Affine Hamiltonians in higher order geometry

TL;DR

This paper introduces affine Hamiltonians in higher order geometry, explores their duality with Lagrangians via Legendre maps, and establishes a variational equivalence with Euler-Lagrange equations, extending classical mechanics concepts.

Contribution

It defines affine Hamiltonians for higher order systems and proves their duality with Lagrangians, including an Ostrogradski type theorem linking Hamilton and Euler-Lagrange equations.

Findings

Affine Hamiltonians are dual to Lagrangians via Legendre maps.

Hamilton equations of affine Hamiltonians are equivalent to Euler-Lagrange equations.

The paper provides non-trivial examples illustrating the theory.

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 $k\geq 2$ are studied, relating them to a Hamilton equation. An Ostrogradski type theorem is proved: the Hamilton equation of an affine familtonian $h$ is equivalent with Euler-Lagrange equation of its dual lagrangian $L$. Zermelo condition is also studied and some non-trivial examples are given.