Higher-order contact mechanics

TL;DR

This paper develops a comprehensive geometric framework for higher-order contact mechanics, extending classical theories to include dissipative systems and providing unified Lagrangian and Hamiltonian formalisms.

Contribution

It introduces a complete geometric theory for higher-order contact systems, including variational principles, dynamical equations, and a unified Lagrangian-Hamiltonian approach.

Findings

Formulated the variational principle for higher-order contact systems.

Derived geometric dynamical equations using extended tangent bundles.

Provided examples illustrating the application of the theory.

Abstract

We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, ${\rm T}^kQ\times{\mathbb R}$, whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formalisms for these kinds of systems, including their variational formulation. The variational principle, the contact forms, and the geometric dynamical equations are obtained by using those structures and generalizing the standard formulation of contact Lagrangian and Hamiltonian systems. As an alternative approach, we develop a unified description that encompasses the Lagrangian and Hamiltonian equations as well as their relationship through the Legendre map; all of them are obtained from the contact dynamical equations and the constraint algorithm that is implemented because, in this formalism, the dynamical systems are always singular. Some interesting examples are finally analyzed using these geometric formulations.