Regularity properties of fiber derivatives associated with higher-order mechanical systems

TL;DR

This paper investigates the properties of fiber derivatives in higher-order mechanical systems, establishing conditions for Hamiltonian functions and exploring their regularity and the dynamics in Lagrangian submanifolds.

Contribution

It provides necessary and sufficient conditions for higher-order Hamiltonian functions and introduces a definition of regularity based on fiber derivatives, advancing the geometric understanding of such systems.

Findings

Characterization of higher-order Hamiltonian functions

Conditions for regularity in fiber derivatives

Dynamics described via Lagrangian submanifolds

Abstract

The aim of this work is to study fiber derivatives associated to Lagrangian and Hamiltonian functions describing the dynamics of a higher-order autonomous dynamical system. More precisely, given a function in $T^*T^{(k-1)}Q$, we find necessary and sufficient conditions for such a function to describe the dynamics of a kth-order autonomous dynamical system, thus being a kth-order Hamiltonian function. Then, we give a suitable definition of (hyper)regularity for these higher-order Hamiltonian functions in terms of their fiber derivative. In addition, we also study an alternative characterization of the dynamics in Lagrangian submanifolds in terms of the solutions of the higher-order Euler-Lagrange equations.