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A new setting for higher order Lagrangians in the time dependent case.

1. INTRODUCTION

The aim of this paper is to construct a general setting of time dependent higher order spaces, that allows to consider some new classes of time dependent Lagrangians. Most authors study the time dependent case considering the product manifold M x R or M x I, where I [subset] R is an interval, when the coordinates on M can be taken independently of the real coordinate t [member of] R or [member of] I. We study here a more case, replacing M x I by the total space [M.sub.I] of a fibered manifold [M.sub.I] [right arrow] T. It allows to consider also fibered manifolds that are not locally trivial ones. This fibered manifold lifts to a higher order manifold [T.sup.k][M.sub.I], that can play the role of a higher order tangent space. It allows to consider some connections and a canonical k-order semispray, as in the time independent case (see [7] and [13]).

2. THE AFFINE SETTING OF THE TIME DEPENDENT CASE, USING FIBERED MANIFOLDS

A fibered manifold is a surjective submersion E [??] M. An affine bundle E [??] M is a fibered manifold which the change rules of the local coordinates on E have the form

[x.sup.i'] = [x.sup.i']([x.sup.J]), [y.sup.[alpha]'] = [g.sup.[alpha]'.sub.[alpha]]([x.sup.j])[y.sup.[alpha]] + [v.sup.[alpha]]'([x.sup.j]), (2.1)

where the coordinates ([x.sup.i]) are on the base M and the coordinates ([y.sup.[alpha]]) are on the fibers. An affine section in the bundle E is a differentiable map M [??] E such that [pi] o s = [id.sub.M]. The local components ([s.sup.[alpha]]) of s change according to the rule [s.sup.[alpha]]' ([x.sup.i]') = [g.sup.[alpha].sub.[alpha]]' ([x.sup.j]) [s.sup.[alpha]] ([x.sup.j]) + [v.sup.[alpha]]'([x.sup.j]). We denote by [??] [??] M the canonically vector bundle associated with the affine bundle E [??] M. More precisely, using local coordinates, the coordinates change on [??] following the rules [x.sup.i]' = [x.sup.i]' ([x.sup.j]), [z.sup.[alpha]]' = [g.sup.[alpha].sub.[alpha]] ([x.sup.j])[z.sup.[alpha]], when the coordinates on E change according the formulas (2.1).

Every vector bundle is an affine bundle, called a central affine bundle. In this case [v.sup.[alpha]] ([x.sup.j]) = 0.

In the time dependent case we consider a particular fibered manifold [M.sub.I] [??] I, where I is a one dimensional manifold (i.e., according to the classification of one dimensional manifolds, it is an open real interval I [subset] R or a circle [S.sup.1]); for sake of simplicity we take I = R. It is possible that [M.sub.I] be not a local trivial fibration (or a bundle); for example, [M.sub.I] = [R.sup.2]\{(x,0) : x < 0} and [pi](x, y) = x. If M is a manifold, then [M.sub.I] can be a bundle [M.sub.I] [right arrow] I, with a typical fiber M, for example the trivial bundle M x I [right arrow] I.

Considering the natural projections T[M.sub.I] [??] [M.sub.I] and T[M.sub.I] [??] TI = I x R, then for every z [member of] [M.sub.I], [pi](z) = t one define [T.sub.z][M.sub.I] = [T.sub.z][M.sub.I] [intersection] [[pi].sup.-1.sub.2], (t,1). Then the natural projection [[pi].sub.I] : T[M.sub.I] [right arrow] T[M.sub.I] is the projection of an affine bundle. Considering local coordinates [{[x.sup.i],t}.sub.i=1,m] on which change according to the rules [x.sup.i'] = [x.sup.i'] {[x.sup.i], t), t' = t. The local coordinates induced on [(T[M.sub.I]).sub.t] and TM change according to the rules [x.sup.i'] = [x.sup.i'] ([x.sup.i],t), [y.sup.i'] = [y.sup.i][partial derivative][x.sup.i']/[partial derivative]t ([x.sup.i], t) [+ or -]. These formulas prove the following interpretation.

Proposition 2.1. The affine bundles T[M.sub.I] [right arrow] [M.sub.I] and [J.sup.1][pi] [right arrow] [M.sub.I] are isomorphic.

It is the classical setting used, for example, in [2, Sect. 33]. We say that [[pi].sub.I] : [J.sup.1][pi] = T[M.sub.I] [right arrow] T[M.sub.I] is the t-tangent bundle of [M.sub.I].

In the case when [M.sub.I] = M x I, where M is a manifold, some local coordinates on M which change according to the rules [x.sup.i'] = [x.sup.i'] ([x.sup.i]) induce a local changes of coordinates on T[M.sub.I] given by, [x.sup.i'] = [x.sup.i'] ([x.sup.i']), t' = t, thus [pi]: T[M.sub.I] [right arrow] T[M.sub.I] is a local vector bundle. In this case we say that M x I [right arrow] M is the trivial time bundle of the manifold M. The t-tangent bundle T[M.sub.I] is in this case the vector bundle [[pi].sup.*.sub.0]TM, where [M.sub.I] = M x I [??] M is the natural projection.

In the general case, the t-tangent bundle T[M.sub.I] is an affine bundle. Thus we can consider its affine dual bundle, that is an affine bundle T[dagger][M.sub.I] [right arrow] T[M.sub.I] with one dimensional fiber. The manifold [T.sup.*][M.sub.I] is called in [2, Sect. 33] the extended phase space; [T.sup.*][M.sub.I] [congruent to] [M.sub.I].

Proposition 2.2. The total space of the cotangent bundle [T.sup.*][M.sub.I] is canonically diffeomorphic with the total space [T.sup.[dagger]][M.sub.I] of the affine dual of T[M.sub.I] [right arrow] [M.sub.I]

Proof. The coordinates (t, [x.sup.i], [y.sup.0], [y.sup.i]) on T[M.sub.I] change according to the rules:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE ASCII.]

(t, [x.sup.i], [p.sub.0], [p.sub.i]) on [T.sup.*][M.sub.I] and (t, [x.sup.i] [OMEGA], [p.sub.i]) on [T.sup.[dagger]][M.sub.I] change according to the rules:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE ASCII.]

respectively, thus the conclusion follows. ?

It follows that we can consider an affine Hamiltonian h on T[M.sub.I] as h: T[M.sub.I] [right arrow] [T.sup.[dagger][M.sub.I] = [T.sup.*][M.sub.I].

Let us find a way to construct non-trivial affine Hamiltonians using Hamiltonians on [M.sub.I]. Let H: [T.sup.*][M.sub.I] [right arrow] R be a Hamiltonian on [M.sub.I]. We suppose that there is [[alpha].sub.0] [member of] R such that [partial derivative]H/[partial derivative][p.sub.0](t, [x.sup.i], [p.sub.0], [p.sub.i]) [not equal to] 0, (for all]P(t, [x.sup.i], [p.sub.0], [p.sub.i] [member of] [T.sup.*][M.sub.I] and [??}(P) = [[alpha].sub.0]. According to the implicit functions theorem, it follows that the local equation [??](t, [x.sup.i], [p.sub.0], [p.sub.i]) = [[alpha].sub.0] has local solutions [p.sub.0] = [H.sub.0][t,[x.sup.1],[p.sub.i], that define a global section [h.sub.0]: [T.sup.*][M.sub.I] [right arrow] [T.sup.[dagger]][M.sub.I] = [T.sup.*][M.sub.I], [h.sub.0](t, [x.sup.i], [p.sub.i]) = (t, [x.sup.i], [p.sub.i], [H.sub.0](t, [x.sup.i], [p.sub.i])) i.e. an affine Hamiltonian on T[M.sub.I].

Let us find under which conditions on a Hamiltonian [??]: [T.sup.*][M.sub.I] [right arrow] R, the solutions of its Hamilton equation comes from the sections of the fibred manifold [T.sup.*][M.sub.I] [??] I, where [pi]' is obtained as the composition [T.sup.*][M.sub.I] [??] [M.sub.I] [right arrow] I. In order to see this, let us write the Hamilton equations of [??} in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The second equation imposes that [??] fulfils [partial derivative][??]/[partial derivative][p.sub.0] = 1, thus it has the local form [??] (t, [x.sup.i], [p.sub.0], [p.sub.i]) = [p.sub.0] + [H.sub.0] (t,[x.sup.i],[p.sub.i). The equation [??](t, [x.sup.i], [p.sub.0], [p.sub.i]) = 0 has the solution [p.sub.0] = [H.sub.0](t, [x.sup.i], [p.sub.i]), thus the formula (t, [x.sup.i], [p.sub.i]) [??] (t, [p.sup.i], [p.sub.i], -[H.sub.0](t, [x.sup.i], [p.sub.i])) define an affine Hamiltonian on T[M.sub.I]. The above Hamilton equations become the Hamilton equations of h (see [13]).

3. A GENERALIZATION OF THE HIGHER TANGENT SPACES

In the general case, the t-tangent bundle T[M.sub.I] is an affine bundle. We are going to define, for [greater than or equal to] > 1, the k-acceleration t-bundle [T.sup.k][M.sub.I], as an affine bundle over [T.sup.k][M.sub.I]. In the case of a trivial time bundle of a manifold M, we find [T.sup.k][M.sub.I] = I x [T.sup.k]M, thus we recover the well known bundle of accelerations of order k.

Let us denote [M.sub.I] = [T.sup.0][M.sub.I], [[pi].sub.1] = [pi] : T[M.sub.I] [right arrow] [T.sup.0][M.sub.I] and [T.sup.1][M.sub.I] = T[M.sub.I] and consider the vector pseudofield [GAMMA] (1) [partial derivative]/[partial derivative]t + [y.sup.i] [partial derivative]/[partial derivative][x.sup.i] on the affine bundle [T.sup.1][M.sub.I] [??] [T.sup.0][M.sub.I]. Let us suppose that the vector pseudofields [GAMMA](r) on [T.sup.r]M and the r-acceleration bundles [T.sup.r][M.sub.I] have been defined for 1 [less than or equal to] r [less than or equal to] k - 1, as affine bundles over [T.sup.r-1][M.sub.I]. Then [T.sup.k][M.sub.I] is defined according to the change rules of the local coordinates given by formulas (3.2) below. The vector pseudofield [[GAMMA].sup.(k)] on [T.sup.k][M.sub.I] is defined by its action [[GAMMA].sup.(k).sub.U] = [[GAMMA].sup.(k-1).sub.U] + [ky.sup.(k)i] [partial derivative]/[[partial derivative]y.sup.(k-1)i] where [[GAMMA].sup.(k-1).sub.U] is considered as a (local) vector field on [T.sup.k]M and U is the domain which corresponds to the coordinates ([x.sup.i]). Notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

On the intersection of two domains corresponding to U and U', we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

Indeed, according to the recursive definitions of [[GAMMA].sup.(r).sub.U'] and [[GAMMA].sup.(r).sub.U'] we have [[GAMMA].sup.(k).sub.U'] ([x.sup.i']) = [[GAMMA].sup.(k).sub.U], ([x.sup.i'), [[GAMMA].sup.(k).sub.U'] ([y.sup.(r)i'] = [[GAMMA].sup.(k).sub.U]([r.sup.(r)i')] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [[GAMMA].sup.(K).sub.U'] = ([y.sup.(k)i'] = 0, [for all] i' = 1, m;

thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proposition 3.1. The fibered manifold ([T.sup.k][M.sub.I], [[pi].sub.k], [T.sup.k-1][M.sub.I]) is an affine bundle, for k [greater than or equal to] 1.

In the case of a trivial time bundle of a manifold M, we find that [T.sup.k][M.sub.I] = I x [T.sup.k]M, where [T.sup.k]M is the bundle of accelerations of order k of M, studied by R. Miron in [7]-[9].

The vector bundle canonically associated with the affine bundle ([T.sup.k][M.sub.I], [[pi].sub.l], [T.sup.k-1][M.sub.I]) is the vector bundle [p.sup.*.sub.k-1]VT[M.sub.I], where [p.sub.k-1] : [T.sup.k][M.sub.I] [right arrow][M.sub.I] is [p.sub.k-1] = [[pi].sub.1] o [[pi].sub.2] o = o [[pi].sub.k-1] and VT[M.sub.I] [right arrow] [M.sub.I] is the vertical vector bundle of the fibered manifold [M.sub.I] [right arrow] M. The total space of the dual [p.sup.*.sub.k-1] [(VT[M.sub.I]).sup.*] of the vector bundle [p.sup.*.sub.k-1]VT[M.sub.I] is the total space of the fibered manifold ([T.sup.k-1][M.sub.I] x [M.sub.I] [(VT[M.sub.I].sup*], [r.sub.k], [M.sub.I]). As in the time independent case, it can be used in the study of the dual geometrical objects of order k on M which are time dependent, in particular to define time dependent Hamiltonians of order k on M. In the sequel we denote [P.sup.*.sub.k-1] [(VT[M.sub.I]).sup.*] = [T.sup.K*][M.sub.I]; it can be viewed as a vector bundle over [T.sup.K-1][M.sub.I] as well as a fibered manifold over [M.sub.I]

Notice that the local coordinates ([y.sup.(k)i]) change according to the rules:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the local coordinates (t, [x.sup.i], [y.sup.(1)i], ..., [y.sup.(k)i]) on [T.sup.k][M.sub.I] change according to the rules:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

The local vector field [partial derivative]/[partial derivative]t changes according to the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.3)

As in the time independent case, there is an affine morphism [X.sub.k] [T.sub.k][[M.sub.I] [right arrow] T[T.sub.k-1][[M.sub.I], that it is injective on fibers. It has the local form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Notice that in the point [y.sup.(k-1)] [member of] T[T.sub.k-1][[M.sub.I] of coordinates (t, [x.sup.i] [y.sup.(1)2], ..., [y.sup.(k-1)i], all the local vectors in the right side belong to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] .

Proposition 3.2. The local definition of [X.sub.k] [T.sub.k][[M.sub.I] [right arrow] T[T.sub.k-1][[M.sub.I]does not depend on coordinates. Then [X.sub.k] is an affine morphism of affines bundles over [T.sub.k-1][[M.sub.I], that is injective on fibers.

The composition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] defines a fibered manifold on [T.sub.k][[M.sub.I] with base [M.sub.I], that it is not an affine bundle. The vector bundle T[T.sub.k][[M.sub.I] = ker [[pi].sup.k.sub.*] is the vertical bundle of [T.sub.k-1][[M.sub.I].

Let s: [T.sub.k][[M.sub.I] [right arrow] [T.sub.k+1][[M.sub.I] be an affine section of the affine bundle [[pi].sub.k+1]: [T.sub.k+1][[M.sub.I] [right arrow] [T.sub.k][[M.sub.I]. The composition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a vector field on [T.sub.k][[M.sub.I], called a k-t-semispray on M. Since [X.sub.+1] is injective on fibers, the following property is true.

Proposition 3.3. There is a one to one correspondence between the sections of the affine bundle [[pi].sub.k+1]: [T.sub.k+1][[M.sub.I] [right arrow] [T.sub.k][[M.sub.I] and the k-t-semisprays on M.

Notice that the local form of a k-t-semispray is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the local functions ([S.sup.i] (x.sup.i], [y.sup.(1)i], ..., [y.sup.(k)i)) change according to the rules:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let us consider the local vector field [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Lemma 3.1. The local vector fields [DELTA] define a global vector field.

Thus the local vector fields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

have not in general global forms on [T.sub.k][[M.sub.I], thus no one of them can play the role of a Liouville vector field.

In the particular case when [M.sub.I] = M x I, the two vector fields have global forms, recovering the Liouville vector field of order k on M, used in the time independent case.

The endomorphism J: T[T.sup.k] [M.sub.I] locally defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

has a global form. Formulas (3.3) and (3.2) show that J [partial derivative]/[partial derivative]t is well defined. Notice that [J.sup.k+1] = 0 and J[[GAMMA].sup.(k).sub.U] = 0. We call J the (k + 1)-tangent structure on [T.sup.k][M.sub.I]. The tensorial form of J is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The adjoint endomorphism [J.sup.*] T[T.sup.k] [M.sub.I] [right arrow] T[T.sup.k][M.sub.I] is locally given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For a (local) real function f [member of] G [T.sup.k][M.sub.I] we denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since J: T[T.sup.k][M.sub.I]is a global endomorphism, it follows that the local form w j(f) is well defined (i.e. its definition does not depend of coordinates).

In the sequel we study a class of Ehresmann connections on the fibered manifold [P.sub.k-1] [T.sup.k][M.sub.I] [right arrow] [M.sub.I], which we call non-linear k-t-connections on [M.sub.I]. Notice that the vertical bundle is V[T.sup.k][M.sub.I] = ker ([p.sub.k-i]) * [contains] T[T.sup.k][M.sub.I].

A non-linear k-t-connection on M is given by

* a 1-dimensional distribution [H.sub.0] [T.sup.k][M.sub.I] [contains] T[T.sup.k][M.sub.I] (called the t-horizontal distribution) that projects isomorphically on fibers by the composition of differentials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

* an m-dimensional distribution [H.sub.1] [T.sup.k][M.sub.I] [contains] T[T.sup.k][M.sub.I] (called the h-horizontal distribution) that projects isomorphically on the fibers of V[M.sub.I] = ker[pi]*, by the differential. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The vector bundle [H.sub.0] [T.sup.k][M.sub.I] [cross product] T[T.sup.k][M.sub.I] is denoted by H[T.sup.k][M.sub.I] and it is called the horizontal distribution. Taking account into the dimension it follows that T[T.sup.k][M.sub.I] V[T.sup.k][M.sub.I] [contains] H[T.sup.k][M.sub.I] (Whitney sum), thus H[T.sup.k][M.sub.I] is an Ehresmann connection.

Let us consider a local section [delta]/[delta]t of [H.sub.0] [T.sup.k][M.sub.I] that projects on [partial derivative]/[partial derivative]t and the local sections [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] that project, element by element, on the local sections [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. These sections have the local forms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Taking account into the dimensions, the set of local sections [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a local base of the sections of H[[theta].sup.(r)]. We say that this base is adapted if

* [delta]/[delta]t = [delta]/[delta]t', ie [delta]/[delta]t comes from a vector field on [T.sup.k][M.sub.I] and

* on the intersection of two local domains on [E.sup.(r)] the system of vectors < [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] change according to the rule

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 3.1. Every k-t-semispray defines at least one non-linear connection on [T.sup.k][M.sub.I] that allows local adapted bases of sections.

A regular Lagrangian on [T.sup.k][M.sub.I] is a map L: [T.sup.k][M.sub.I] [right arrow] R such that its vertical Hessian [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is non-singular. The next result shows that a regular Lagrangian gives rise to a k-t-semispray.

Proposition 3.4. If L is a regular Lagrangian w of order k, then there is a k-t-semispray defined by L.

Notice that the fc-t-semispray is given by the local functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Finally, the above constructions give a new approach of the time dependent case.

Invited paper to celebrate Professor Constantin Vdriste, on the occasion of his seventies

Received: June 30, 2009.

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MARCELA POPESCU AND PAUL POPESCU

Department of Applied Mathematics

13; Al. I. Cuza st., Craiova, 200585; Romania

E-mail address: marcelacpopescu@yahoo.com

University of Craiova

Department of Applied Mathematics

13; Al. I. Cuza st., Craiova, 200585; Romania

E-mail address: paul_p_popescu@yahoo.com
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Author:Popescu, Marcela; Popescu, Paul
Publication:Journal of Advanced Mathematical Studies
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Geographic Code:4EXRO
Date:Jan 1, 2010
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