🔬 condensed matter

Elastodynamics from Eulerian Poisson-bracket formalism: application to chiral odd solids

This paper develops a systematic Eulerian Poisson-bracket formalism for elastic solids that captures essential geometric nonlinearities arising from coordinate transformations and particle flow, demonstrating its ability to derive the odd elastic modulus in chiral active solids driven by internal torques.

Original authors: Cheng-Tai Lee, Tomer Markovich

Published 2026-08-03

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CC BY 4.0

Original authors: Cheng-Tai Lee, Tomer Markovich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Invisible Dance of Materials

Imagine you are watching a crowd of people at a concert. If you want to understand how the crowd moves, you have two main ways to look at it. You could follow one specific person from start to finish, watching exactly where they step and how they bump into others; this is like the "Lagrangian" view, where you track individual particles through their journey. Or, you could stand still at a fixed spot on the floor and watch everyone who passes by you, noting the flow of the crowd as it rushes past; this is the "Eulerian" view, where you focus on the space itself rather than the travelers.

In the world of physics, scientists use these two perspectives to understand how materials behave, from the flow of honey to the stretching of rubber. A powerful mathematical tool called the "Poisson-bracket formalism" acts like a rulebook for predicting how these materials move and react. For a long time, physicists thought this rulebook had to be written differently depending on whether you were tracking particles (solids) or watching a flow (fluids). Solids were usually studied by following the particles, while fluids were studied by watching the space. But what happens when a solid starts acting a bit like a fluid, or when it has a "handedness" that makes it twist in weird ways? This is where things get tricky, and where a new way of looking at the old rules becomes necessary.

The Paper's Discovery: A New Lens for Twisting Solids

In this paper, the authors, Cheng-Tai Lee and Tomer Markovich, tackle a specific puzzle: how to apply the "Eulerian" (fixed-space) rulebook to elastic solids, which are usually studied with the "Lagrangian" (particle-tracking) method. They wanted to see if they could describe solids using the fixed-space view without losing any of the physics, especially when the solids are doing something unusual.

The authors found that while both methods agree when things are simple and the material isn't moving much, they start to tell different stories when the material gets stretched, twisted, or pushed by internal forces. Specifically, when they applied the Eulerian method to a special type of material called a "chiral active solid," they discovered that this approach naturally generates extra, complex terms in the equations. These extra terms are not mistakes; they are real physical effects that arise because the "volume" of the material changes as it deforms, and because particles are constantly flowing in and out of the tiny regions scientists use to measure the material.

To prove their point, the authors used a model of a "chiral active solid." Imagine a jelly made of tiny, spinning gears. In this material, the gears are driven by internal motors (active torques) that make them spin. This spinning causes the whole jelly to twist and deform in a way that is not symmetrical. The authors showed that by using their new Eulerian method, they could directly calculate a strange property called "odd elasticity." This is a mechanical behavior where pushing the material in one direction doesn't just squeeze it; it also makes it twist in a specific, non-reciprocal way. For example, if you squeeze it, it might tilt; if you twist it, it might expand.

The paper explicitly argues against the idea that the Eulerian and Lagrangian methods are always interchangeable, especially for nonlinear effects. They demonstrate that if you try to force the Eulerian view to look exactly like the Lagrangian one by ignoring these extra terms, you miss the very phenomenon that makes these active solids special: the odd elastic modulus. By successfully deriving this modulus directly from the Eulerian framework, the authors show that this method is not just a mathematical curiosity but a necessary tool for understanding materials that are driven by internal forces and measured in real, deformed space. They didn't just suggest this might work; they worked through the math to prove that the Eulerian approach captures the same physical reality as the traditional method, but with a different, often more natural, set of equations for these specific, active materials.

In short, the paper provides a new, consistent way to describe how "living" or active solids behave when they are pushed and pulled. It shows that looking at the material from a fixed point in space (Eulerian) reveals hidden, nonlinear twists that are essential for understanding how these materials generate their own motion and unique mechanical responses. This opens the door to better understanding complex materials like biological tissues or synthetic gels that contain active, spinning components.

Technical Summary: Elastodynamics from Eulerian Poisson-bracket formalism: application to chiral odd solids

Problem Statement
The Poisson-bracket (PB) formalism is a standard method for deriving dynamic equations of coarse-grained (CG) fields from microscopic particle mechanics. Conventionally, the Lagrangian formulation (using reference/undeformed coordinates) is applied to elastic solids, while the Eulerian formulation (using real/deformed coordinates) is reserved for fluids. However, there is growing interest in Eulerian descriptions for solids, particularly for phenomena defined naturally in real space, such as viscoelastic responses, moving boundaries, and active materials where elastic order coexists with flow. A gap exists in the systematic application of the Eulerian PB formalism to elastic systems whose potentials are typically defined in Lagrangian space. Specifically, it is unclear how the Eulerian framework handles the nonlinearities arising from coordinate transformations and particle flow, and whether it remains consistent with the established Lagrangian approach when nonlinear elastic effects are significant.

Methodology
The authors develop a systematic procedure to apply the Eulerian PB formalism to elastic systems, bridging the gap between Lagrangian and Eulerian frameworks. The methodology involves:

Field Definition and Classification: Coarse-grained fields are categorized into "density-type" (e.g., mass, momentum densities) and "particle-averaged" (e.g., displacement, internal rotation). The authors establish the transformation rules between these fields in Lagrangian ( $r$ ) and Eulerian ( $R$ ) spaces, accounting for the Jacobian $J$ of the deformation gradient.
Hamiltonian Transformation: The Lagrangian Hamiltonian is transformed into the Eulerian framework. This involves changing coordinates from $r$ to $R$ and substituting Lagrangian density fields with their Eulerian counterparts. Crucially, this transformation introduces a factor of $J^{-1}$ , which generates higher-order nonlinear terms in the potential energy density that are absent in the linear approximation.
Poisson-Bracket Derivation: The authors derive the Eulerian PBs for the relevant fields. Unlike Lagrangian PBs, where the same particle pack is tracked, Eulerian PBs account for particle flow across neighboring CG volumes. This results in additional spatial-derivative terms in the brackets (e.g., $\{g_c, u\}$ ), which physically represent the exchange of particles between volumes.
Dynamic Equation Transformation: A direct transformation method is provided to convert dynamic equations derived in one framework to the other. This involves relating the total time derivative ($d/dt$) to the partial time derivative ( $\partial/\partial t$ ) plus a convective (streaming) term, and utilizing the Piola identity to relate stress measures.
Application to Chiral Active Solids: To illustrate the formalism, the authors apply it to a 2D model of disordered chiral active solids. In this model, active torques drive internal particle rotations, inducing geometric nonlinearities. They formulate the Lagrangian and Eulerian elastic potentials, derive the equations of motion for center-of-mass momentum and angular momentum, and analyze the resulting stress tensors.

Key Contributions and Results

Identification of Nonlinear Sources: The paper demonstrates that the Eulerian formulation generates additional nonlinearities absent in the Lagrangian counterpart. These arise from two distinct sources:
1. Hamiltonian Transformation: The change in CG volume under coordinate transformation ( $J^{-1}$ ) introduces nonlinear terms directly into the Hamiltonian.
2. Eulerian PBs: The non-vanishing derivatives of the Dirac delta function in the Eulerian PBs (due to particle flow) generate additional spatial-gradient terms in the dynamic equations.
Consistency of Frameworks: The authors prove the consistency between the Lagrangian and Eulerian PB formalisms. By applying the Eulerian PB formalism directly to the Eulerian Hamiltonian, they recover the same Cauchy stress tensor and odd elastic modulus previously derived using the Lagrangian formalism followed by a stress transformation.
Derivation of Odd Elasticity: In the specific case of chiral active solids, the geometric nonlinearities induced by active torques are shown to be essential for the emergence of the odd elastic modulus ( $K_o$ ). This modulus non-reciprocally couples two different shear modes in the stress-strain response.
Direct Recovery of Stress: The study successfully recovers the standard relation between the Cauchy stress and the first Piola-Kirchhoff stress tensors through the direct transformation of dynamic equations, validating the Eulerian approach for real-space stress measurements.

Significance
The paper claims that the Eulerian PB formalism is a robust and consistent tool for studying elastic systems, particularly those involving large deformations or active driving forces where nonlinear effects are non-negligible. By demonstrating that the Eulerian framework can directly capture emergent nonlinear elastic behavior (specifically odd elasticity) in driven active solids, the work provides a theoretical foundation for studying systems where the reference configuration is dynamic or where stresses are naturally measured in the deformed (real) space. This approach offers a unified route to connect microscopic particle dynamics (such as torque-driven internal rotations) to macroscopic elastic behavior without relying solely on post-hoc stress transformations.

The Invisible Dance of Materials

The Paper's Discovery: A New Lens for Twisting Solids

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