[Paper Review] Pseudomomentum: origins and consequences

H. Singh, J. A. Hanna|arXiv (Cornell University)|Jul 12, 2020

Experimental and Theoretical Physics Studies4 citations

TL;DR

This paper establishes a unified variational framework for deriving balance laws of momentum, energy, and pseudomomentum in continuum mechanics by simultaneously varying the action with respect to spatial position, time, and material labels. It demonstrates that pseudomomentum balance underpins key conservation laws in fluids and elastic structures, including vorticity, circulation, and helicity, and enables direct computation of propulsive forces in confined elastic rods via path-independent integrals.

ABSTRACT

The balance of pseudomomentum is discussed and applied to simple elasticity, ideal fluids, and the mechanics of inextensible rods and sheets. A general framework is presented in which the simultaneous variation of an action with respect to position, time, and material labels yields bulk balance laws and jump conditions for momentum, energy, and pseudomomentum. The example of simple elasticity of space-filling solids is treated at length. The pseudomomentum balance in ideal fluids is shown to imply conservation of vorticity, circulation, and helicity, and a mathematical similarity is noted between the evaluation of circulation along a material loop and the J-integral of fracture mechanics. Integration of the pseudomomentum balance, making use of a prescription for singular sources derived by analogy with the continuous form of the balance, directly provides the propulsive force driving passive reconfiguration or locomotion of confined, inhomogeneous elastic rods. The conserved angular momentum and pseudomomentum are identified in the classification of conical sheets with rotational inertia or bending energy.

Motivation & Objective

• To unify the derivation of momentum, energy, and pseudomomentum balance laws through simultaneous variation of action with respect to spatial coordinates, time, and material labels.
• To clarify the physical origin and consequences of pseudomomentum in conservative mechanical systems governed by an action principle.
• To demonstrate that pseudomomentum balance in ideal fluids implies conservation of vorticity, circulation, and helicity, with a mathematical analogy to the J-integral in fracture mechanics.
• To show that integration of the pseudomomentum balance yields the propulsive force in passive reconfiguration of confined, inhomogeneous elastic rods.
• To identify conserved angular momentum and pseudomomentum in conical sheets with rotational inertia or bending energy.

Proposed method

• Derives bulk balance laws and jump conditions by varying the action with respect to spatial position, time, and material coordinates, yielding momentum, energy, and pseudomomentum balances.
• Uses the Lagrangian density $\bar{\mathcal{L}}$ depending on material coordinates $\bar{\mathbf{x}}$, spatial position $\mathbf{x}$, velocity $d_t\mathbf{x}$, and deformation gradient $\mathbf{F}$.
• Applies the chain rule to the referential gradient of $\bar{\mathcal{L}}$ to express the pseudomomentum balance in terms of $\partial\bar{\mathcal{L}}/\partial\mathbf{F}$, $\partial\bar{\mathcal{L}}/\partial d_t\mathbf{x}$, and $\mathbf{F}$.
• Derives the Eshelby tensor as $\left[\frac{\partial\bar{\mathcal{L}}}{\partial\mathbf{F}}\right]^T \cdot \mathbf{F} - \bar{\mathcal{L}} \mathbf{I}$, the referential flux of pseudomomentum.
• Applies the balance laws to specific systems: simple elasticity, ideal fluids, inextensible rods and sheets, and conical structures with bending energy or rotational inertia.
• Uses path-independent integrals and jump conditions across discontinuities to compute forces and conserved quantities, including the J-integral analogy in circulation evaluation.

Experimental results

Research questions

• RQ1 How does the simultaneous variation of the action with respect to spatial, temporal, and material coordinates yield consistent balance laws for momentum, energy, and pseudomomentum?
• RQ2 What is the role of pseudomomentum in the conservation of vorticity, circulation, and helicity in ideal fluids?
• RQ3 How can the pseudomomentum balance be integrated to compute the propulsive force in passive locomotion of confined elastic rods?
• RQ4 What conserved quantities emerge in conical sheets with rotational inertia or bending energy, and how are they related to pseudomomentum?
• RQ5 How does the Eshelby tensor arise from the variational framework, and what is its physical interpretation as the referential flux of pseudomomentum?

Key findings

• The pseudomomentum balance in ideal fluids directly implies conservation of vorticity, circulation, and helicity, with circulation evaluation mathematically analogous to the J-integral in fracture mechanics.
• Integration of the pseudomomentum balance, using a prescription for singular sources derived from the continuous form, yields the propulsive force driving passive reconfiguration of inhomogeneous elastic rods confined in a channel.
• The Eshelby tensor, defined as $\left[\frac{\partial\bar{\mathcal{L}}}{\partial\mathbf{F}}\right]^T \cdot \mathbf{F} - \bar{\mathcal{L}} \mathbf{I}$, emerges as the referential flux of pseudomomentum in the balance law.
• In conical sheets with rotational inertia or bending energy, both angular momentum and pseudomomentum are conserved, and their classification is clarified through the variational framework.
• The balance laws for momentum, energy, and pseudomomentum are derived from a single variational principle, with jump conditions across material discontinuities ensuring consistency at interfaces.
• The framework reconciles conflicting interpretations of pseudomomentum by showing it arises from variation with respect to material coordinates, independent of spatial momentum balance.