Unified formalism for non-autonomous mechanical systems

TL;DR

This paper introduces a comprehensive geometric framework that unifies Lagrangian and Hamiltonian formalisms for time-dependent mechanical systems, ensuring consistency and broad applicability.

Contribution

It develops a unified formalism based on Skinner and Rusk's approach, applicable to both regular and non-regular systems, with detailed analysis of equations of motion and an example application.

Findings

Framework recovers all characteristics of classical formalisms.

Ensures compatibility and consistency of dynamical equations.

Demonstrates applicability through nonlinear wave equation semidiscretization.

Abstract

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk (1983). The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.