Phase space flows for non-Hamiltonian systems with constraints

TL;DR

This paper extends Dirac's formalism to non-Hamiltonian systems with constraints, providing a unified approach to phase space flows using generalized brackets and operators, and addressing response functions.

Contribution

It introduces a Nosé-Dirac bracket and generalizes Dirac's projection method for non-Hamiltonian constrained systems, improving response function accuracy.

Findings

Dirac's formalism avoids spurious response terms.

Phase space measure corrections are necessary for general perturbations.

Generalized brackets describe non-Hamiltonian phase space flows.

Abstract

In this paper, non-Hamiltonian systems with holonomic constraints are treated by a generalization of Dirac's formalism. Non-Hamiltonian phase space flows can be described by generalized antisymmetric brackets or by general Liouville operators which cannot be derived from brackets. Both situations are treated. In the first case, a Nos\'e-Dirac bracket is introduced as an example. In the second one, Dirac's recipe for projecting out constrained variables from time translation operators is generalized and then applied to non-Hamiltonian linear response. Dirac's formalism avoids spurious terms in the response function of constrained systems. However, corrections coming from phase space measure must be considered for general perturbations.