Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 509–521

REDUCTION OF SINGULAR LAGRANGIANS OF HIGHER-ORDER

M. DE LEÓN, MARIA H. MELLO, AND P. R. RODRIGUES

Abstract. Given a singular non-autonomous Lagrangian of higher order we construct a reduced evolution space of the same order and a local regular Lagrangian on it which is gauge equivalent to the original one. The general geometrical framework used is the theory of almost stable tangent geometry of higher order introduced in [17]. We show that higher order almost stable manifols can be endowed with a local Poincaré-Cartan form via the inverse problem of Lagrangian Dynamics. Keywords. Singular, non-autonomous higher order Lagrangians, connections, inverse problem, reduction, almost stable tangent geometry. MS classification. Primary 58F05, 53C15, Secondary 53C05, 70H35, 70H05. PACS: 0320, 0240.

1. Introduction

Let Q be an n-dimensional manifold, with local coordinates (q^{i}) and J^{k}(IR, Q) the stable tangent bundle of order k of Q, with local coordinates (t_{0}, q_{0}^{i}, q_{1}^{i}, ⋯, q_{k}^{i}) where t is the Cartesian coordinate in IR, q_{r}^{i} = (d^{r}/dt^{r})_{t_{0}}(q^{i} ∘ σ), 1 ≤ r ≤ k, 1 ≤ i ≤ n. Sometimes we shall put q_{0}^{i} = q^{i}. Consider a non-autonomous Lagrangian of order k on Q, i.e., a function L : J^{k}(IR, Q) → IR. We know (see [19]) that the dynamics of L is developed in the stable tangent bundle of order 2k − 1 of Q, say the jet manifold J^{2k−1}(IR, Q). In fact we may construct a 2-form Ω_{L} (the Poincaré-Cartan 2-form) and the energy function E_{L} on J^{2k−1}(IR, Q). This construction is obtained by translating the canonical geometric structures (higher order almost tangent geometry and the Liouville vector field) on the higher order tangent bundle T^{k}Q to J^{k}(IR, Q) via the canonical identification J^{k}(IR, Q) ≅ IR×T^{k}Q. Then we obtain a canonical geometry on J^{k}(IR, Q), called almost stable tangent geometry of higher order (see [17]). If L is regular, then (Ω_{L}, dt) is a cosymplectic structure and hence there exists a unique vector field ξ_{L} on J^{2k−1}(IR, Q) such that

i_{ξ_{L}} Ω_{L} = 0, i_{ξ_{L}} dt = 1. (1.1)

The vector field ξ_{L}, called Euler-Lagrange vector field, is a 2k-th order non-autonomous differential equation on J^{2k−1}(IR, Q) and its solutions are precisely the solutions of the Euler-Lagrange equations and they may be considered as the intrinsic version of the motion equations.

Research of the first author was supported in part by DGICYT-Spain, Proyecto PB88-0012.

Research of the second author was supported in part by CNPq-Brazil, Proc. 30.2125/85.

Research of the third author was supported in part by CNPq-Brazil, Proc. 30.1115/79.

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

If L is singular (or degenerate) then the pair (Ω_{L}, dt) is not cosymplectic and thus these equations has no solution in general, and even if it exists it will not be unique nor a 2k-th order differential equation. The aim of this paper is to reduce the evolution space J^{2k−1}(IR, Q) to an integrable almost stable-tangent manifold of the same order on which exists a local regular Lagrangian gauge equivalent to the original one, having the same dynamical information. To do this we use the notion of almost stable-tangent structure of higher order. This reduction procedure may be compared with the Marsden-Weinstein one [23, 24] by means of the constraint algorithm developed in [6] (see [5] for the order one, [9] for the autonomous case of order one and [10] for the autonomous case of higher order).

The autonomous (resp. non-autonomous) case of order one was studied by Cantrijn et al. [3] (resp. de León et al. [16] (see also [14]). The autonomous higher order situation was studied in [15]. In the present article we are assuming that a global dynamics exists - i.e., only primary Lagrangian constraints are involved, and thus the constraint algorithm is not used. Indeed, “the constraint algorithm is of completely different nature as the geometric structures on evolution space and to make complatible both processes requires further refinements of the theory” [14]. Concerning the constraint algorithm we notice a recent study of higher-order autonomous Lagrangians by Gráxia et al [10] using what they call intermediate spaces.

The paper is structured as follows. In Section 2 we prove that higher order almost stable tangent manifolds may serve as evolution spaces. By the way the inverse problem of Lagrangian dynamics is examined. The reduction procedure is presented in Section 3. An Appendix gives some basic results.

2. Higher order Lagrangians on Integrable Almost Stable Tangent Manifolds. The Inverse Problem

This section is devoted to show that almost s-tangent manifolds endowed with an appropriate cosymplectic structure admit locally a Lagrangian function given by a Poincaré-Cartan 2-form. Moreover, underlying the geometric procedure it will be clear that the inverse problem for higher order Lagrangian dynamics is also developed – given a 2k-th order differential equation ξ, does there exist a Lagrangian function L such that ξ is precisely the Euler-Lagrange vector field for L? (see [7, 8, 12, 16, 22]).

Let L be a non-autonomous Lagrangian with energy E_{L}. Consider the 1-form α_{L} = Δ(L), where

Δ = ∑_{r=1}^{k} (−1)^{r−1} \frac{1}{r!} d_{T}^{r−1}d_{J_{r}} , (2.1)

(for a definition of d_{T} and d_{J_{r}} see [18] and for J_{r} see below). The Poincaré-Cartan 1form (resp. 2-form) is θ_{L} = α_{L} −E_{L}dt (resp. Ω_{L} = ω_{L} +dE_{L} ∧dt if we set ω_{L} = −dα_{L}). Its local expressions are

α_{L} = ∑_{r=1}^{k} p_{i}^{r}dq_{r−1}^{i}, ω_{L} = ∑_{r=1}^{k} dq_{r−1}^{i} ∧ dp_{i}^{r}, p_{i}^{r} = ∑_{s=0}^{k−r}(−1)^{s} \frac{d^{s}}{dt^{s}} (\frac{∂L}{∂q_{r+s}^{i}}).

Notice that p_{i}^{r} depends at most in q_{2k−r}, i.e., p_{i}^{r} = p_{i}^{r}(q_{0}, q_{1}, ⋯, q_{2k−r}), and we have (∂p_{i}^{r}/∂q_{2k−r}^{j}) = (−1)^{k−r}(∂^{2}L/∂q_{k}^{i}∂q_{k}^{j}). If we set θ_{r}^{i} = dq_{r}^{i} − q_{r+1}^{i}dt; 0 ≤ r ≤ k − 1, then we can writte θ_{L} = ∑_{r=1}^{k} p_{i}^{r} θ_{r−1}^{i} + Ldt, and so Ω_{L} = ∑_{r=1}^{k}{dp_{i}^{r} ∧ θ_{r−1}^{i} + p_{i}^{r} dθ_{r−1} + dL ∧ dt}, from which we deduces

i_{\frac{∂}{∂q^{j}_{s}}} Ω_{L} = −\frac{∂p_{i}^{r}}{∂q_{s}^{j}} θ_{r−1}^{i}, if k ≤ s ≤ 2k − 1

i_{\frac{∂}{∂q^{j}_{s}}} Ω_{L} = −\frac{∂p_{i}^{r}}{∂q_{s}^{j}} θ_{r−1}^{i} + dp_{j}^{s+1} + p_{j}^{s} dt − \frac{∂L}{∂q_{s}^{j}} dt , if 0 ≤ s ≤ k − 1 .

Proposition 2.1. Let ξ be a 2k-th order differential equation on J^{2k−1}(IR, Q) and Ω be a closed 2-form on J^{2k−1}(IR, Q) such that: (1) (Ω, dt) is a cosymplectic structure; (2) L_{ξ}Ω = 0 ; (3) (J^{k})^{∗}Ω = 0 . Then there exists a local Lagrangian L : J^{k}(IR, Q) ⟶ IR which is a solution of the inverse problem for ξ.

Proof. (J^{k})^{∗}Ω_{L} = 0 implies that Ω is semibasic with respect to the fibration ρ_{k−1}^{2k−1} : J^{2k−1}(IR, Q) ⟶ J^{k−1}(IR, Q). Then there exists a ρ_{k−1}^{2k−1} -semibasic 1-form α on J^{2k−1}(IR, Q) such that Ω = −dα. Hence α is locally expressed as follows: α = Ddt + ∑_{r=0}^{k−1} B_{r}^{i} (∂/∂q_{r+1}^{i}). Thus we obtain α(ξ) = D + ∑_{r=0}^{k−1} B_{r}^{i}q_{r+1}^{i}. Also, since L_{ξ} Ω = −dL_{ξ} α = 0, we deduce that there exists a locally defined function L on J^{2k−1}(IR, Q) such that L_{ξ} α = dL, from which we deduce, by applying to d/dt, ∂/∂q_{s}^{i}, 1 ≤ s ≤ 2k − 1, that (∂L/∂q_{s}^{i}) = 0 , k + 1 ≤ s ≤ 2k − 1, which imply that L is in fact locally defined on J^{k}(IR, Q) and

ξ(B_{0}^{i}) = \frac{∂L}{∂q_{0}^{i}}; B_{s}^{i} = \frac{∂L}{∂q_{s+1}^{i}} − ξ(B_{s+1}^{i}), 0 ≤ s ≤ k − 2; B_{k−1}^{i} = \frac{∂L}{∂q_{k}^{i}} .

Hence p_{i}^{r} = B_{r+1}^{i}, for all r, 1 ≤ r ≤ k−1. Therefore we have E_{L} = α(ξ)−D−L, Δ(L) = α − D dt. Define h = L − α(ξ). Then we have α_{L} = α + hdt. Thus Ω_{L} = Ω − dh ∧ dt and hence i_{ξ}Ω_{L} = i_{ξ}Ω − (ξh)dt + dh = −dh − (ξh)dt + dh = −(ξh)dt = 0, since i_{ξ}dα = dh implies ξh = i_{ξ}dh = i_{ξ}i_{ξ}dα = 0. We remark that L is a regular function since (Ω_{L}, dt) is a cosymplectic structure. Thus, we have ξ = ξ_{L}.

Corollary 2.2. A necessary and sufficient condition for a 2k-th order differential equation ξ on J^{2k−1}(IR, Q) be a regular Euler-Lagrange vector field is the existence of a closed 2-form on J^{2k−1}(IR, Q) like in Proposition 2.1.

Next, we shall extend the above result for almost s-tangent manifolds of order 2k − 1. First of all let us recall that from the natural identification J^{k}(IR, Q) ≅ IR × T^{k}Q (using the mapping j_{t}^{k}σ → (t, j_{0}^{k}τ), where τ(s) = σ(s + t)) we may transport all the geometric structures on T^{k}Q to J^{k}(IR, Q). For instance, the set of canonical tensor fields J_{r} of type (1,1) can be extended to J^{k}(IR, Q) (and also denoted by J_{r}), with local expression

J_{r} = ∑_{s=0}^{k−r} \frac{(r + s)!}{s!} \frac{∂}{∂q_{r+s}^{i}} ⊗ dq_{s}^{i}; 1 ≤ r ≤ k.

For r = 1 one has the canonical almost tangent structure J_{1} of order k of T^{k}Q Also, we have the family of vector fields C_{r}, locally given by

C_{r} = ∑_{s=1}^{k−r+1} \frac{(r + s − 1)!}{(s − 1)!} q_{s}^{i} \frac{∂}{∂q_{r+s−1}^{i}}; 1 ≤ r ≤ k,

(for k = 1 one has the canonical Liouville vector field C_{1}). We define the following (1, 1)-tensor fields:

J̄_{1} = J_{1} + \frac{∂}{∂t} ⊗ dt , J̃_{1} = J_{1} − C_{1} ⊗ dt ,

and one has: J̄_{r} = J̄_{1}^{r} and J̃_{r} = J̃_{1}^{r}, i.e.,

J̄_{r} = J_{r} + \frac{∂}{∂t} ⊗ dt, J̃_{r} = J_{r} − C_{r} ⊗ dt (J̃_{r} = J̄_{r} − (C_{r} + \frac{∂}{∂t}) ⊗ dt) .

Now, (J̄_{1}, dt, ∂/∂t) is an almost s-tangent structure on J^{k}(IR, Q) according to the following definition.

Definition 2.1. ([17])- Let V be an (k + 1)n + 1-dimensional manifold. If there is a triple (F̄, τ, T), where F̄ is a tensor field of type (1.1), τ is a 1-form and T a vector field on V such that (i) τ(T) = 1 ; (ii) F̄^{k+1} = τ ⊗ T ; (iii) rank F̄ = kn + 1, then we say that V is endowed with an almost s-tangent structure of order k. In such a case V is called an almost s-tangent manifold of order k.

In [17] an almost s-tangent structure (F̄, τ, T) of order k was interpreted as a Gstructure. Then (F̄, τ, T) is integrable as a G-structure if and only if there exist local coordinates (t, q_{0}^{i}, q_{1}^{i}, ⋯, q_{k}^{i}) around each point of V such that

F̄ = ∑_{r=0}^{k} (r + 1) \frac{∂}{∂q_{r+1}^{i}} ⊗ dq_{r}^{i} + \frac{∂}{∂t} ⊗ dt, τ = dt, T = \frac{∂}{∂t} .

(we refer [27] for an application of stable tangent bundles to the unified field theory). Hence the canonical almost s-tangent structure (J̄_{1}, dt, ∂/∂t) is integrable. Furthermore every almost s-tangent structure of order k is integrable if and only if is locally isomorphic to the canonical one.

We also recall that a vector field ξ on J^{k}(IR, Q) is said to be a (k + 1)-th order differential equation if J̄_{1}ξ = (∂/∂t) + C_{1}. We deduce that a (k + 1)-th order differential equation ξ on J^{k}(IR, 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}, q_{1}, ⋯, q_{k}).

Definition 2.2. A semispray of type r on J^{k}(IR, Q) is a vector field ξ_{r} verifying J̄_{r}(ξ_{r}) = (∂/∂t) + C_{r}.

Locally:

ξ_{r} = \frac{∂}{∂t} + q_{1}^{i} \frac{∂}{∂q_{0}^{i}} + q_{2}^{i} \frac{∂}{∂q_{1}^{i}} + ⋯ + q_{k−r+1}^{i} \frac{∂}{∂q_{k−r}^{i}} + ξ_{k−r+1}^{i} \frac{∂}{∂q_{k−r+1}^{i}} + ⋯ + ξ_{k}^{i} \frac{∂}{∂q_{k}^{i}} .

It is obvious that a (k + 1)-th differential equation on T^{k}Q is in fact a semispray of type 1. Locally the action of J_{r−1} on ξ_{r} is given by:

J_{r−1}(ξ_{r}) − C_{r−1} = \frac{k!}{(k − r + 1)!} [ξ_{k−r+1}^{i} − q_{k−r+2}^{i}]\frac{∂}{∂q_{k}^{i}} .

It is easy to see that a vector field ξ is a (k + 1)-th differential equation iff ⟨ξ, θ_{r}^{i}⟩ = 0 and ⟨ξ, dt⟩ = 1.

Definition 2.3. A dynamical connection on J^{k}(IR, Q) is a tensor field Γ of type (1, 1) on J^{k}(IR, Q) such that (i) J_{1}Γ = J̃_{1} ; (ii) J̃_{1}Γ = J̃_{1} ; (iii) ΓJ̃_{k} = −J̃_{k}; (iv) ΓJ_{k} = −J_{k}.

Then the local expressions of Γ are

Γ(\frac{∂}{∂t}) = − ∑_{r=1}^{k} q_{r}^{i} \frac{∂}{∂q_{r−1}^{i}} + Γ^{i} \frac{∂}{∂q_{k}^{i}} ,

Γ(\frac{∂}{∂q_{r}^{i}}) = \frac{∂}{∂q_{r}^{i}} + ∑_{r=1}^{k} (Γ^{r−1})_{j}^{i} \frac{∂}{∂q_{k}^{j}} , 0 ≤ r ≤ k − 1 , Γ(\frac{∂}{∂q_{k}^{i}}) = −\frac{∂}{∂q_{k}^{i}} .

The vector field ξ = Γ^{2}(∂/∂t) is locally expressed by

ξ = \frac{∂}{∂t} + ∑_{r=0}^{k−1} q_{r}^{i} \frac{∂}{∂q_{r−1}^{i}} + (Γ^{i} + ∑_{r=1}^{k} q_{r}^{j}(Γ^{r−1})_{j}^{i})\frac{∂}{∂q_{k}^{i}}

i.e. it is a (k + 1)-th order differential equation, called canonically associated to Γ. It is possible to show ([19]) that if ξ is a given (k + 1)-th order differential equation on J^{k}(IR, Q) then Γ defined by

Γ_{ξ} = −\frac{2}{k + 1}L_{ξ} J̃_{1} + \frac{k − 1}{k + 1}(Id − ξ ⊗ dt)

is a dynamical connection on J^{k}(IR, Q) whose associated (k + 1)-th order differential equation is precisely ξ. It follows that if L is a regular Lagrangian and ξ_{L} the Euler-Lagrange vector field, then there exists a dynamical connection Γ_{L} on ρ_{2k−2}^{2k−1} : J^{2k−1}(IR, Q) ⟶ J^{2k−2}(IR, Q) whose paths are solutions of the motion equations. This connection is given by Γ_{L} = −(1/k)L_{ξ_{L}} J̃_{1} + (k − 1/k)(Id − ξ_{L} ⊗ dt) and ξ̃ is the 2k-th order differential equation associated to Γ.

We show now how to extend Proposition 2.1 to an almost integrable s-tangent manifold of higher-order. Let X be a vector field on a manifold V endowed with an integrable almost s-tangent structure (F̄, τ, T) of order 2k − 1. We set C_{X} = F̄(X) − T , F = F̄ − T ⊗ τ , F̃ = F − C_{X} ⊗ τ.

In adapted coordinates (t, q_{0}^{i}, q_{1}^{i}, ⋯, q_{2k−1}^{i}) we have

F̃(\frac{∂}{∂t}) = −C_{X} ; F̃(\frac{∂}{∂q_{r}^{i}}) = (r + 1) \frac{∂}{∂q_{r+1}^{i}} , 0 ≤ r ≤ 2k − 2 ;

F̃(\frac{∂}{∂q_{2k−1}^{i}}) = 0 ; F (\frac{∂}{∂t}) = 0 ;

F (\frac{∂}{∂q_{r}^{i}}) = (r + 1) \frac{∂}{∂q_{r+1}^{i}} , 0 ≤ r ≤ 2k − 2; F (\frac{∂}{∂q_{2k−1}^{i}}) = 0 ; (2.2)

Now, let us define the following operator:

Γ_{X} = −\frac{1}{k} L_{X} F̃ + \frac{k − 1}{k}(Id − X ⊗ τ) .

We suppose that Γ_{X} behaves as a dynamical connection in the fibration ρ_{2k−2}^{2k−1} : J^{2k−1}(IR, Q) → J^{2k−2}(IR, Q), i.e. ,

(i) F Γ_{X} = F̃; (ii) F̃Γ_{X} = F̃; (iii) Γ_{X} F̃^{2k−1} = −F̃^{2k−1}; (iv) Γ_{X}F^{2k−1} = −F^{2k−1}, (2.3)

as well as

(v) F [X, T] = 0. (2.4)

Let us express X locally as X = u (∂/∂t) + ∑_{r=0}^{2k−1} X_{r}^{i} (∂/∂q_{r}^{i}), 1 ≤ i ≤ n. Then, from (2.2) one has C_{X} = (u − 1) (∂/∂t) + ∑_{r=0}^{2k−2} (r + 1) X_{r}^{i} (∂/∂q_{r+1}^{i})

From the conditions (i - iv) one obtains

u − 1 = 0 , \frac{∂X_{0}^{i}}{∂q_{1}^{j}} = δ_{j}^{i} ;

\frac{∂X_{0}^{i}}{∂q_{s+1}^{j}} = 0 , 1 ≤ s ≤ 2k − 2 , \frac{∂X_{s}^{i}}{∂q_{2k−11}^{j}} = 0 , 0 ≤ s ≤ 2k − 3 ;

\frac{∂X_{2k−2}^{i}}{∂q_{2k−1}^{j}} = δ_{j}^{i} ;

\frac{∂X_{s}^{i}}{∂q_{r}^{j}} = \frac{r + 1}{s + 1} \frac{∂X_{s+1}^{i}}{∂q_{r+1}^{j}} , 0 ≤ s ≤ 2k − 3 , 0 ≤ r ≤ 2k − 2 , r ≠ s + 1 ;

\frac{∂X_{s}^{i}}{∂q_{s+1}^{j}} = \frac{∂X_{s+1}^{i}}{∂q_{s+2}^{j}} , i ≠ j;

−\frac{1}{s + 1} \frac{∂X_{s}^{i}}{∂q_{s+1}^{i}} + \frac{1}{s + 2} \frac{∂X_{s+1}^{i}}{∂q_{s+2}^{j}} = 1 , 0 ≤ s ≤ 2k − 3 .

From these equalities we deduce that:

u = 1 ; \frac{∂X_{s}^{i}}{∂q_{s+1}^{j}} = δ_{j}^{i} , 0 ≤ s ≤ 2k − 2 ;

\frac{∂X_{s}^{i}}{∂q_{r}^{j}} = 0 , 0 ≤ s ≤ 2k − 3 , s + 2 ≤ r;

\frac{s + 1}{r + 1} \frac{∂X_{s}^{i}}{∂q_{r}^{j}} = \frac{∂X_{s+1}^{i}}{∂q_{r+1}^{j}} , r ≠ s + 1 , 0 ≤ s ≤ 2k − 3 , 0 ≤ r ≤ 2k − 2 ;

and from (v) we obtain (∂X_{s}^{i}/∂t) = 0, 0 ≤ s ≤ 2k − 2. Thus the functions X_{s}^{i} take the following form: X_{s}^{i} = q_{s+1}^{i} + g_{s}(q_{0}^{j}, q_{1}^{j}, ⋯, q_{s}^{j}). For instance: X_{0}^{i} = q_{1}^{i} + g_{0}(q_{0}^{j}) , X_{1}^{i} = q_{2}^{i} +g_{1}(q_{0}^{j}, q_{1}^{j}), and so on. Define a new coordinate system (t̄, q̄_{0}^{i}, q̄_{1}^{i}, ⋯, q̄_{2k−1}^{i}) by setting t̄ = t , q̄_{0}^{i} = q_{0}^{i} , q̄_{r}^{i} = ∑_{s=0}^{r−1} X_{s}^{j} (∂q̄_{r−1}^{i}/∂q_{s}^{j}) , 1 ≤ r ≤ 2k − 1, where X_{s}^{j} = q_{s+1}^{j} + g_{s}(q_{0}, q_{1}, ⋯, q_{s}). A direct computation shows that (t̄, q̄_{0}^{i}, q̄_{1}^{i}, ⋯, q̄_{2k−1}^{i}) is such that the vector field X reads locally as a 2k-th order differential equation, say

X = \frac{∂}{∂t̄} + q̄_{1}^{i} \frac{∂}{∂q̄_{0}^{i}} + q̄_{2}^{i} \frac{∂}{∂q̄_{1}^{i}} + ⋯ + q̄_{2k−1}^{i} \frac{∂}{∂q̄_{2k−2}^{i}} + ξ^{i} \frac{∂}{∂q̄_{2k−1}^{i}} ,

and such coordinate system is also adapted to the integrable almost s-tangent structure (F̄, T, τ).

Definition 2.4. A vector field X verifying the above (i)−(v) conditions is called a nonautonomous regular Lagrangian dynamical system of order 2k on an integrable almost s-tangent manifold V of order 2k − 1 if there exists a closed 2-form Ω such that:

(1) (Ω, τ) is a cosymplectic structure; (2) L_{X}Ω = 0 ; (3) (F^{k})^{∗}Ω = 0.

As the arguments used in Proposition 2.1 (and the Corollary 2.2) are local, we use a similar procedure to show the following

Theorem 2.3. Let X be a non-autonomous regular Lagrangian dynamical system of order 2k on an integrable almost s-tangent manifold V of order 2k −1. Then there exists a local function L : U ⊂ V ⟶ IR such that Ω = Ω_{L} and X is the Euler-Lagrange vector field associated to L.

3. Reduction of Singular Lagrangians of Higher-order

As we know the singularity of L implies that Ω_{L} is no longer of maximal rank. Also, (Ω_{L}, dt) is not in general a precosymplectic structure. Nevertheless, we need some weak regularity condition. In the autonomous case, the geometrical structure which support the Lagrangian dynamics is a presymplectic structure. Thus, for each t ∈ IR let us define the autonomous Lagrangian L_{t} : T^{k}Q ⟶ IR, L_{t}(x) = L(t, x). Suppose that the rank of ω_{L_{t}} is constant, say 2a and independent of t. If we set ϖ_{L} = ω_{L} − Δ(∂L/∂t) ∧ dt then we have Ω_{L} = ϖ_{L} + (Δ(∂L/∂t) + dE_{L}) ∧ dt. We see that that the restriction of ϖ_{L} to each level {t} × T^{2k−1}Q is precisely ω_{L_{t}}. Then our assumption is equivalent to say that ϖ_{L} is of constant rank 2a. Notice that ϖ_{L} is not presymplectic since it is not closed. Then we have

Ω_{L}^{b} = ϖ_{L}^{b} + bϖ_{L}^{b−1} ∧ (Δ(\frac{∂L}{∂t}) + dE_{L}) ∧ dt , ∀b ≥ 1.

Since ϖ_{L}^{a} ≠ 0, ϖ_{L}^{a+1} = 0, we obtain (Ω_{L})^{a} ∧ dt ≠ 0, (Ω_{L})^{a+1} ∧ dt = 0, (Ω_{L})^{a+2} = 0, and so 2a ≤ rank Ω_{L} ≤ 2a+2. According to [16] we know that the rank of Ω_{L} at a point x ∈ J^{2k−1}(IR, Q) is 2a if and only if there exists a tangent vector ξ ∈ T_{x}(J^{2k−1}(IR, Q)) such that i_{ξ}(Ω_{L})_{x} = 0, i_{ξ}(dt)_{x} = 1.

If L : J^{k}(IR, Q) ⟶ IR is a degenerate Lagrangian, then the dynamical equations (1.1) will not posses a globally defined solution in general, and even if it exists it will not be unique. However, we confine ourselves to those Lagrangians for which (1.1) posseses a global solution (see the introduction).

Definition 3.1. A degenerate non-autonomous Lagrangian L : J^{k}(IR, Q) ⟶ IR admits a global dynamics if there exists a globally defined vector field X on J^{2k−1}(IR, Q) satisfying i_{X}Ω_{L} = 0, i_{X}dt = 1.

In such a case, Ω_{L} has constant rank 2a and further (Ω_{L}, dt) is a precosymplectic structure on J^{2k−1}(IR, Q) (of rank 2a). The converse is also true.

Definition 3.2. The gauge distribution K of L is defined by K = Ker Ω_{L} ∩ Ker dt.

Proposition 3.1. If L : J^{k}(IR, Q) ⟶ IR admits a global dynamics and J_{r}(K) = K ∩ Im J_{r}, for all r, 1 ≤ r ≤ 2k − 1, then there exists a 2k-th order differential equation ξ such that i_{ξ}Ω_{L} = 0, i_{ξ}dt = 1.

Proof. Suppose that X is a vector field on J^{2k−1}(IR, Q) such that i_{X} Ω_{L} = 0, i_{X} dt = 1. Then X is locally expressed by X = (∂/∂t) + A_{0}^{i}(∂/∂q_{0}^{i}) + ⋯ + A_{2k−1}^{i} (∂/∂q_{2k−1}^{i}) and we have

(i_{X}Ω_{L})(\frac{∂}{∂q_{2k−1}^{j}}) = \frac{∂p_{i}^{1}}{∂q_{2k−1}^{j}} (A_{0}^{i} − q_{1}^{i}) ,

which implies (∂p_{i}^{k}/∂q_{2k−1}^{j}) (A_{0}^{i} − q_{1}^{i}) = 0. On the other hand we have

J_{2k−1}X − C_{2k−1} = (2k − 1)! (A_{0}^{i} − q_{1}^{i})\frac{∂}{∂q_{2k−1}^{i}} .

Hence

i_{J_{2k−1}−C_{2k−1}} Ω_{L} = (2k − 1)!(A_{0}^{i} − q_{1}^{i})i_{\frac{∂}{∂q{i \atop 2k−1}}} Ω_{L} = −(2k − 1)!(A_{0}^{i} − q_{1}^{i})\frac{∂p_{i}^{1}}{∂q_{2k−1}^{j}} = 0 .

Thus we deduce that J_{2k−1}X −C_{2k−1} ∈ Ker Ω_{L}∩ Ker dt = K. Also, J_{2k−1}X −C_{2k−1} ∈ Im J_{2k−1} = J_{2k−1}(K). Then there exists a vector field Y ∈ K such that J_{2k−1}X − C_{2k−1} = J_{2k−1}Y. If we put ξ_{2k−1} = X − Y we have J_{2k−1}ξ_{2k−1} = (∂/∂t) + C_{2k−1}, i.e., ξ_{2k−1} is a semispray of type 2k − 1 and i_{ξ_{2k−1}} Ω_{L} = 0, i_{ξ_{2k−1}}dt = 1.

Next we proceed by induction. Suppose that there exists a semispray of type r, k+1 ≤ r ≤ 2k − 1 such that i_{ξ_{r}} Ω_{L} = 0, i_{ξ_{r}} dt = 1. Then ξ_{r} is locally expressed by

ξ_{r} = \frac{∂}{∂t} + q_{1}^{i} \frac{∂}{∂q_{0}^{i}} + ⋯ + q_{2k−r}^{i} \frac{∂}{∂q_{2k−r−1}^{i}} + ⋯ + B_{2k−r}^{i} \frac{∂}{∂q_{2k−r}^{i}} + ⋯ + B_{2k−1}^{i} \frac{∂}{∂q_{2k−1}^{i}} .

We obtain

J_{r−1}ξ_{r} − C_{r−1} = \frac{(2k − 1)!}{(2k − r)!} (B_{2k−r}^{i} − q_{2k−r+1}^{i})\frac{∂}{∂q_{2k−1}^{i}} ,

from which we deduce

i_{J_{r−1}ξ_{r}−C_{r−1}} Ω_{L} = −\frac{(2k − 1)!}{(2k − r)!} (B_{2k−r}^{i} − q_{2k−r+1}^{i})\frac{∂p_{j}^{1}}{∂q_{2k−1}^{i}} θ_{0}^{j} .

On the other hand, we have

(i_{ξ_{r}} Ω_{L})(\frac{∂}{∂q_{r−1}^{j}}) = ∑_{s=1}^{k} \frac{∂p_{i}^{s}}{∂q_{r−1}^{j}} θ_{s−1}^{i}(ξ_{r})

= \frac{∂p_{i}^{2k−r+1}}{∂q_{r−1}^{j}} θ_{2k−r}^{i}(ξ_{r}) = (−1)^{2k−r+1} \frac{∂p_{i}^{1}}{∂q_{2k−1}^{j}} (B_{2k−r}^{i} − q_{2k−r+1}^{i}) ,

Thus, J_{r−1}ξ_{r} − C_{r−1} ∈ K ∩ Im J_{r−1} = J_{r−1}(K). Then there exists a vector field Y ∈ K such that J_{r−1}ξ_{r} − C_{r−1} = J_{r−1}Y. If we put ξ_{r−1} = ξ_{r} − Y we deduce that ξ_{r−1} is a semispray of type r − 1 and i_{ξ_{r−1}} Ω_{L} = 0, i_{ξ_{r−1}}dt = 1. Consequently, there exists a semispray ξ_{k−1} of type k − 1 on J^{2k−1}(IR, Q) such that i_{ξ_{k−1}} Ω_{L} = 0, i_{ξ_{k−1}}dt = 1.

Next, we shall apply another time an induction on the type of semispray. Suppose that there exists a semispray ξ_{r}, 1 ≤ r ≤ k − 1 of type r on J^{2k−1}(IR, Q) such that i_{ξ_{r}} Ω_{L} = 0, i_{ξ_{r}}dt = 1. Then ξ_{r} is locally expressed by

ξ_{r} = \frac{∂}{∂t} + q_{1}^{i} \frac{∂}{∂q_{0}^{i}} + ⋯ + q_{2k−r}^{i} \frac{∂}{∂q_{2k−r−1}^{i}} + ⋯ + D_{2k−r}^{i} \frac{∂}{∂q_{2k−r}^{i}} + ⋯ + D_{2k−1}^{i} \frac{∂}{∂q_{2k−1}^{i}} .

and we obtain

J_{r−1}ξ_{r} − C_{r−1} = \frac{(2k − 1)!}{(2k − r)!} (D_{2k−r}^{i} − q_{2k−r+1}^{i})\frac{∂}{∂q_{2k−1}^{i}} ,

From i_{ξ_{r}} Ω_{L} = 0, we have

(i_{ξ_{r}} Ω_{L})(\frac{∂}{∂q_{r−1}^{j}}) = (dp_{j}^{r} + p_{j}^{r−1}dt − \frac{∂L}{∂q_{r−1}^{j}} dt)(ξ_{r})

= ξ_{r}(p_{j}^{r}) + p_{j}^{r−1} − \frac{∂L}{∂q_{r−1}^{j}}

= (D_{2k−r}^{i} − q_{2k−r+1}^{i})\frac{∂p_{j}^{r}}{∂q_{2k−r}^{i}} = (−1)^{2k−r+1}(D_{2k−r}^{i} − q_{2k−r+1}^{i})\frac{∂p_{j}^{1}}{∂q_{2k−1}^{i}} .

Hence J_{r−1}ξ_{r} − C_{r−1} ∈ K ∩ Im J_{r−1} = J_{r−1}K and there exists a vector field Y ∈ K such that J_{r−1}ξ_{r} − C_{r−1} = J_{r−1}Y. If we put ξ_{r−1} = ξ_{r} − Y we deduce that ξ_{r−1} is a semispray of type r − 1 and i_{ξ_{r−1}} Ω_{L} = 0, i_{ξ_{r−1}}dt = 1. Consequently, there exists a semispray ξ = ξ_{1} of type 1 on J^{2k−1}(IR, Q) such that i_{ξ} Ω_{L} = 0, i_{ξ}dt = 1.

In what follows we shall consider a degenerate non-autonomous Lagrangian L satisfying the following conditions:

(D1) ϖ_{L} has constant rank, say 2a; (D2) L admits a global dynamics; (D3) J_{r}(K) = K ∩ Im J_{r}, for all r, 1 ≤ r ≤ 2k − 1; (D4) the foliation defined by K = Ker Ω_{L} ∩ Ker dt is a fibration.

The assumptions (D1) and (D2) are equivalent to say that (Ω_{L}, dt) is a precosymplectic structure. The assumption (D3) implies that there exists a 2k-th order differential equation ξ such that i_{ξ} Ω_{L} = 0 , i_{ξ} dt = 1. Further, (D3) implies that dim K = 2k dim (K ∩ Im J_{2k−1}) = 2kd, where d = dim K ∩ Im J_{2k−1}. We denote by π_{L} : J^{2k−1}(IR, Q) ⟶ J^{2k−1}(IR, Q)/K the canonical projection. For what follows we need the following result (see [3] for a proof).

Proposition 3.2. Let D be an integrable distribution on a manifold M such that the foliation defined by D is a fibration and let M/D be the leaf space.

(1) A vector field X on M projects onto M/D iff [X, Z] ∈ D, for all Z ∈ D.

(2) A p-form ω projects onto M/D iff i_{Z}ω = 0, L_{Z}ω = 0, for all Z ∈ D. When ω is closed, then it projects onto M/D iff i_{Z}ω = 0, for all Z ∈ D.

(3) A tensor field R of type (1,1) on M projects onto M/D iff R(D) ⊂ D and Im (L_{Z}R) ⊂ D, for all Z ∈ D.

In our case, it is clear that, under the given assumptions (D1)-(D4), Ω_{L} and dt project onto (J^{2k−1}(IR, Q))/K, i.e. there exist a closed 2-form Ω̃_{L} and a closed 1-form τ on (J^{2k−1}(IR, Q))/K such that π_{L}^{∗}Ω̃_{L} = Ω_{L}, π_{L}^{∗}τ = dt. We set J̄^{2k−1}(IR, Q) = J^{2k−1}(IR, Q)/K. Then we deduce that J̄^{2k−1}(IR, Q) has dimension 2kc + 1, where c = n − d, a = kc, and further Ω̃_{L}^{a} ∧ τ ≠ 0 , Ω̃_{L}^{a+1} = 0. Thus (Ω̃_{L}, τ) is a cosymplectic structure on J̄^{2k−1}(IR, Q).

Proposition 3.3. (1) J̄_{1} projects onto J̄^{2k−1}(IR, Q) iff (L_{Z} J̄_{1})(K) ⊂ K, for all Z ∈ K;

(2) ∂/∂t projects onto J̄^{2k−1}(IR, Q) iff [∂/∂t, Z] ∈ K, for all Z ∈ K.

Theorem 3.4. Suppose that J̄_{1} and ∂/∂t project onto J̄^{2k−1}(IR, Q). Then (F̄, τ, T) is an integrable almost s-tangent structure of order 2k − 1 on J̄^{2k−1}(IR, Q), where F̄, and T are the projections of J̄_{1} and ∂/∂t, respectively.

Proposition 3.5. (1) Each vector field X such that i_{X}Ω_{L} = 0, i_{X}dt = 1 projects onto J̄^{2k−1}(IR, Q). (2) If J̄_{1} is projectable onto F̄, then J̄_{r} is also projectable, for all r, 2 ≤ r ≤ 2r − 1 and further its projection is F̄^{r}. (3) If J̄_{1} and ∂/∂t are projectable, then the Liouville vector field C_{1} on J^{2k−1}(IR, Q) is also projectable. (4) Hence if J̄_{1} and ∂/∂t are projectable onto F̄ and T, respectively, then J̃_{1} is projectable and its projection is precisely F̃ = F − C ⊗ τ, where C denotes the projection of C_{1}.

Suppose that J̄_{1} and ∂/∂t project onto J̄^{2k−1}(IR, Q) and let (F̄, τ, T) be the projected integrable almost s-tangent structure of order 2k − 1 on J̄^{2k−1}(IR, Q). From Proposition 3.1 one has a 2k-th order differential equation ξ and Proposition 3.5 shows that ξ also projects onto ξ̄. Therefore we have Cξ̄ = C.

Define a tensor field of type (1, 1) on J̄^{2k−1}(IR, Q) by

Γξ̄ = −\frac{1}{k} Lξ̄F̃ + \frac{k − 1}{k}(Id − ξ̄⊗ τ) .

Proposition 3.6. Γ_{ξ} projects onto Γξ̄. Further, Γξ̄ satisfies the following properties:

(1) F Γξ̄ = F̃; (2) F̃Γξ̄ = F̃; (3) Γξ̄F̃^{2k−1} = −F̃^{2k−1}; (4) Γξ̄F^{2k−1} = −F^{2k−1};

(5) Lξ̄Ω̃_{L} = 0; (6) (F^{k})^{∗}Ω̃_{L} = 0; (7) F [ξ̄, T] = 0.

The next Theorem can be shown using the results stated above and the properties about equivalence of Lagrangians (for the equivalence problem we remit to the Appendix).

Theorem 3.7. ξ̄ is a non-autonomous regular Lagrangian dynamical system of order 2k on J̄^{2k−1}(IR, Q). Then there exists a local function L̄ : U ⊂ J̄^{2k−1}(IR, Q) ⟶ IR such that Ω = ΩL̄ and ξ̄ is the Euler-Lagrange vector field associated to L̄. If we set L̃ = L̄∘π_{L}, then L̃ and L are geometrically equivalent (that is, the corresponding Poincaré-Cartan 2form are the same) and hence equivalent (their gauge distributions are equal) and gauge equivalent, i.e, there exists a closed 1-form α on J^{k−1}(IR, Q) such that L = L̃ + α̂.

Next we characterize a class of singular Lagrangians giving a condition for the projections of J̄_{1} and ∂/∂t. Denote by j_{k} : J^{k}(IR, Q) ≅ IR × T^{k}Q ⟶ T^{k}(IR × Q) ≅ IR^{k+1} × T^{k}Q the canonical injection; in local coordinates we have j_{k}(t, q_{0}^{i}, ⋯, q_{k}^{i}) = (t, q_{0}^{i}, 0, q_{1}^{i}, ⋯, 0, q_{k}^{i}). Let s be a section of π : IR × T Q ⟶ J^{1}(IR, Q). If s is locally given by s(t, q_{0}^{i}) = (t, q_{0}^{i}, s^{i}(t, q_{0})), then ŝ = j_{k} ∘ s is a vector field on IR × Q locally given by ŝ(t, q_{0}^{i}) = s^{i}(∂/∂q_{0}^{i}) i.e. , ŝ is a time-dependent vector field on Q. Thus we deduce that the restriction of the r-th lift ŝ^{r} of ŝ to T^{k}(IR × Q) is not tangent in general to the submanifold J^{k}(IR, Q). Consider the natural identification T^{k}(IR×Q) ≅ IR^{k+1}×T^{k}Q locally given by (t, q_{0}^{i}, τ_{1}, q_{0}^{i}, ⋯, τ_{k}, q_{k}^{i}) = (t, τ_{1}, ⋯, τ_{k}, q_{0}^{i}, q_{1}^{i}, ⋯, q_{k}^{i}), in such a way that t and τ_{1}, ⋯, τ_{k} are global coordinates on T^{k}(IR × Q). We set s̃^{r} = ŝ^{r} − ∑_{a=1}^{k} dτ_{a}(ŝ^{r})(∂/∂τ_{a}). Then s̃^{r} is tangent to J^{k}(IR, Q) and it is called the r-th lift of s to J^{k}(IR, Q).

Now, let D be a distribution on IR × Q which admits a local basis in a neighbourhood of each point of the form {ŝ_{1}, ⋯, ŝ_{a}}, where {s_{1}, ⋯, s_{a}} are a linearly independent local sections of IR × T Q ⟶ IR × Q, 1 ≤ a ≤ n + 1. Hence {ŝ_{1}, ..., ŝ_{a}} are linearly independent vector fields on IR × Q. We may construct a new (k + 1)a -dimensional distribution D̃^{k} on J^{k}(IR, Q) as follows: D̃^{k} is locally spanned by the local bases of the form {s̃_{1}^{r}, ⋯, s̃_{a}^{r}, 0 ≤ r ≤ k}.

Definition 3.3. A distribution K on J^{k}(IR, Q) is said to be an s-tangent distribution if there exists a distribution D on IR × Q such that K = D̃^{k} and [∂/∂t, D] ⊂ D.

Theorem 3.8. Suppose that L : J^{k}(IR, Q) ⟶ IR is a non-autonomous singular Lagrangian such that its gauge distribution K = Ker Ω_{L} ∩ Ker dt is an s-tangent distribution on J^{2k−1}(IR, Q). Then J̄_{1}, C_{1} and ∂/∂t project onto J̄^{2k−1}(IR, Q).

Appendix A. Auxiliary Results

A.1. r-lifts of vector fields. If X is a vector field on a manifold Q, then X may be lifted to a vector field X^{r}, 0 ≤ r ≤ k to T^{k}Q; X^{r} is called the r-th lift of X. In particular, X^{k} is the complete lift of X. If ϕ_{t} is the flow generated by X then T^{k}ϕ_{t} is the flow generated by X^{k}. Then we have X^{r} = J_{k−r}X^{k}. Further we see that X^{r}, 0 ≤ r ≤ k − 1 is vertical with respect to the canonical fibration T^{k}Q ⟶ T^{r−1}Q.

A.2. Paths of a dynamical connection. From the equations given in definition 2.3 of a dynamical connection we deduce that Γ^{3}−Γ = 0, i.e. Γ is an f-structure on J^{k}(IR, Q) (see [20]). The associated complementary projectors are m = Id−Γ^{2} and l = Γ^{2}. If we set N = l(J^{k}(IR, Q)) and M = m(J^{k}(IR, Q)), then they are complementary distributions. We deduce that N (resp. M) is (k + 1)n-dimensional (resp. one-dimensional), locally spanned by {∂/∂q_{r}^{i}, 0 ≤ r ≤ k} (resp. globally spanned by m(∂/∂t)). If we now set H = (1/2)(Id+Γ)(N), V = (1/2)(Id−Γ)(N), then H and V are complementary distributions in N, that is N = H ⊕ V. Moreover the distribution V is locally spanned by {∂/∂q_{k}^{i}} and then it is precisely the vertical distribution of the fibration ρ_{k−1}^{k} : J^{k}(IR, Q) ⟶ J^{k−1}(IR, Q). Hence Γ defines in fact a connection in ρ_{k−1}^{k} : J^{k}(IR, Q) ⟶ J^{k−1}(IR, Q) which horizontal distribution is M ⊕ H.

A path of a dynamical connection Γ on J^{k}(IR, Q) is a curve s in Q such that the k-jet s^{k} of s is a horizontal curve in J^{k}(IR, Q), that is the k-jet s^{k}(t) belongs to (M ⊕ H)_{s^{k}(t)} for all t. It is possible to show that a dynamical connection and its associated semispray have the same paths.

A.3. Equivalence of higher-order Lagrangians. Let α be a 1-form on J^{k−1}(IR, Q). Then we may define a function α̂ on J^{k}(IR, Q) by α̂(j_{0}^{k}σ) = ⟨T_{k}(j_{0}^{k}σ), α(j_{0}^{k−1}σ)⟩. If α = Adt + ∑_{r=0}^{k−1} (A_{r})_{i} dq_{r}^{i} , then we have α̂(t, q_{0}^{i}, q_{1}^{i}, ⋯, q_{k}^{i}) = A + ∑_{r=0}^{k−1} (A_{r})_{i} q_{r+1}^{i}. Notice that Ω_{L} is semibasic with respect to the fibration ρ_{k−1}^{2k−1} : J^{2k−1}(IR, Q) ⟶ J^{k−1}(IR, Q).

We propose several kinds of equivalence of regular non-autonomous higher order Lagrangians. For those having the same dynamics:

Definition A.1. We say that two regular non-autonomous Lagrangians L_{1} and L_{2} are equivalent if ξ_{L_{1}} = ξ_{L_{2}}.

For those determinating the same cosymplectic structure:

Definition A.2. We say that two regular non-autonomous Lagrangians L_{1} and L_{2} are geometrically equivalent if Ω_{L_{1}} = Ω_{L_{2}}.

From the above Definitions one has: If L_{1} and L_{2} are geometrically equivalent then they are equivalent.

Since Ω_{aL_{1}+bL_{2}} = aΩ_{L_{1}} + bΩ_{L_{2}}, for all a, b ∈ IR, we deduce that L_{1} and L_{2} are geometrically equivalent iff Ω_{L} = 0, where L = L_{1} − L_{2}. Also, it is not hard to show the following

Proposition A.1. A Lagrangian L : J^{k}(IR, Q) ⟶ IR is such that Ω_{L} = 0 iff there exists a closed 1-form α on J^{k−1}(IR, Q) such that L = α̂.

Corollary A.2. Two regular non-autonomous Lagrangians L_{1} and L_{2} are geometrically equivalent iff L_{2} = L_{1} + α̂, where α is a closed 1-form on J^{k−1}(IR, Q).

Definition A.3. Two regular non-autonomous Lagrangians L_{1}, L_{2} : J^{k}(IR, Q) ⟶ IR are said to be gauge equivalent if L_{2} = L_{1} +α̂, where α is a closed 1-form on J^{k−1}(IR, Q).

According to the Corollary, we deduce that two regular non-autonomous Lagrangians L_{1} and L_{2} are gauge equivalent iff they are geometrically equivalent.

Now, we consider degenerate non-autonomous Lagrangians. We introduce the following definitions.

Definition A.4. Let L_{1}, L_{2} : J^{k}(IR, Q) ⟶ IR be two degenerate non-autonomous Lagrangians. We say that L_{1} and L_{2} are :

1. equivalent if their gauge distributions are equal: Ker Ω_{L_{1}} ∩ Ker dt = Ker Ω_{L_{2}} ∩ Ker dt.

2. geometrically equivalent if Ω_{L_{1}} = Ω_{L_{2}}.

3. gauge equivalent if L_{2} = L_{1} + α̂, where α is a closed 1-form on J^{k−1}(IR, Q).

It is clear that, if L_{1} and L_{2} are regular, these definitions coincide with the previous ones. Since Corollary A.2 holds true, we deduce the following properties:

(1) If L_{1} and L_{2} are geometrically equivalent, then they are equivalent.

(2) L_{1} and L_{2} are geometrically equivalent iff they are gauge equivalent.

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(Manuel de León) Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain.

E-mail address, M. de León: CEEML02@CC.CSIC.ES

(M. H. Mello) Departamento de Matemática Aplicada, Instituto de Matematica, Universidade Federal Fluminense, 24.020-300 Niteroi, RJ, Brazil.

(Paulo R. Rodrigues) Departamento de Geometria, Instituto de Matemática, Universidade Federal Fluminense, 24.020-300 Niterói, RJ, Brazil.

E-mail address, P. R. Rodrigues: GPMPRR@BRUFF.BITNET

Received 29 October 1992