The classical mechanics of non-conservative systems

TL;DR

This paper introduces a new formulation of Hamilton's principle compatible with initial value problems, enabling the analysis of non-conservative systems like viscous drag and dissipative oscillators within classical mechanics.

Contribution

It provides a novel Hamiltonian and Lagrangian framework for non-conservative systems, addressing a long-standing gap in classical mechanics.

Findings

New formalism for non-conservative systems

Application to viscous drag and dissipative oscillators

Enables study of dissipative effects with Hamiltonian methods

Abstract

Hamilton's principle of stationary action lies at the foundation of theoretical physics and is applied in many other disciplines from pure mathematics to economics. Despite its utility, Hamilton's principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data. This subtlety can have undesirable effects. I present a formulation of Hamilton's principle that is compatible with initial value problems. Remarkably, this leads to a natural formulation for the Lagrangian and Hamiltonian dynamics of generic non-conservative systems, thereby filling a long-standing gap in classical mechanics. Thus dissipative effects, for example, can be studied with new tools that may have application in a variety of disciplines. The new formalism is demonstrated by two examples of non-conservative systems: an object moving in a fluid with viscous drag forces and a harmonic oscillator coupled to a dissipative environment.