Variational principle for contact Tonelli Hamiltonian systems

TL;DR

This paper develops a variational principle for contact Hamiltonian systems, extending Mather's method from classical Hamiltonian dynamics to contact dynamics under certain growth conditions.

Contribution

It introduces the first implicit variational principle for contact Hamiltonian systems with Tonelli and Osgood growth assumptions, broadening the scope of variational methods.

Findings

Established an implicit variational principle for contact flow equations.

Extended Mather's variational approach to contact Hamiltonian systems.

Provides foundational step for further research in contact dynamics.

Abstract

We establish an implicit variational principle for the equations of the contact flow generated by the Hamiltonian $H(x,u,p)$ with respect to the contact 1-form $\alpha=du-pdx$ under Tonelli and Osgood growth assumptions. It is the first step to generalize Mather's global variational method from the Hamiltonian dynamics to the contact Hamiltonian dynamics.