Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems

TL;DR

This paper develops a unified geometric Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems, including singular Lagrangians, with detailed examples like the Duffing equation and particles with time-dependent parameters.

Contribution

It introduces a novel geometric formalism for time-dependent contact systems that handles singular Lagrangians and constraints, extending previous approaches.

Findings

Formalism effectively describes dissipative systems with singular Lagrangians

Successfully applied to complex examples like the Duffing equation and particles with time-dependent mass

Provides a geometric framework for analyzing time-dependent constrained systems

Abstract

In this paper we present a unified Lagrangian--Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.