Pith. sign in

REVIEW 2 major objections 4 minor 2 cited by

Hamiltonian treatment of non-conservative systems

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Nonconservative systems, usually intractable to action principles, are embedded here in a doubled symplectic phase space with an explicit Hamiltonian whose physical slice reproduces the original dissipative equations.

desk verdict Solid formal extension of Galley's doubled-variable formalism to Hamiltonians, with a genuine gauge freedom and reconstruction recipe, but the 'any IVP' claim needs sharpened regularity qualifications. read the letter →

arxiv 2507.18658 v1 pith:4LSVMRHU submitted 2025-07-23 physics.class-ph math-phmath.MP

Christopher Aykroyd , Adrien Bourgoin , Christophe Le Poncin-Lafitte This is my paper · ORCID
classification physics.class-phmath-phmath.MP MSC 70H0570H2537J0534A34 PACS 45.20.Jj
keywords double-variableactionnonconservativeHamiltonianmechanicsphysicallimitcanonicalgaugeshiftsymplecticembeddingdissipativedynamicsinitial-valueproblemLiederivativeformulation
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 seeks to remove a long-standing blockage: dissipative or otherwise nonconservative systems have no single-variable action principle of the usual kind, because classical actions are boundary-value problems and dissipation is an initial-value phenomenon. Its proposal is to double every coordinate into a physical copy and a virtual copy, couple them through a nonconservative term, and then take the physical limit where the two copies coincide. In that doubled setting the paper constructs a genuine Hamiltonian system on a symplectic manifold, shows the physical slice contracts in phase space while the doubled volume is preserved, and demonstrates a momentum-shift gauge freedom that leaves the dynamics unchanged. The central payoff is a reconstruction recipe: from essentially any second-order equation $\ddot{q}=U(q,\dot{q},t)$ one can write down, explicitly, a doubled Lagrangian and a doubled Hamiltonian whose physical limit reproduces the equation. A sympathetic reader would care because this imports Hamiltonian methods — perturbation theory, canonical transformations, Noether-style flux balance — into systems previously outside their reach.

What carries the argument

The load-bearing object is the doubled phase space with coordinates $(q_+,\pi_-,q_-,\pi_+)$ and the plus-minus variable parametrization, on which the Poisson bracket reads $\{\{q_+,\pi_-\}\}=\{\{q_-,\pi_+\}\}=\mathbb{1}_N$, paired with an antisymmetric doubled Hamiltonian. The central identity is the Lie-form Hamiltonian $A=\pi_-\cdot V(q_+,\pi_+,t)-q_-\cdot G(q_+,\pi_+,t)$, whose linear dependence on the virtual variables is exact for the dynamics; together with the on-shell time-derivative operator $D_t=\partial_t+\{\{\cdot,A\}\}$, it encodes the whole physical content of the theory. The Legendre transform of the doubled Lagrangian, with regularity condition $\det(\partial^2\Lambda/\partial\dot{q}_+\partial\dot{q}_-)\neq 0$, converts this into Hamilton's equations, and the canonical gauge shift $\phi$ (via $\pi_+\mapsto\pi_++\phi$) generates the family of equivalent Hamiltonians.

What would settle it

Take a constrained or non-smooth second-order ODE such as the pencil-on-a-cusp problem the paper cites, apply the reconstruction recipe to build the doubled Lagrangian and Hamiltonian, and look for two distinct solutions with identical physical initial data: if the physical limit fails to select a unique trajectory, or if the reconstructed Hamiltonian's physical-slice flow does not match the original $\ddot{q}=U(q,\dot{q},t)$, the central claim fails. A less singular check: pick a smoothly dissipative system, numerically integrate the doubled Hamiltonian with a symplectic integrator, and verify that the divergence of the physical-slice vector field equals the predicted contraction rate while the full doubled divergence remains zero.

Watch

Extended reading notes

Core claim

The paper's central claim is that nonconservative dynamics can be embedded, without approximation, into a conservative Hamiltonian system of twice the dimension. The doubled Hamiltonian has the linear 'Lie' form $A(q_\pm,\pi_\pm,t)=\pi_-\cdot V(q_+,\pi_+,t)-q_-\cdot G(q_+,\pi_+,t)$, where $V$ is an arbitrarily chosen momentum-to-velocity map and $G$ is the force expressed in phase-space variables; Hamilton's equations in the doubled variables are canonical, and on the physical slice $q_-=\pi_-=0$ they reduce precisely to the original equations of motion $\ddot{q}=U(q,\dot{q},t)$. The physical limit is not imposed by hand: with identical initial data for the two copies and an antisymmetric doubled Lagrangian, the standard uniqueness theorem for first-order ordinary differential equations forces the copies to coincide. The paper also exhibits a canonical gauge shift $\phi$ that changes the definition of momentum without changing the trajectories, and it uses this freedom both to clarify the Legendre transform and to give an explicit reconstruction recipe for arbitrary second-order ODEs. In this picture energy and phase-space volume are globally conserved in the doubled space, while the lower-dimensional physical slice dissipates.

Load-bearing premise

The construction works only if the doubled equations of motion can be written as a smooth first-order system whose right-hand side is Lipschitz-continuous (and the momentum map is invertible), so that the two copies starting from the same initial data are forced by uniqueness to stay identical; against singular, constrained, or non-smooth forces, such as the pencil-on-a-cusp example the paper cites, the physical limit may cease to be unique.

Editorial extensions

If this is right

  • Any second-order initial-value problem $\ddot{q}=U(q,\dot{q},t)$ admits an exact doubled-Lagrangian and doubled-Hamiltonian description, so Hamiltonian perturbation methods can be applied to nonconservative systems.
  • The physical slice dynamics can contract phase-space volume even though the global doubled flow is incompressible, reconciling dissipation with Liouville's theorem.
  • The momentum identity is not unique: shifting $\pi_+$ by an arbitrary function $\phi(q_+,\pi_+,t)$ leaves the equations of motion invariant, so one can choose the most convenient momentum definition, for instance kinetic momentum in electromagnetic contexts.
  • Conservative and nonconservative parts of a doubled Hamiltonian separate through a Helmholtz-type decomposition, giving a criterion for what counts as genuinely dissipative and enabling perturbative treatments of small dissipation.
  • Energy flux balance equations follow from the doubled Poisson bracket, generalizing Noether reasoning to systems that dissipate.

Reading between the lines

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

  • Editorial inference: the gauge freedom suggests that symplectic and variational integrators for dissipative systems could be constructed by discretizing the doubled action on the physical slice, with the gauge chosen to keep the integrator well-conditioned.
  • Editorial inference: the reconstruction recipe makes the inverse problem of Lagrangian mechanics nearly algorithmic for any smooth second-order ODE, which may extend to regularized or constrained formulations if the smoothness obstruction identified in the paper can be handled piecewise.
  • Editorial inference: connecting the Helmholtz decomposition to forces suggests a quantitative measure of 'dissipativity' of a force field as the size of its divergence-free part, which could be used to compare different nonconservative formalisms.
  • Editorial inference: because the doubled Hamiltonian has zero Noether energy on physical trajectories, conservation laws must be read as flux balances rather than constants of motion; this may alter how one assigns physical energy to open systems.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a doubled-variable (Schwinger-Keldysh-Galley) action principle for nonconservative systems, with the goal of extending Hamiltonian mechanics to dissipative dynamics. It clarifies boundary conditions for an initial-value formulation, argues that the physical limit q↑=q↓ emerges from uniqueness of the doubled Euler-Lagrange equations, introduces a canonical gauge freedom that shifts momenta without changing physical dynamics, and constructs doubled Hamiltonians via Legendre transform. It also presents a linearized 'Lie' form of the action and Hamiltonian, and gives a reconstruction recipe that builds Lagrangians and Hamiltonians from second-order ODEs of the form ẍ=U(x,ẋ,t). The main advertised result is that virtually any classical initial-value problem can be embedded on an enlarged symplectic manifold with explicit Hamiltonian and Lagrangian functions.

Significance. If it holds, the paper provides a concrete, explicit mechanism for embedding broad classes of nonconservative second-order systems into a higher-dimensional Hamiltonian framework, with a physical-limit selection mechanism and a gauge family. The derivations are internally consistent for smooth, regular data, and the reconstruction formulas (Sec. V) are checkable by direct substitution; the paper is also honest in its concluding remarks about regularity limitations. The inverse-problem construction is definitional rather than predictive, since the Hamiltonian is built from a prescribed acceleration function U, but it is nevertheless a useful canonicalization tool. The clarity on boundary conditions and the linear Lie formulation are genuine contributions, although the advertised breadth of the central claim is broader than what the proofs establish.

major comments (2)
  1. [Abstract and Sec. V, Eqs. (119)–(133)] The central claim that 'virtually any classical initial-value problem' (abstract) and 'any second-order ODE' (Sec. V) can be embedded is stated without the regularity hypotheses that the construction itself requires. The reconstruction Lagrangian (121) and Hamiltonian (133) are well defined as formal expressions, but the Euler-Lagrange equations require differentiability of U, the physical-limit uniqueness argument in Sec. II (Eqs. (26)-(29)) requires the doubled first-order system to have a locally Lipschitz right-hand side, and the Legendre transform requires det H ≠ 0 in Eq. (55). For non-smooth U, such as Coulomb friction, U(q)=|q|^{1/2}, or impact-type nonsmoothness, the variational principle does not select a unique physical trajectory, and the conclusion (Sec. VI) concedes that regularity assumptions 'might not hold globally' (pencil-on-a-cusp). The abstract and Sec. V should carry the same qualification, ideally as a precise theorem stating U ∈ C^1 (or Lipschitz) and P a local diffeomorphism; otherwise the advertised breadth overreaches the proved statement.
  2. [Sec. II, Eqs. (22)–(29)] The 'physical limit emerges naturally' argument should be stated as a theorem with explicit hypotheses. The variational problem imposes both the initial conditions (23) and the final equality conditions (25); the claim that the final conditions are automatically satisfied by the unique solution of the IVP relies on Picard-Lindelöf uniqueness for the doubled Euler-Lagrange system, which in turn requires the EL equations to be reducible to standard first-order form with a Lipschitz vector field. The authors mention 'usual conditions' but never specify the required smoothness of Λ and the non-degeneracy of the Hessian needed to solve for the doubled accelerations. For non-regular or constrained Lagrangians, q↑ = q↓ may fail to be forced, so the physical limit is not 'natural' in those cases; the scope of the uniqueness claim should be made precise.
minor comments (4)
  1. [Sec. II D] The sentence 'A full demonstration in the non-linearized formulation is also provided in Sec. IIIC' is inaccurate; Sec. IIIC works to first order in the minus variables and discards O(−3). Since higher-order terms do not affect the physical equations this is sufficient, but the claim of a full nonlinear demonstration should be revised.
  2. [Throughout] There are several typos and grammatical slips that should be corrected: 'altough' (Sec. II A), 'varing' (Sec. IIC), 'constrined' (Fig. 2 caption), 'recognised as is as the' (footnote 2), and the introduction's 'arbitrary discrete initial-value equations' (probably should be 'dissipative' or 'given').
  3. [Eq. (22) and Eq. (33)] Eq. (22) uses δ_ab for the boundary term while Eq. (33) later introduces the path-label metric η_ab; a brief note clarifying why the metric does not appear in Eq. (22) would prevent confusion.
  4. [Sec. III A 1, Eq. (78)] The notation f± for the ± combination of f↑ and f↓ can be confusing because f+ is also used to denote evaluation at (q+,π+); consider a different symbol, such as f^±, to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; Sec. V is an explicitly inverse construction and the paper's load-bearing derivations are self-contained.

full rationale

The paper is transparent about the direction of its main construction: Sec. V is titled 'Reconstruction of variational principles from second-order equations of motion' and builds the Lagrangian and Hamiltonian from the given acceleration function. Equations (120), (126), and (133) place U, or the momentum-space force G derived from U, directly into the generator, so recovering q-double-dot = U from Hamilton's equations is a verification of the construction, not an independent prediction. The paper does not present this as a first-principles derivation, so the reconstruction is not a circular step. The physical-limit emergence is argued from the anti-symmetry of the doubled Lagrangian plus Picard-Lindelof uniqueness under explicitly stated regularity assumptions; the conclusion even qualifies the result for constrained or non-smooth systems such as the pencil-on-a-cusp problem. The gauge freedom, symplectic structure, and Liouville-volume reconciliation are derived algebraically from the Legendre transform and the doubled Poisson bracket. The only self-reference is a pointer to a companion paper on Lie perturbation methods, which is not used as evidence for any claim in this work. Therefore no circular step meeting the enumerated patterns is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central reconstruction theorem is a representation result: it presupposes the target ODE and constructs a Hamiltonian that contains the acceleration function U as an input. The true new content of the paper, the canonical gauge freedom and the physical-limit argument, rests on standard ODE uniqueness and regularity assumptions rather than on fitted parameters. No empirical constants are introduced.

free parameters (3)
  • Gauge function φ(q_+, ˙q_+, t) = arbitrary (e.g., φ = -∂K/∂˙q_- for the conservative gauge)
    Introduced in Secs. IID and IIIC, it shifts the physical momentum π_+ -> π_+ + φ without changing the physical dynamics. The whole family of gauge-related Hamiltonians is parametrized by φ, so the Hamiltonian representation is not unique.
  • Momentum map P(q, ˙q, t) = arbitrary local diffeomorphism in ˙q (e.g., P = ˙q)
    Used in the reconstruction, Eq. (122), to define the physical momentum when building a Lagrangian or Hamiltonian from a given ODE. Different choices of P give different Lagrangians and Hamiltonians for the same equations of motion.
  • Conservative/dissipative split (L, K) = non-unique up to harmonic functions and boundary terms
    Appendix A shows the separation of Λ into a conservative L and nonconservative K is unique only if decay conditions are imposed; otherwise harmonic additions leave the dynamics invariant. This is a free functional choice in the formalism.
assumptions (4)
  • standard math Picard-Lindelöf existence and uniqueness for the doubled Euler-Lagrange system
    Sec. II after Eq. (26): the physical limit q↑ = q↓ is concluded from identical initial data plus uniqueness. The theorem requires a Lipschitz right-hand side, which is assumed but not proven for generic K.
  • domain assumption Regularity of the doubled Lagrangian: det H = det(∂²Λ/∂˙q_+∂˙q_-) ≠ 0 in the physical limit
    Eq. (55): needed for the Legendre transform to be a local diffeomorphism and for the Hamiltonian to exist. The reconstruction section requires this to hold globally.
  • domain assumption Antisymmetry of the coupling term K under label exchange
    Eq. (20): imposed so that the action is antisymmetric; it is essential for the physical-limit argument and for the cancellation of boundary terms, but it is a modeling restriction on admissible K.
  • standard math Helmholtz decomposition with decay at infinity, giving a unique split into gradient and divergence-free parts
    Appendix A: used to distinguish conservative from nonconservative contributions; uniqueness requires decay or specified boundary conditions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hamiltonian treatment of non-conservative systems." pith.science (2026). https://pith.science/paper/4LSVMRHU

@misc{pith2026250718658,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian treatment of non-conservative systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4LSVMRHU}},
  note         = {Machine review of arXiv:2507.18658}
}
read the original abstract

We present a novel extension of Hamiltonian mechanics to nonconservative systems built upon the Schwinger-Keldysh-Galley double-variable action principle. Departing from Galley's initial-value action, we clarify important subtleties regarding boundary conditions, the emergence of the physical-limit trajectory, and the decomposition of the Lagrangian into conservative and dissipative sectors. Importantly, we demonstrate that the redundant doubled configuration space admits a gauge freedom at the level of the canonical momenta that leaves the physical dynamics unchanged. From a Legendre transform, we construct the corresponding family of gauge-related nonconservative Hamiltonians; we show that virtually any classical initial-value problem can be embedded on our enlarged symplectic manifold, supplying the associated Hamiltonian and Lagrangian functions explicitly. As a further contribution, we derive a completely equivalent linear ``Lie'' formulation of the double-variable action and Hamiltonian which streamlines computations and renders transparent many structural properties of the formalism.

Figures

Figures reproduced from arXiv: 2507.18658 by the authors.

Figure 1
Figure 1. Illustration of the principle of stationary action [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Phase-space slices for the damped harmonic oscillator in the doubled-variable Hamiltonian formalism. On the [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Equilibrium Propagation for Non-Conservative Systems

    cs.LG 2026-02 conditional novelty 6.0 of 10

    A modified Equilibrium Propagation with an antisymmetric-Jacobian correction computes exact cost gradients for non-conservative neural dynamics.

  2. First Principles Quantization of a Non-Conservative Scalar Field

    hep-th 2025-09 reject novelty 4.0 of 10

    A damped scalar field is quantized via doubled variables; the path-integral propagators match in-in results, but the canonical quantization has an internal inconsistency.

Reference graph

Works this paper leans on

33 extracted references · 29 canonical work pages · cited by 2 Pith papers

  1. [1]

    Thisnotationhelpsisolategenuinephysicalcontributions of the dynamics—those surviving in the physical limit— from redundant “virtual” terms

    Additional Notation and Perturbative Expansions To complete our discussion of the symplectic frame- work on the doubled phase space, it proves convenient to introduce a set of notational conventions that make per- turbative expansions in the minus variables transparent. Thisnotationhelpsisolategenuinephysicalcontributions of the dynamics—those surviving i...

    work page

  2. [2]

    Fractional derivatives Fractional derivatives, which generalize ordinary dif- ferentiation to non-integer orders, offer another path- way for formulating nonconservative action principles. By incorporating terms in the Lagrangian that depend on fractional time-derivatives of the dynamical variables [5, 6, 11, 12], the resulting Euler-Lagrange equations na...

    work page

  3. [3]

    extended phase space approach

    Contact mechanics From a Hamiltonian standpoint, “contact” geometry provides a higher-dimensional phase space (a contact manifold) in which dissipative processes behave as flows preserving a contact form [19]. The contact manifold can be seen as an extension of the traditional symplec- tic phase space, where the extra dimension is not the time variable (a...

    work page

  4. [4]

    mirror” variable that effectively absorbs the lost energy. Each mirror variable evolves according to an “ad- joint

    Extra degrees of freedom Finally, one can circumvent Bauer’s corollary by ex- panding the system’s configuration space, and thus em- bedding it into a conservative framework of higher- dimension. A classical example is Bateman’s trick [7]: pair each dissipative degree of freedom with an auxil- iary “mirror” variable that effectively absorbs the lost energ...

    work page

  5. [5]

    Liouville volume One of the most remarkable implications of the doubled-variable Hamiltonian formulation is its ability to reconcile nonconservative dynamics with the incom- pressibleflowsmandatedbyLiouville’stheorem. Naively, this looks contradictory, since these systems must gener- ally lose energy and contract their phase space, whereas Hamiltonian sys...

    work page

  6. [6]

    gauge freedom

    Example: As an illustrative example, we consider the damped harmonic oscillator, which can be described by the Hamiltonian (see Sec. V for details on how to construct it from the equations of motion): A(q±, π±) = 1 m π−π+ + q− γ m π+ + kq+ , (89) and the following equations of motion: ˙q+ = π+ m, ˙q− = π− m + γ m q−, (90a) ˙π+ = − γ m π+ − kq+, ˙π− = −kq−...

    work page

  7. [7]

    P. S. Bauer, Proceedings of the National Academy of Sci- ences 17, 311 (1931)

    work page 1931

  8. [8]

    M. E. Gurtin,Variational principles for linear elastody- namics, Vol. 16 (Brown University, 1963) pp. 34–50

    work page 1963

Show all 33 references
  1. [9]

    M. E. Gurtin, Quarterly of Applied Mathematics22, 252 (1964)

    1964

  2. [10]

    Tonti, Annali di Matematica pura ed applicata95, 331 (1973)

    E. Tonti, Annali di Matematica pura ed applicata95, 331 (1973)

    1973

  3. [11]

    G. F. Dargush and J. Kim, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 85, 066606 (2012)

    2012

  4. [12]

    G. F. Dargush, Phys. Rev. E86, 066606 (2012)

    2012

  5. [13]

    Bateman, Phys

    H. Bateman, Phys. Rev.38, 815 (1931)

    1931

  6. [14]

    Schwinger, Journal of Mathematical Physics 2, 407 (1961)

    J. Schwinger, Journal of Mathematical Physics 2, 407 (1961)

    1961

  7. [15]

    L. V. Keldysh, Sov. Phys. JETP20, 1018 (1965)

    1965

  8. [16]

    C.R.Galley,PhysicalReviewLetters 110,10.1103/phys- revlett.110.174301 (2013)

    2013 doi

  9. [17]

    Riewe, Phys

    F. Riewe, Phys. Rev. E53, 1890 (1996)

    1996

  10. [18]

    Riewe, Phys

    F. Riewe, Phys. Rev. E55, 3581 (1997)

    1997

  11. [19]

    Caldirola, Il Nuovo Cimento18, 393 (1941)

    P. Caldirola, Il Nuovo Cimento18, 393 (1941)

    1941

  12. [20]

    Kanai, Progress of Theoretical Physics3, 440 (1948)

    E. Kanai, Progress of Theoretical Physics3, 440 (1948)

    1948

  13. [21]

    E.A.CalzettaandB.-L.B.Hu, Nonequilibrium Quantum Field Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2008)

    2008

  14. [22]

    J. S. B. Rayleigh and S. Baron, The theory of sound, vol. 2, revised and enlarged macmillan, Vol. 132 (London: Macmillan and Co., 1896)

  15. [23]

    E. G. Virga, Phys. Rev. E91, 013203 (2015)

    2015

  16. [24]

    Damour, P

    T. Damour, P. Jaranowski, and G. Schäfer, Phys. Rev. D 93, 084014 (2016)

    2016

  17. [25]

    Bravetti, H

    A. Bravetti, H. Cruz, and D. Tapias, Annals of Physics 376, 17 (2017)

    2017

  18. [26]

    Dekker, Phys

    H. Dekker, Phys. Rep.80, 1 (1981)

    1981

  19. [27]

    Kosyakov,Introduction to the classical theory of par- ticles and fields (Springer Science & Business Media, 2007)

    B. Kosyakov,Introduction to the classical theory of par- ticles and fields (Springer Science & Business Media, 2007)

    2007

  20. [28]

    P. M. Morse and H. Feshbach,Methods of Theoretical Physics. Vol. 1-2(McGraw-Hill, 1953)

    1953

  21. [29]

    Um, K.-H

    C.-I. Um, K.-H. Yeon, and T. F. George, Phys. Rep.362, 63 (2002)

    2002

  22. [30]

    C. R. Galley and M. Tiglio, Phys. Rev. D79, 124027 (2009)

    2009

  23. [31]

    C. R. Galley, D. Tsang, and L. C. Stein, The principle of stationary nonconservative action for classical mechanics and field theories (2014), arXiv:1412.3082 [math-ph]

    2014 arXiv

  24. [32]

    Smale, Inventiones Mathematicae11, 45 (1970)

    S. Smale, Inventiones Mathematicae11, 45 (1970)

    1970

  25. [33]

    Tsang, C

    D. Tsang, C. R. Galley, L. C. Stein, and A. Turner, ApJL 809, L9 (2015). 21 Appendix A: Helmholtz decomposition Consider a smooth function, antisymmetric under label-exchange, decomposed into the form f (x±) = x− · F (x+) + O(−)3, (A1) assuming F a square-integrable vector fie...

    2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.