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REVIEW 2 major objections 3 minor 69 references

Applying the Poisson-bracket formalism in real space directly reproduces the Cauchy stress and odd modulus K_o = τ/4 of chiral active solids, bypassing the Lagrangian stress transform.

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2026-08-03 05:19 UTC pith:73Z7YABE

load-bearing objection The formal Eulerian-to-Lagrangian dynamics transformation (Eq. 14) is a genuine methodological step forward; the odd-elasticity recovery is a consistency check that inherits an asserted adiabatic assumption rather than providing a new physical prediction. the 2 major comments →

arxiv 2607.29521 v1 pith:73Z7YABE submitted 2026-07-31 cond-mat.soft physics.bio-phphysics.flu-dyn

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

Cheng-Tai Lee , Tomer Markovich This is my paper
classification cond-mat.soft physics.bio-phphysics.flu-dyn
keywords Poisson-bracket formalismEulerian elasticityodd elasticitychiral active solidsmicropolar (Cosserat) elasticityCauchy stressgeometric nonlinearitiescoarse-grained dynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to establish that the Poisson-bracket (PB) formalism, a standard route for deriving coarse-grained field dynamics, can be applied directly to elastic solids described in Eulerian (real-space) variables even though their elastic potentials are naturally written in Lagrangian (reference-space) variables. It shows that the Eulerian formulation is not a passive coordinate change: transforming the Hamiltonian and evaluating Eulerian brackets produces extra nonlinear terms, from coarse-grained volume changes and from particle flow between neighbouring volumes, that the Lagrangian truncation omits. The illustrative case is a two-dimensional chiral active solid with rotating particles, where active torques create geometric nonlinearities that yield an odd elastic modulus coupling the two shear modes non-reciprocally. Recovering this modulus directly from Eulerian brackets, rather than by transforming the Lagrangian stress, demonstrates the consistency of the two frameworks and provides a route for materials whose measured responses live in real space. The paper thus argues that the Eulerian PB formalism is a valid and useful tool for nonlinear elastodynamics of driven solids.

Core claim

The paper's central claim is that the Eulerian PB formalism, applied to the Eulerian Hamiltonian obtained from the Lagrangian one through the Jacobian factor J, generates the full real-space stress tensor and the odd elastic modulus K_o = τ/4 in a chiral active solid, matching the result of a previous Lagrangian derivation. The key step is the fast-relaxation (adiabatic) elimination of the internal rotation θ, which reduces the rotational dynamics to the algebraic relation ϕ ≈ (τ/2κ_c)(1 + ((λ̃+μ̃)/κ_c) ∇·u). Substituting this into the Eulerian momentum balance (Eq. 25) produces the Cauchy stress with an active prestress and an elasticity tensor that contains the odd modulus proportional to

What carries the argument

The load-bearing object is the Eulerian Poisson bracket, computed with canonical pairs defined in the real/deformed space, so that derivatives of delta functions generate the convective terms that the Lagrangian bracket misses; the most important examples are {g_i^c(R), u_j(R′)} = (δ_ij − ∇′_i u_j) δ(R−R′) and {g_i^c(R), θ(R′)} = −∇′_i θ(R′) δ(R−R′). The second load-bearing identity is Eq. (14), dΨ°/dt = J[∂Ψ/∂t + ∇_j(v_j^c Ψ)], which converts Eulerian dynamics into Lagrangian dynamics, carries the nonlinear streaming and volume-change contributions, and yields the Cauchy-to-first-Piola–Kirchhoff stress relation. In the chiral-solid example, these brackets, combined with the adiabatic elimin

Load-bearing premise

The argument assumes that the internal rotation θ relaxes so quickly that it can be set to local mechanical equilibrium, ∂H/∂θ = 0, turning it into an algebraic function of the displacement gradient; if θ does not relax fast, the quasistatic odd modulus K_o = τ/4 becomes a frequency-dependent quantity and the central stress-strain claim no longer holds as stated.

What would settle it

Measure the shear-stress response of a chiral active solid in the regime |∇u| ≪ θ ≪ 1 as a function of driving frequency, or solve the full coupled dynamics of displacement and internal rotation numerically without imposing ∂H/∂θ = 0: if the effective odd modulus deviates from τ/4 or shows strong frequency dependence where the paper predicts a quasistatic constant, the central reduction is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The real-space (Cauchy) stress becomes directly accessible from the Eulerian PB formalism, so experimental or computational studies that measure stress in deformed coordinates can compare with theory without reference-frame transformations.
  • In any elastic system with large internal rotations, the extra Eulerian nonlinearities—neglected in linear elasticity—must be kept; ignoring them hides the odd response and misstates the stress-strain relation.
  • The recovered odd modulus K_o = τ/4 implies non-reciprocal shear coupling: a simple shear strain induces pure shear stress, and a pure shear strain induces negative simple shear stress, with consequences such as tilting under uniaxial compression and growing wave modes as established in the broader odd-elasticity literature.
  • The transformation Eq. (14) gives a general dictionary between Eulerian and Lagrangian dynamics, allowing the two formulations to be cross-checked even in systems such as viscoelastic, elastoplastic, or active materials where a fixed reference frame is not always available.
  • For three-dimensional micropolar solids, the axis-angle canonical-momentum construction in Appendix A extends the same Eulerian PB scheme, so the approach is not limited to two-dimensional chiral solids.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the fast-variable assumption on θ is relaxed, the quasistatic modulus τ/4 should become a frequency-dependent odd response; measuring the shear modulus as a function of driving frequency in a chiral active solid would provide a clean test the paper does not itself perform.
  • Eq. (14) is likely a general tool beyond elasticity: any coarse-grained theory with a dynamic reference configuration, such as plastic flow, growth, or active remodeling, could use this transformation to connect Eulerian observations to Lagrangian material behavior.
  • The derivation suggests that odd elastic response does not require a microscopic model of contact forces, only a Hamiltonian with an active torque potential and geometric nonlinearities, so the same Eulerian PB route could predict odd moduli in other torque-driven field theories such as chiral liquid crystal elastomers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper develops a systematic Eulerian Poisson-bracket (PB) formalism for elastic systems whose Hamiltonians are originally written in Lagrangian variables. It derives the transformation between Lagrangian and Eulerian field dynamics (Eq. 14), identifies two sources of additional nonlinearities in the Eulerian description (the Jacobian volume factor in the Hamiltonian and the spatial-gradient terms in Eulerian PBs), and verifies the relation between Cauchy and first Piola–Kirchhoff stresses. As an illustration, the formalism is applied to a 2D chiral active solid with internal particle rotations; the resulting stress and elasticity tensors (Eqs. 28–29) recover the odd elastic modulus K_o=τ/4 of the authors' previous Lagrangian treatment (Ref. [21]).

Significance. If the formal claims hold, Eq. (14) and the PB list in Appendix B provide a useful and clean bridge between Lagrangian and Eulerian descriptions of solids with internal rotation. The derivation of Eq. (14) and its use to recover the standard stress transformation are elegant, and the presentation of the Eulerian PBs with 3D generalization is valuable. However, the illustrative recovery of K_o=τ/4 is a consistency check against Ref. [21] rather than an independent prediction: the input potential Eq. (18) already contains the geometric nonlinearities that produce the odd modulus, and the result depends on an adiabatic elimination of the internal rotation that is asserted rather than derived. These points do not undermine the formal methodology, but they temper the strength of the 'recovering' claim.

major comments (2)
  1. [§2.3, Eq. (26)] The coefficient of the φ(∇·u) term is printed as λ+μ, but the potential V in Eq. (20) contains −(λ̃+μ̃)φ²(∇·u), whose derivative with respect to φ is −2(λ̃+μ̃)φ(∇·u). Equation (27) then uses λ̃+μ̃, so Eq. (26) as printed is not the equation that yields Eq. (27). Since Eq. (27) is substituted into Eq. (25) to obtain the central stress tensor Eq. (28), please correct the coefficient and check the signs in the intermediate algebra.
  2. [§2.3, Eqs. (26)–(28)] The central quantitative result K_o=τ/4 is obtained by replacing the full angular-momentum equation by the algebraic relation Eq. (27) (∂H/∂θ=0). This is an adiabatic elimination of the internal rotation θ. The paper asserts that θ relaxes fast and cites Refs. [1,20,63–65], but no timescale separation is derived for the present model, and the reactive PB equations (25)–(26) contain no dissipative relaxation. If the rotational degree of freedom is not fast at the frequencies of interest, the effective modulus is frequency-dependent, K_o(ω), rather than the static τ/4. Please state the validity regime (e.g., damping vs κ_c) or derive the elimination from a dissipative model.
minor comments (3)
  1. [Abstract and §2.3] The wording 'recovering' and 'demonstrate its ability to capture emergent nonlinear elastic behavior' overstates the example. The potential Eq. (18) already contains the geometric nonlinearities that produce K_o, so the computation is a consistency check of the Eulerian route against Ref. [21], not an independent prediction. Please qualify the language accordingly.
  2. [§2.2, Eq. (20)] Please show the intermediate steps in transforming V° to V; the origin of the −κ_c ε_li δ_jk term is not transparent from the one-line description following Eq. (20). A fuller derivation would aid reproducibility.
  3. [§2.1, Eq. (18)] The notation λ, μ versus λ̃, μ̃ is confusing. The tilde coefficients are imported from Ref. [21] but are not defined in this paper; please define them explicitly at the point of introduction.

Circularity Check

0 steps flagged

No significant circularity: the Eulerian-PB transformation is independently derived and the odd-modulus example is an explicit consistency check with the authors' prior model, not a new prediction forced by the formalism.

full rationale

The load-bearing formal result is Eq. (14), obtained from the field transformation Psi^o = J Psi and d ln J/dt = div v_c; it is not a restatement of any later result. Equations (25)-(28) follow by substituting the Eulerian PBs and the explicit Hamiltonian into Eq. (7), and K_o = tau/4 emerges only after adiabatic elimination of phi, not from a fitted coefficient or from assuming K_o in the input. The potential (18) is imported from Ref. [21] by the same research group, and the paper openly labels the outcome as "recover[ing] the stress and elasticity tensors of Ref. [21]" (Sec. 2.3), so the example is a consistency check rather than an independent prediction. The fast-relaxation condition behind Eq. (27) is an assumption cited to Chaikin-Lubensky and prior work, but it is a stated regime condition and not circular: replacing the angular-momentum balance by an algebraic relation does not reduce the result to its input by construction. The only notable self-citation is Ref. [21] serving as the source of the model potential; because the new derivation follows an independent Eulerian-PB route and the earlier result is peer-reviewed prior work, the central derivation retains independent content. No reduction-by-construction step was found.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central derivation is a deterministic algebra of PB identities given a Hamiltonian; the physical content enters through the imported potential Eq. (18), the 2D canonical pair, the scale separation |∇u|≪θ≪1, and the fast relaxation of θ. No new entities are postulated. The free parameters are constitutive inputs; none are fitted to data in this preprint, but their values determine the 'recovered' odd modulus.

free parameters (3)
  • Active torque density τ
    Input driving field; appears linearly in the result K_o=τ/4. Not fitted to data here, but its microscopic value is not derived.
  • Cosserat coupling κ_c
    Controls relaxation relation ϕ≈τ/(2κ_c) (Eq. 27); the odd modulus is proportional to τ/(2κ_c)×κ_c/2, so κ_c determines the modulus scale. Value imported from Ref [21].
  • Nonlinear elastic coefficients λ̃, μ̃
    Appear in geometric-nonlinearity terms of V^◦ (Eq. 18) and in the correction to ϕ (Eq. 27); their values are not derived here.
axioms (6)
  • domain assumption The 2D internal rotation θ and angular momentum ℓ form a canonical pair, {ℓ,θ}=δ(R−R′) (Eq. 24).
    The central PB algebra relies on this; in 3D the canonical momentum is M^{-1}ℓ, not ℓ, so the simple 2D structure does not carry over.
  • ad hoc to paper The elastic potential has the specific Cosserat form Eq. (18), including geometric nonlinearities with coefficients λ̃, μ̃, κ_c, taken from Ref. [21].
    This imported potential is the source of the nonlinear terms that later produce K_o=τ/4; a different coarse-grained potential would change the recovered elastic tensor.
  • ad hoc to paper Internal rotation relaxes fast to local torque balance, ∂H/∂θ=0 (Eq. 27).
    This adiabatic elimination converts angular-momentum dynamics into an algebraic relation; the odd modulus is a static modulus only under this assumption.
  • domain assumption Small-displacement, small-rotation expansion with |∇u|≪θ≪1; ∇θ and higher-order strain terms are negligible.
    Justifies truncating the potential at leading nonlinear order (Sec. 2.1); the separation of scales is an input.
  • domain assumption The deformation map R=r+u is one-to-one and the reference coordinate r_α is a fixed particle label, so ∂r_α/∂R_α=0 in Eq. (38).
    Underlies the non-gradient part of the PB {g^c,u}=δ_ijδ(R−R′) (Eq. 47); fails for flows with topological rearrangements or defect-mediated plasticity.
  • standard math Piola identity ∇^◦_j(JF^{-1}_{jl})=0 and standard continuum identities are used in Eq. (16).
    Standard differential geometry of the deformation gradient; no independent verification needed.

pith-pipeline@v1.3.0-daily-deepseek · 18786 in / 20774 out tokens · 205737 ms · 2026-08-03T05:19:45.382881+00:00 · methodology

0 comments
read the original abstract

The Poisson-bracket (PB) formalism is widely used to derive dynamics of coarse-grained (CG) fields to capture large-scale physics, extending the role of PBs in classical particle mechanics to macroscopic fields. It has been applied to fluctuations in critical phenomena, hydrodynamics of liquid crystals, liquid crystal elastomers, tissues, and the emergence of odd viscosity from spinning particles. The PB formalism can be formulated in either the Lagrangian framework, using reference space, or the Eulerian framework, using real space. Conventionally, the Lagrangian formulation is used for elastic solids, and the Eulerian one for fluids. However, growing interest in Eulerian descriptions of solids has emerged for phenomena naturally defined in real space, such as viscoelastic responses, moving interfaces, and field-induced structural changes in particles. Here we develop a systematic formulation for applying the Eulerian PB formalism to elastic systems with potentials typically written in Lagrangian space, and clarify its consistency with the Lagrangian counterpart. We show that the Eulerian formulation generates additional nonlinearities absent in the Lagrangian framework. Such nonlinearities originate from CG volume changes under coordinate transformation and from particle flow across neighboring CG volumes. They must be retained when nonlinear effects are important. To illustrate, we study chiral active solids of finite-sized particles, where active torques drive internal particle rotations and generate geometric nonlinearities. These nonlinearities give rise to the odd elastic modulus, which non-reciprocally couples two different shear modes in stress-strain response. By recovering this modulus directly from the Eulerian PB formalism, we demonstrate its ability to capture emergent nonlinear elastic behavior in driven active solids, whose stresses are naturally measured in real space.

Figures

Figures reproduced from arXiv: 2607.29521 by Cheng-Tai Lee, Tomer Markovich.

Figure 1
Figure 1. Figure 1: Transformation between Eulerian and Lagrangian frameworks. Reference [21] applied Lagrangian PB formalism to a Lagrangian elastic potential and then used the stress transformation to get the Cauchy stress (red dashed arrow). Here we first find the Eulerian Hamiltonian and directly apply Eulerian PB (blue solid arrow). Note that the Eulerian framework typically generates non-linearities. For example, a line… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Schematic of the change in the CG volume from Lagrangian (reference) to Eulerian (deformed) space. Consider tracing the same pack of particles within the Lagrangian CG volume ∆V ◦ (black solid square). In Lagrangian space, these particles remain confined within this CG volume, and crossing between neighboring CG volumes is not allowed. The particles have other properties such as their linear momenta (i… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Model of the isotropic, disordered chiral active solid studied here, represented as a network of interconnected rod-like particles in the spirit of Cosserat elasticity theory. (b) Each particle α has a center-of-mass (CM) displacement uα and can internally rotate by an angle θ α away from its equilibrium orientation, indicated by the blue line. This internal rotation can be driven by an active torque τ… view at source ↗

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