Classical Lagrange formalism for non-conservative dynamical systems

TL;DR

This paper extends the classical Lagrange formalism to non-conservative systems, deriving equations of motion without extra degrees of freedom and exploring related concepts like energy and Hamiltonian analogs.

Contribution

It introduces a generalized Lagrange formalism for stationary non-conservative systems, including new equations and conserved quantities, without additional degrees of freedom.

Findings

Derived equations of motion for non-conservative systems

Identified generalized integrals of motion related to energy

Applied formalism to damped harmonic oscillator and solvable models

Abstract

The classical Lagrange formalism is generalized to the case of arbitrary stationary (but not necessarily conservative) dynamical systems. It is shown that the equations of motion for such systems can be derived in the standard ways from the Lagrange functions which are linear in system velocities, have no explicit dependence on time, and do not require the introduction of any additional degrees of freedom. We show that time-symmetry of such Lagrange functions leads to the integrals of motion naturally generalizing the notion of energy but not coinciding with it in non-conservative cases. The non-conservative analogs of Hamilton equations, Poisson brackets, Hamilton-Jacobi equations, Liouville theorem and Principle of Stationary Action are discussed as well. As an example, we consider two cases demonstrating the work of the proposed schema: the classical model of one-dimensional damped harmonic oscillator and a special class of non-physical exactly-solvable multi-dimensional dynamical models.