Hamiltonian treatment of non-conservative systems

TL;DR

This paper extends Hamiltonian mechanics to non-conservative systems using a double-variable action principle, clarifying boundary conditions, gauge freedoms, and providing explicit Hamiltonian and Lagrangian formulations for initial-value problems.

Contribution

It introduces a novel Hamiltonian framework for non-conservative systems based on the Schwinger-Keldysh-Galley formalism, including gauge freedoms and a linear Lie formulation.

Findings

Constructed gauge-related nonconservative Hamiltonians.

Embedded classical initial-value problems in an enlarged symplectic manifold.

Derived a linear Lie formulation simplifying the formalism.

Abstract

We present a novel extension of Hamiltonian mechanics to nonconservative systems built upon the Schwinger-Keldysh-Galley double-variable action principle. Departing from Galley's initial-value action, we clarify important subtleties regarding boundary conditions, the emergence of the physical-limit trajectory, and the decomposition of the Lagrangian into conservative and dissipative sectors. Importantly, we demonstrate that the redundant doubled configuration space admits a gauge freedom at the level of the canonical momenta that leaves the physical dynamics unchanged. From a Legendre transform, we construct the corresponding family of gauge-related nonconservative Hamiltonians; we show that virtually any classical initial-value problem can be embedded on our enlarged symplectic manifold, supplying the associated Hamiltonian and Lagrangian functions explicitly. As a further contribution, we derive a completely equivalent linear ``Lie'' formulation of the double-variable action and Hamiltonian which streamlines computations and renders transparent many structural properties of the formalism.