hence we proceed as follows. Let ξ_{2} be a vector field on P_{2,F +2} such that (i_{ξ_{2}}ΩH̃_{1} = 0, i_{ξ_{2}} η̃_{1} = 1)_{|_{P_{2,F +2}}} and consider a vector field Y on P_{1,F +1} such that T α_{1,F +2}(Y) = ξ_{2}. Then, since α_{1,F +2} ∘ j_{1} = id_{P_{2,F +2}} and using that V_{1} ⊆ Ker T α_{1} and the fact that α_{1,F +2} : S_{[Y]} ⟶ P_{2,F +2} is a diffeomorphism, we deduce that ξ_{1} = T j_{1}(ξ_{2}) is the unique vector field on S_{[Y]} which verifies (i_{ξ_{1}}Ω_{E_{1}} = 0, i_{ξ_{1}}τ_{1} = 1, J_{1}(T γ_{1}(ξ_{1})) = C_{1} ∘ γ_{1})_{|_{S_{[Y ]}}} .

Therefore there exists a submanifold S_{1} of P_{1,F +1} and an unique vector field ξ_{1} on it such that ξ_{1} is a solution of the partial motion equations and satisfies the third order differential equation condition.

From the construction of section 3 and proceeding by induction we deduce that for all p_{0} ∈ P_{0F} , α_{0}^{−1}(α_{0}(p_{0})) ⊆ P_{0F} . Thus, S̃_{0} = α_{0}^{−1}(S_{1}) is a submanifold of P_{0F} in such a way that the involutive distribution Ker T α_{0} restricts to S̃_{0} and the induced mapping α̃_{0} : S̃_{0} → S_{1} is a submersion. Suppose that ξ̃_{0} is an α̃_{0}−projectable vector field onto ξ_{1}. Then (iξ̃_{0}Ω_{L} = 0, iξ̃_{0}dt = 1, J_{2}ξ̃_{0} = C_{2})_{|S̃_{0}} and J_{1}ξ̃_{0} − C_{1} ∈ Ker T α_{0}. Denote by x̃ the leaf of the distribution Ker T α_{0} passing through x. By a similar procedure to that used above we can find an unique point W_{[ξ̃_{0}]}(x̃) on each leaf x̃ on which ξ̃_{0} satisfies the fourth order differential equation condition. In fact, W_{[ξ̃_{0}]}(x̃) = (T ρ̃_{2}^{3})(ξ̃_{0}(x)). We remark that W_{[ξ̃_{0}]}(x̃) ∈ R × T^{3}Q since ξ̃_{0} satisfies the third order differential equation

condition and iξ̃_{0}dt = 1. Set Sξ̃_{0} = \{W_{[ξ̃_{0}]}(x̃)\}. Notice that _{\frac{S̃_{0}}{(Ker T α_{0})_{|S̃_{0}}}} ≅ S_{[ξ_{1}]} = S_{1}.

Consider the canonical inclusion j_{0} : S_{1} ≅ _{\frac{S̃_{0}}{(Ker T α_{0})_{|S̃_{0}}}} ⟶ S̃_{0}, x̃ ⟶ W_{[ξ̃_{0}]}(x̃). Then

Sξ̃_{0} = Im j_{0} and Sξ̃_{0} is a submanifold of S̃_{0} diffeomorphic to S_{1}. Thus we have (iξ̃_{0}Ω_{L} = 0, iξ̃_{0}dt = 1, J_{1}ξ̃_{0} = C_{1})_{|_{Sξ̃_{0}}} . But ξ̃_{0} is not necessarily tangent to Sξ̃_{0}. Now, it is sufficient

to define ξ_{0} = T j_{0}(ξ_{1}). In fact, since α_{0,F +1} ∘ j_{0} = id_{S_{1}} and using that the mapping α_{0,F +1} : Sξ̃_{0} ⟶ S_{1} is a diffeomorphism, we deduce that ξ_{0} is the unique vector field tangent to Sξ̃_{0} which verifies (i_{ξ_{0}}Ω_{L} = 0, i_{ξ_{0}}dt = 1, J_{1}ξ_{0} = C_{1})_{|_{Sξ̃_{0}}} .

Therefore there exists a submanifold S_{0} of P_{0F} and an unique vector field ξ_{0} on it such that ξ_{0} is a solution of the motion equations and satisfies the fourth order differential equation condition.

5 Linear Lagrangians on the accelerations

Let f, h two functions on R × T Q. We consider a Lagrangian L : R × T^{2}Q ⟶ R defined by L = d_{T} f +(ρ̃_{1}^{2})^{∗}h. We suppose that f does not depends on t, since in this case then the term \frac{∂f}{∂t} will be included in h. In local coordinates we have L = \frac{∂f}{∂q_{0}} q_{1}+ \frac{∂f}{∂q_{1}} q_{2}+h(t, q_{0}, q_{1}), i.e., L is linear on the accelerations (see Pais, Uhlenbeck [34] and Müller-Hoissen [30]). The Ostrogradskii generalized momenta are given by p̂_{0} = \frac{∂f}{∂q_{0}} + \frac{∂h}{∂q_{1}} and p̂_{1} = \frac{∂f}{∂q_{1}} . Hence E_{L} = (ρ̃_{1}^{3})^{∗}E_{h}, where E_{h} = Ch − h.Thus Ω_{L} = (ρ̃_{1}^{3})^{∗}Ω_{h}, Ω_{h} = −dd_{J}h + dE_{h} ∧ dt. The intermediate space P_{1} is:

P_{1} = \{(t, q_{0}, q_{1}, q_{2}, p_{0}, p_{1}) ∈ (R × T^{2}Q)_{×(R×T Q)}(R × T^{∗}T Q) | p_{1} = p̂_{1} = \frac{∂f}{∂q_{1}} \}.

Further α_{0}(t, q_{0}, q_{1}, q_{2}, q_{3}) = (t, q_{0}, q_{1}, q_{2}, \frac{∂f}{∂q_{0}} + \frac{∂h}{∂q_{1}} , \frac{∂f}{∂q_{1}} ). Therefore dim Ker T α_{0} = n. If we define a 1-form on R×T Q by θ = df+d_{J}h, then P_{11} = (R×T^{2}Q)_{×(R×T Q)}θ̃(R×T Q), where θ̃(t, v) = (t, θ(t, v)) ∈ R × T^{∗}T Q. Notice that θ is a 1-form on T Q depending on t. Then P_{11} is a (3n + 1)-dimensional submanifold of P_{1}. Moreover, the projection γ_{1} : P_{1} ⟶ R × T^{2}Q induces a diffeomorphism γ_{1} : P_{11} ⟶ R × T^{2}Q in such a way that γ_{1} ∘α_{01} = ρ̃_{2}^{3}. Thus α_{01} is a submersion. On the other hand we have α_{1}(t, q_{0}, q_{1}, q_{2}, p_{0}) = (t, q_{0}, q_{1}, p_{0}, \frac{∂f}{∂q_{1}} ). Then P_{21} = Im α_{1} is a (3n+1)-dimensional submanifold of R×T^{∗}T Q and α_{11} : P_{1} ⟶ P_{21} is also a submersion. Finally P_{22} = M_{1} = θ̃(R×T Q) and then it is a (2n+1)-dimensional submanifold of R×T^{∗}T Q. The projection γ_{2} : R×T^{∗}T Q ⟶ R×T Q induces a diffeomorphism γ_{2} : P_{22} ⟶ R × T Q. Since γ_{2} ∘ Leg_{1} = ρ̃_{1}^{3} then we deduce that Leg_{1} is also a submersion. Thus, we obtain that dim Ker T Leg = 2n. Furthermore we have γ_{2}^{∗}Ω_{h} = Ω_{H_{1}}, E_{h} ∘ γ_{2} = H_{1}.

Consequently the set of motion equations becomes: i_{X}Ω_{L} = 0, i_{X}dt = 1, J_{3}X = C_{3}, on R × T^{3}Q ; i_{Y} Ω̃_{1} = 0, i_{Y} dt = 1, J_{2}Y = C_{2}, on R × T^{2}Q with Ω̃_{1} = (ρ̃_{1}^{2})^{∗}Ω_{h} and i_{Z}Ω_{h} = 0, i_{Z}dt = 1, JZ = C , on R × T Q.

We remark that γ_{1}^{∗}Ω̃_{1} = j_{11}^{∗}Ω_{E_{1}}. We have an equivalence between the corresponding solutions through the canonical projections ρ̃_{2}^{3} and ρ̃_{1}^{2}:

[figure: P_{0} = R × T^{3}Q; P̃_{1} = R × T^{2}Q; P̃_{2} = R × T Q; P_{01}; P̃_{11}; P̃_{21}; P_{0r}; P̃_{1r}; P̃_{2r}]

Next we discuss the fourth order differential equation problem. Suppose that h : R × T Q ⟶ R is a regular Lagrangian of order 1. Then (Ω_{h}, dt) is a cosymplectic structure and denote by ξ_{h} its Reeb vector field. Notice that Jξ_{h} = C. Thus (Ω_{H_{1}}, η_{1}) is a cosymplectic structure on P_{22} with Reeb vector field ξ_{2} such that T γ_{2}(ξ_{2}) = ξ_{h}. Hence ξ_{2} is the unique vector field on P_{22} such that (i_{ξ_{2}}ΩH̃_{1} = 0, i_{ξ_{2}} η̃_{1} = 1)_{|_{P_{22}}} . Now, take on P_{11} a α_{1}-projectable vector field Y onto ξ_{2}. A direct computation shows that S_{1} = S_{[Y]} = (ξ_{h} × θ̃)(R × T Q). Furthermore we have that ξ_{h} ∘ γ_{2} = γ_{1} ∘ j_{1}. Thus, if we put ξ_{1} = T j_{1}(ξ_{2}) then T γ̃_{1}(ξ_{1}) = (ξ_{h}^{C})_{|_{ξ_{h}(R×T Q})}, where ξ_{h}^{C} is the complete lift of ξ_{h} and γ̃_{1} : S_{1} ⟶ ξ_{h}(R × T Q) is the diffeomorphism induced by γ_{1}. Notice that ξ_{h}(R × T Q) ⊂ R × T^{2}Q since Jξ_{h} = C and i_{ξ_{h}}dt = 1.

Next we put S̃_{0} = α_{0}^{−1}(S_{1}). Hence S̃_{0} = (ρ̃_{2}^{3})^{−1}(ξ_{h}(R × T Q)). Let ξ̃_{0} be a vector field on S̃_{0} which is ρ̃_{2}^{3}-projectable onto (ξ_{h}^{C})_{|_{ξ_{h}(R×T Q})} (or equivalently, α̃_{0}-projectable onto ξ_{1}). A direct computation shows that S_{0} = Sξ̃_{0} = ξ_{h}^{C}(ξ_{h}(R × T Q)). Furthermore, since ξ_{h}^{C} ∘ γ̃_{1} = j_{0}, we obtain that the the vector field ξ_{0} is precisely (ξ_{h}^{C})_{|_{ξ{C \atop h} (ξ_{h}(R×T Q))}}^{C}.

We conclude that linear Lagrangians on the accelerations are in fact equivalent to Lagrangians of order 1.

6 Outlook

In a next paper we shall extend these results to the arbitrary higher order, say k. For a singular Lagrangian L : J^{k}(R, Q) ⟶ R the Hamiltonian equations must be verify a k-th order differential equation condition which may be geometrically expressed as J_{1}(T γ_{k}(Y)) = C_{1}. Here J_{1} is the canonical almost tangent structure and C_{1} the Liouville vector field on T^{k−1}Q and γ_{k} : R×T^{∗}T^{k−1}Q ⟶ R×T^{k−1}Q is the canonical projection. Then we must consider on the Lagrangian side the equations

i_{X} Ω_{L} = 0, i_{X} dt = 1, J_{k+1}X = C_{k+1}.

An extension of the formalism of Skinner and Rusk following Cariñena and López [6] can be useful. Also time-evolution and intermediate evolution operators may be constructed for higher order degenerate time-dependent Lagrangians.

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M. de León: Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, SPAIN. e-mail: CEEML02@CC.CSIC.ES J. C. Marrero: Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de La Laguna, La Laguna, Tenerife, Canary Islands, SPAIN. e-mail: JSC@IAC.ES

Received 29 September 1992