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REVIEW 6 major objections 5 minor 12 references

Alternative writings of classical elastodynamics equations as a first order symmetric system

T0 review · 6 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Classical elastodynamics can be written as a first-order symmetric system for eleven different choices of variables.

desk verdict The 1D systems and the simpler 2D systems check out, but the four-variable 2D matrices in §3.4 have concrete symmetry and assembly errors, so the paper is a conditionally useful working draft rather than a finished result. read the letter →

arxiv 2501.04743 v1 pith:NIKJFAFH submitted 2025-01-08 physics.class-ph

Dimitris Sfyris This is my paper · ORCID
classification physics.class-ph MSC 74B0574B9935L0235Q74
keywords LineartheoryFirstordersymmetricsystemElastodynamicsAnisotropicelasticityCompatibilityequationsTime-differentiatedconstitutivelawhyperbolicMomentum-velocityconjugacy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that the equations of classical linear anisotropic elastodynamics, in one and two space dimensions, admit first-order symmetric writings of the form $A \frac{\partial q}{\partial t} + \sum_i B_i \frac{\partial q}{\partial x_i}=0$ for a menu of variable sets: velocity with displacement gradient or strain, velocity with stress, all three together, and variants in which momentum $\rho_0 u_t$ replaces or accompanies velocity. The construction is inverse: the author writes the desired system, forces the spatial-derivative matrices $B_i$ to be symmetric by combining compatibility equations, the momentum equation, and the time-differentiated constitutive law, and then checks that the time-derivative matrix $A$ is symmetric and positive definite. If the central claim is right, every listed writing is a legitimate symmetric hyperbolic formulation of classical elastodynamics, and that matters because symmetric form is the standard route to energy estimates, well-posedness arguments, and finite-element treatments of wave propagation.

What carries the argument

The machinery is an inverse symmetrization procedure. Starting from the first-order form, the author fixes a variable vector $q$ and writes candidate field equations drawn from three sources: the momentum equation, the compatibility equations linking strain components to velocity gradients, and the time-differentiated constitutive law $\dot\sigma = C:\dot e$ (written through the compliance $C^{-1}$ for stress-based systems). These equations are multiplied and added so that every spatial-derivative matrix $B_i$ becomes symmetric; the same additions alter the time-derivative matrix $A$, and the writing is accepted when $A$ comes out symmetric and, for hyperbolicity, positive definite. The compatibility equations carry the strain- or displacement-gradient-based writings, while the time-differentiated constitutive law carries the stress-based writings.

What would settle it

Solve the 1D three-variable system of eqs. (30)-(32) with constant $\alpha$ and initial data $u_t(0,x)=0$, $u_x(0,x)=0$, $\sigma(0,x)=1$. If the solution develops $\sigma-\alpha u_x\neq 0$ while satisfying the system, the symmetric writing admits non-elastodynamic motions and full equivalence fails; if the constraint is preserved, equivalence is confirmed.

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Extended reading notes

Core claim

The paper's central claim is a catalog: for the one-dimensional case it produces symmetric writings with respect to $(u_t,u_x)$, $(u_t,\sigma)$, $(u_t,u_x,\sigma)$, $(\rho_0 u_t,u_x)$, $(\rho_0 u_t,\sigma)$, $(\rho_0 u_t,u_x,\sigma)$, and $(\rho_0 u_t,u_t,u_x,\sigma)$; for the two-dimensional case it produces symmetric writings with respect to $(u_t,\mathbf e)$, $(u_t,\sigma)$, $(u_t,\mathbf e,\sigma)$, and $(\rho_0 u_t,u_t,\mathbf e,\sigma)$. In each writing the matrices $A$ and $B_i$ are explicitly symmetric, and $A$ is positive definite when the material parameters satisfy the stated conditions ($\alpha>0$ in 1D; the usual major and minor symmetries plus the displayed invertibility and determinant conditions in 2D). The paper also reports a correction to an earlier two-dimensional $(u_t,\mathbf e)$ writing: the earlier version had dependent compatibility equations and only positive semidefinite $A$, while the new five-equation version has independent equations and an invertible $A$.

Load-bearing premise

The rewrite replaces the material law (stress equals elasticity times strain) by its time-derivative form, so the new systems agree with classical elastodynamics only for initial data that already satisfy the material law, and the paper does not prove that the evolution keeps those data on it.

Editorial extensions

If this is right

  • Each of the eleven writings is a symmetric first-order system, so Friedrichs' theory of symmetric hyperbolic systems applies and energy estimates can be obtained in every variable set.
  • The combined writings that use velocity, strain or displacement gradient, and stress tie the compatibility equations and the differentiated constitutive law into one system, which the paper suggests as a setting for studying hyperbolicity of implicit constitutive theories.
  • The momentum-based writings make the velocity-momentum conjugacy explicit, connecting the symmetric forms directly to kinetic energy and to candidate Hamiltonian structures.
  • The corrected two-dimensional $(u_t,\mathbf e)$ writing has five independent equations with an invertible symmetric $A$, replacing the earlier six-equation version whose redundant compatibility equations left $A$ only positive semidefinite.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the stress- and combined-variable systems use the time-differentiated constitutive law, a reader should test numerically whether the algebraic constraint $\sigma=C:\mathbf e$ is preserved along solutions; if it drifts, the equivalence to classical elastodynamics holds only on the constrained initial-data submanifold.
  • The same inverse symmetrization recipe should extend to three dimensions and to nonlinear or implicit constitutive laws, at the cost of heavier bookkeeping; the paper stops at 2D and describes 3D as a straightforward generalization.
  • The four-variable writings with both momentum and velocity are deliberately redundant, so they are better read as templates for variational or port-Hamiltonian formulations than as minimal evolution systems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 5 minor

Summary. The paper proposes an inverse construction of first-order symmetric systems for classical linear elastodynamics. In 1D it presents symmetric writings for seven variable sets, including (u_t,u_x), (u_t,σ), (u_t,u_x,σ), momentum-based versions, and a four-variable set (ρ0u_t,u_t,u_x,σ); in 2D it presents four writings, the most complex being (ρ0u_t,u_t,e,σ). The construction forms linear combinations of the momentum equation, compatibility equations, and the time-differentiated constitutive law, then verifies symmetry of the resulting matrices A and B_i. The paper also contains a remark correcting a mistake in the author's earlier work [8].

Significance. If the construction is correct, the paper offers a useful catalog of symmetric first-order formulations of anisotropic linear elastodynamics, with potential applications to finite element methods, Hamiltonian formulations, and semigroup theory. The approach is fully algebraic and parameter-free, and the 1D constructions together with the simpler 2D systems are largely plausible. The main weakness is that the most complex 2D writing, §3.4, is not correct as printed, due to numerous inconsistencies between the displayed equations and the assembled matrices; these appear to be repairable typographical errors rather than a fundamental flaw, but they must be fixed before the central claim can be accepted.

major comments (6)
  1. [§3.4, Eq. (158)] Equation (158) is missing the factor 2 multiplying v2,t. Adding Eqs. (156) and (157) yields 2v2,t − ... = 0, and the (2,2) entry of A in Eq. (184) is 2. As printed, the equation does not match the assembled system.
  2. [§3.4, Eqs. (168)–(170)] The coefficient of v1,1 in Eq. (168) and of v5,1 in Eq. (169) is printed as −C1222, but the compatibility equation (138) from §3.3 requires −C1122. The error is carried into the sum (170) and then into the B1 matrix.
  3. [§3.4, Eq. (185)] Matrix B1 is not symmetric as printed and does not reproduce the stated field equations. Row 5, column 6 should be −C1122 (from Eq. (167)) but is printed as −C1222; row 6, columns 1 and 5 should be −C1122 (from the corrected Eq. (170)) but are printed as −C1222; row 3, column 7 should be −C2111 (from Eq. (161)) but is printed as −C1211; row 4, column 7 should be −2C2112 (from Eq. (164)) but is printed as −2C2122. With these corrections B1 becomes symmetric.
  4. [§3.3, Eq. (141)] The terms multiplying v7,t and v8,t are interchanged. The equation should read C−1_2211 v6,t + C−1_2222 v7,t + 2C−1_2212 v8,t − v5,2 = 0 to agree with the A matrix (144) and with the structure of the time-differentiated constitutive law.
  5. [§3.2, Eqs. (127)–(129)] The terms C−1_1122 v4,2, C−1_2222 v4,2, and C−1_1222 v4,2 must be time derivatives v4,t. As printed, these equations are inconsistent with the A matrix (131) and with the fact that v4 = σ22.
  6. [§2.2–§3.4, constraint preservation] The systems that use the time-differentiated constitutive law as a field equation are equivalent to classical elastodynamics only for initial data satisfying σ = C:e (or σ = α ux in 1D). The paper does not state or prove that this constraint is preserved. For time-independent coefficients preservation is immediate (wt = 0); for the 1D case with α = α(t) one obtains wt = (αt/α)w. This gap should be closed explicitly. Relatedly, Eq. (119) in §3.2 omits the (C−1)t σ term that appears in the 1D treatment (Eqs. (15)–(16)); the paper should state whether C is assumed time-independent.
minor comments (5)
  1. [§3.3, Eq. (136)] The last term should be −2C2212 v5,2, not −C2212 v5,2, to match the A and B matrices.
  2. [§3.4, notation] The expressions 'C11122' and 'C12122' in Eqs. (153)–(158), (167), and (173) should be written as '2C1112' and '2C1212' to avoid ambiguity.
  3. [§3.4, Eq. (178)] Equation (178) is garbled: '2C−1_12122ρ0v10,t' should be '2ρ0 C−1_2212 v10,t' (or equivalently '2ρ0 C−1_2122 v10,t') to match Eq. (177) multiplied by ρ0.
  4. [§3.1 and §3.3] The paper verifies only that det A is nonzero, but for a symmetric hyperbolic system in the sense of Friedrichs the matrix A must be positive definite. The relevant blocks are Hessians of the (positive definite) elastic energy and complementary energy, so positive definiteness holds, but this should be stated explicitly.
  5. [§3.2, Eq. (119)] The time-differentiated constitutive law is written as C−1 σ̇ = ė, which silently assumes time-independent C; if time-dependent coefficients are allowed, as the 1D treatment does, the term (C−1)t σ must be included.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the constructions are explicit algebraic derivations from the stated equations, with self-citation only as methodological background.

full rationale

The paper's central claim is that each listed variable set admits a symmetric first-order system A q_t + sum B_i q_{x_i} = 0. The derivation is self-contained: for each writing, the paper states the starting equations (momentum, compatibility, time-differentiated constitutive law), declares the linear combination used, and displays the resulting A and B_i matrices. Symmetry is then a direct check, not an imported result. The self-citation to [8] is used only to motivate the 'inverse' strategy and is explicitly corrected in the Remark in Section 3.1 ('This is due to a mistake made in [8]'), so it is not load-bearing. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported, and no ansatz is smuggled via citation. The time-differentiated constitutive law used in Sections 2.2, 2.3, 3.2, 3.3 and 3.4 does introduce a constraint issue: equivalence to classical elastodynamics holds only for initial data satisfying sigma = C:e, and the paper does not prove constraint preservation. This is a mathematical assumption or gap, not circularity. Likewise, the apparent asymmetry and entry mismatches in the Section 3.4 matrices (e.g., B1 entry (1,6) = -C1122 versus (6,1) = -C1222) are correctness defects in the printed matrices, not a circular reduction of the claimed result to its inputs. The central derivation therefore has independent mathematical content: each proposed symmetric system can be verified directly from the displayed equations, and if an individual matrix is wrong, it is a computational error rather than a circular argument. Score 1 merely acknowledges the paper's reliance on the author's own prior work [8] for the overall strategy.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No numbers are fitted to data and no new entities are introduced. The central claim rests only on standard elasticity assumptions (symmetries, invertibility, positive definiteness) and on the constraint-manifold condition needed because constitutive laws enter solely through their time derivatives.

assumptions (3)
  • domain assumption The elasticity tensor C (or the 1D coefficient α) has major and minor symmetries and is positive definite, hence invertible.
    The symmetrization of the 2D matrices uses C_ijkl = C_jikl = C_ijlk = C_klij, and the writings involving stress require C^{-1}. Invertibility is implied by positive definiteness but is not explicitly stated where C^{-1} is introduced (§3.2, §3.3, §3.4).
  • domain assumption The solution is constrained to the constitutive manifold σ = C:e (or σ = αu_x in 1D), and this constraint is preserved by the new evolution equations.
    The systems replace the algebraic constitutive law by its time derivative, so they are equivalent to elastodynamics only for initial data satisfying the constitutive relation. Preservation of the constraint is standard for these systems but is never stated or proved in the paper (§2.3, §3.2, §3.3, §3.4).
  • domain assumption In 1D, α ≠ 0 for writings involving σ, and ρ0 > 0 for momentum writings.
    The inversion of α is explicitly required in §2.2, and positive definiteness of the A matrix is checked for α>0 and ρ0>0 in §2.1-§2.7. These are reasonable physical assumptions for classical elasticity.

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Pith. "Pith review of Alternative writings of classical elastodynamics equations as a first order symmetric system." pith.science (2026). https://pith.science/paper/NIKJFAFH

@misc{pith2026250104743,
  author       = {Pith},
  title        = {Pith review of: Alternative writings of classical elastodynamics equations as a first order symmetric system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIKJFAFH}},
  note         = {Machine review of arXiv:2501.04743}
}
abstract

We explore alternative writings of the equations of classical elastodynamics as a first order symmetric system. In the one dimensional case we present symmetric writings with respect to: i) the velocity ($u_t$) and the displacement gradient ($u_x$), ii) the velocity and stress ($\sigma$), iii) all three quantities: the velocity, the displacement gradient and the stress, and finally iv) the momentum ($\rho_0 u_t$), the velocity, the displacement gradient and the stress. In the two dimensional case we present similar writings with respect to: i) the velocity (${\bf u}_t$) and the strain tensor ($\bf e$), ii) the velocity and the stress tensor ($\boldsymbol \sigma$), iii) all three variables $({\bf u}_t, {\bf e}, \boldsymbol \sigma)$, and finally iv) one more writing utilizing the momentum as well, i.e. $(\rho_0 {\bf u}_t, {\bf u}_t, {\bf e}, \boldsymbol \sigma)$. We accomplish our goal by judiciously using the compatibility equations as well as the momentum equation and the time differentiated constitutive law. This is done in an inverse way: we start by writing our initial equations as a first order system of the form ($\bf q$ being the vector representing the variables in each writing) $$ A \frac{\partial {\bf q}}{\partial t}+\sum_{i=1}^n B_i \frac{\partial {\bf q}}{\partial x_i}=0, $$ with $n=1, 2$ depending on whether we are in 1 or 2 dimensions. We then check what are the symmetric forms of matrices $B_i$ and which combinations of the compatibility equations, the momentum equations and the time differentiated constitutive law should be used in order the symmetric form of matrices $B_i$ to appear into the system. This "symmetrization" process alters matrix $A$ and if the resulting matrix $A$ is symmetric our goal is accomplished. Our analysis is confined to classical elastodynamics, namely geometrically and materially linear anisotropic elasticity.

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Reference graph

Works this paper leans on

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