Affine Hamiltonians in Higher Order Geometry

Abstract

Affine Hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of Lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine Hamiltonians and Lagrangians of order k ≥ 2 are studied, relating them to a Hamilton equation. An Ostrogradski type theorem is proved: the Hamilton equation of an affine Hamiltonian h is equivalent with Euler-Lagrange equation of its dual Lagrangian L. Zermelo condition is also studied and some non-trivial examples are given.

Higher Order Transverse Bundles and Riemannian Foliations

“…In particular, according to the case of trivial foliation of M by points in [9], ν 1. (r−1) * F = ν r−1 F × M ν * F is denoted by ν r * M and play the role of the vectorial dual of the affine bundle ν r F → ν r−1 F .…”

“…A transverse slashed lagrangian of order r is a map L r : ν r F → IR that is differentiable on an open subset ν r * F ⊂ν r F , where ν r F \ν r * F contains I r r−1 (ν r−1 F ). We denote by ν r ′ * F = π r r ′ (ν r * F ) ⊂ν r ′ F and we consider the slashed bundles ν * * F = ν * F \{0} and ν F → IR of the dual affine hamiltonian of L (see [9] for its classical definition and [11] for a coordinate description of the whole construction in the non-foliate case). Analogous, for 0 ≤ j < r − 1, we suppose, step by step, backward from r − 1 from 0, that there the usual partial derivatives of L (j+1) : ν j+1,(r−j−1) * * F = ν r−j−1 * F × M (ν * * F ) j+1 → IR in the highest order transverse coordinates (of order j + 1) define a well-defined Legendre map L (j+1) : ν j+1,(r−j−1) * *…”

“…In order to prove b), one construct a transverse riemannian metric H in the fibers of νF r , using its decomposition given by formula (4) and taking into account that all the summands are isomorphic with V r r−1 , where h is a riemannian metric on fibers. ✷ According to the case of trivial foliation of M by points in [9], ν r−1 F × M ν * F not.…”

“…We denote by H 1,r−1 = L r • L (r) −1 : ν 1,(r−1) * * F → IR its pseudo-hamiltonian. (See [9] for its classical definition and [11] for a coordinate description of the whole construction in the non-foliate case). Analogous, for 0 ≤ j < r − 1, we suppose, step by step, backward from r − 1 from 0, that there the usual partial derivatives of L (j+1) : ν j+1,(r−j−1) * * F = ν r−j−1 * F × M (ν * * F ) j+1 → IR in the highest order transverse coordinates (of order j + 1) define a well-defined Legendre map L (j+1) : ν j+1,(r−j−1) * *…”

“…The existence of a lifted metric, from the base space to the higher order tangent bundle, is an well-known fact in the non-foliate case (see, for example [3, Sect. 9.2]); we have to consider a simpler construction in the foliated case, that it is also vertically exact, as in [9,11]. Proposition 2.5 Any transverse metric g on νF gives canonically a positively admissible lagrangian L (r) of order r and a canonical vertically exact transverse riemannian metric g (r) on ν r F , for any r ≥ 1.…”