The evolution operator connecting the Lagrangian and Hamiltonian formalisms for contact systems

TL;DR

This paper extends the evolution operator concept to contact mechanics, linking Lagrangian and Hamiltonian formalisms for dissipative systems, and explores its properties through examples like the pendulum.

Contribution

It introduces a geometric evolution operator for contact systems, connecting Lagrangian and Hamiltonian formalisms in dissipative mechanics.

Findings

The evolution operator provides a geometric description of contact system dynamics.

It relates Hamiltonian and Lagrangian constraints in contact mechanics.

Examples demonstrate the operator's application to modified pendulum systems.

Abstract

Some mechanical systems with dissipation can be described within the framework of the so-called contact mechanics: a modified form of the Euler-Lagrange equations stemming from Herglotz's variational principle, which admits a geometric formulation in terms of contact geometry. On the other hand, the study of singular Lagrangian systems and Dirac's theory of constraints can be enhanced by using the evolution operator $K$ that connects the Lagrangian and Hamiltonian formalisms. The main purpose of this paper is to transpose this evolution operator to the case of contact mechanics, and to study some of its main properties. In particular, we show that it provides a geometric description of the evolution equations and it relates the Hamiltonian and Lagrangian constraints. To illustrate the theory, we provide examples of singular contact systems based on modified versions of the simple pendulum and the Cawley Lagrangian.