A Lagrangian form of tangent forms

TL;DR

This paper explores the relationship between tangent forms and second order Lagrangians, providing a Hamiltonian framework and solutions for certain cases, advancing the geometric understanding of dynamic systems.

Contribution

It introduces a Lagrangian-Hamiltonian framework for tangent forms, extending quantization formulas and analyzing equivalence relations with gauge considerations.

Findings

Expresses Euler-Lagrange equations as second order Lagrange derivatives of tangent forms.

Provides Hamiltonian formulations extending previous quantization approaches.

Offers local solutions for specific cases using semi-sprays.

Abstract

The aim of the paper is to study some dynamic aspects coming from a tangent form, i.e. a time dependent differential form on a tangent bundle. The action on curves of a tangent form is natural associated with that of a second order Lagrangian linear in accelerations, while the converse association is not unique. An equivalence relation of tangent form, compatible with gauge equivalent Lagrangians, is considered. We express the Euler-Lagrange equation of the Lagrangian as a second order Lagrange derivative of a tangent form, considering controlled and higher order tangent forms. Hamiltonian forms of the dynamics generated are given, extending some quantization formulas given by Lukierski, Stichel and Zakrzewski. Using semi-sprays, local solutions of the E-L equations are given in some special particular cases.