The principle of stationary nonconservative action for classical mechanics and field theories

TL;DR

The paper presents a unified variational framework for classical systems with nonconservative, time-irreversible dynamics by doubling the degrees of freedom and introducing a nonconservative potential density K. This approach yields causal, initial-value compatible equations of motion and generalizes Noether's theorem to account for dissipation, while enabling a broad range of discrete and continuum examples, including Navier–Stokes fluids and viscoelastic materials. The formalism naturally embeds energy and entropy accounting through closure relations and internal energy, and it links to the classical limit of a quantum theory via an in-in density matrix construction. Overall, the work provides a principled, action-based foundation for dissipative continuum mechanics and field theories with practical implications for modeling irreversible processes and developing variational numerical schemes.

Abstract

We further develop a recently introduced variational principle of stationary action for problems in nonconservative classical mechanics and extend it to classical field theories. The variational calculus used is consistent with an initial value formulation of physical problems and allows for time-irreversible processes, such as dissipation, to be included at the level of the action. In this formalism, the equations of motion are generated by extremizing a nonconservative action $\mathcal{S}$, which is a functional of a doubled set of degrees of freedom. The corresponding nonconservative Lagrangian contains a potential $K$ which generates nonconservative forces and interactions. Such a nonconservative potential can arise in several ways, including from an open system interacting with inaccessible degrees of freedom or from integrating out or coarse-graining a subset of variables in closed systems. We generalize Noether's theorem to show how Noether currents are modified and no longer conserved when $K$ is non-vanishing. Consequently, the nonconservative aspects of a physical system are derived solely from $K$. We show how to use the formalism with examples of nonconservative actions for discrete systems including forced damped harmonic oscillators, radiation reaction on an accelerated charge, and RLC circuits. We present examples for nonconservative classical field theories. Our approach naturally allows for irreversible thermodynamic processes to be included in an unconstrained variational principle. We present the nonconservative action for a Navier-Stokes fluid including the effects of viscous dissipation and heat diffusion, as well as an action that generates the Maxwell model for viscoelastic materials, which can be easily generalized to more realistic rheological models. We show that the nonconservative action can be derived as the classical limit of a more complete quantum theory.

Figures (4)

• Figure 1: Left: A schematic of a trajectory ${\bm q} (t)$ for conservative mechanics. Dashed lines represent varied paths and the solid line indicates the path for which the action is stationary. Hamilton's principle requires variations with fixed endpoints. Right: Same as the left schematic but for the doubled degrees of freedom from Galley:2012hx. Using this variational principle allows for variations where only the initial data are specified, which is consistent with initial value problems and "breaks" the time symmetry inherent in conservative actions. In both figures, arrows indicate the direction for the time integration of the Lagrangian (i.e., the action).
• Figure 2: A) Schematic of a forced, damped harmonic oscillator. The oscillator mass is connected to a massless spring with spring constant $k$ and a massless dashpot with damping factor $\lambda$ in parallel. B) Schematic of the Maxwell element. The mass is connected to a spring and dashpot in series. Unlike the forced, damped harmonic oscillator above, the displacement of the center of mass is determined by the spring's displacement, which is elastic, but also by the "plastic" deformation of the dashpot.
• Figure 3: A simple circuit composed of a resistor (R), inductor (L), and capacitor (C). An external voltage $V(t)$ is applied to complete the circuit.
• Figure 4: We take a manifold ${\cal M}$ which has natural foliations by the surfaces of constant time (middle figure). World lines describing paths with fixed coordinates $a$ (e.g., for a given fluid element) and $q$ (e.g., for an Eulerian observer) are shown. In the Lagrange picture, this manifests as the flow of the Eulerian ($q$) coordinates (right figure) relative to the fixed material ($a$) coordinates. When another species of fluid is included, such as for describing the flow of entropy Prix:2002jn2010RSPSA.466.1373A, there is an additional fixed coordinate $\alpha$ that traces a path in the manifold (middle) and flows relative to the fixed material ($a$) coordinates (left).