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REVIEW 5 major objections 5 minor 15 references

The principle of stationary action and Lagrangian for dissipative dynamics with velocity-proportional frictional force

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Pith's one-line read A Lagrangian with an imaginary quadratic term built from a contour Caputo half-derivative yields the linear-friction Euler-Lagrange equation, Hamilton equations, and energy-dissipation law without the short-time limit required by earlier…

desk verdict Original contour-fractional idea, but the central δS argument ignores the complex nature of the action and collapses; not publishable without major revision. read the letter →

arxiv 2608.13413 v1 pith:N5JBZA52 submitted 2026-08-13 cond-mat.mes-hall math-phmath.MPphysics.class-ph

Georgii Koniukov , Dmitry Nerukh This is my paper · ORCID
classification cond-mat.mes-hallmath-phmath.MPphysics.class-ph MSC 26A3370H2570H03
keywords fractionalcalculusCaputoderivativecontourstationaryactionprincipledissipativesystemsviscousfrictionEuler-Lagrangeequationenergydissipation
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

This paper tackles the old problem that velocity-proportional friction cannot be derived from the classical stationary-action principle using ordinary derivatives. It adds to the free Lagrangian an imaginary quadratic term built from a contour version of the Caputo fractional derivative of order $1/2$. Varying the resulting complex action gives the damped Euler-Lagrange equation $-\gamma\dot q = \frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q}$, a Hamilton-like system, and the energy law $dE/dt = -\gamma \dot q^2$. The point is that this is achieved without the unphysical collapse of the time interval that earlier fractional approaches required, and it comes with a geometric picture of dissipation as energy moving between complex branches of the square-root function.

What carries the argument

The contour Caputo fractional derivative of order $1/2$, defined for $t\in(t_0,t_1)$ by $$ {}_C $D^{{1/2}}$_{\Gamma_\varepsilon} q(t) = \frac{1}{\sqrt{\pi}}\int_{\Gamma_\varepsilon} \frac{\dot q(\Re\tau)}{(\tau-t)^{1/2}}\,d\tau\bigg|_{\arg\in[-\pi,\pi)}, $$ with the contour running along $s-i\varepsilon$ and $s+i\varepsilon$. In the $\varepsilon\to0^+$ limit it becomes $2i$ times the ordinary left-sided Caputo half-derivative, and the jump across the contour turns a half-plus-half composition into an ordinary first derivative, either through the semigroup property of fractional integrals or, in the appendix, through the standard Cauchy-kernel jump formula. This object carries the argument: it makes left and right fractional derivatives interchangeable without collapsing the interval $[t_0,t_1]$, the step that earlier fractional Lagrangians had to take.

What would settle it

Evaluate $\operatorname{Im}\delta S$ from equation (69) for a simple trajectory such as $q(t)=A\sin(\pi(t-t_0)/(t_1-t_0))$ and a generic allowed perturbation $\eta(t)=\sin(\pi(t-t_0)/(t_1-t_0))$, keeping the branch arguments explicit. If $\operatorname{Im}\delta S\neq0$ for such real variations, or if $\delta S=0$ only after discarding the imaginary part by an unstated rule, then the fundamental lemma has been applied to a complex condition and equation (71) is not the full stationarity condition.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the friction term $\gamma\dot q$ follows from a genuine variational principle once the Lagrangian is taken as $$L = \frac{m\dot $q^{2}$}{2} - U(q) + \frac{i\gamma}{8}\bigl( {}_C $D^{{1/2}}$_{\Gamma_\varepsilon} q(t)\bigr)^2,$$ with the contour Caputo half-derivative integrated along a contour that passes below and above the real axis. Varying the complex action $S[q]$ with the integration element on the branch $\arg\in[0,2\pi)$ and the fractional kernel on the branch $\arg\in[-\pi,\pi)$ gives, after the $\varepsilon\to0$ limit and the standard jump formula for the Cauchy-type kernel, the Euler-Lagrange equation $$-\gamma\dot q = \frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q},$$ the Hamilton-like system (81), and the energy law $dE/dt = -\gamma \dot q^2$. The paper interprets the intersection $[0,\pi)$ of the two argument intervals modulo the $4\pi$ periodicity of the square root as the geometric locus where the point system and the medium exchange energy.

Load-bearing premise

The whole construction depends on the claim that making a complex action stationary yields one real equation for the motion; the calculation never evaluates the imaginary part of the action's change, and it switches branch conventions for the square root without proving that the switch is harmless.

Editorial extensions

If this is right

  • For a particle with linear friction, the paper's Lagrangian yields $m\ddot q+\gamma\dot q=-dU/dq$ directly, so the friction term is obtained without auxiliary coordinates or a dissipation function.
  • The Hamiltonian system (81) has two canonical momenta, $p=\partial L/\partial\dot q$ and $p_{1/2}=\partial L/\partial q^{(1/2)}$, and reproduces the same dynamics, giving a phase-space description without a short-time limit.
  • The derived energy balance $dE/dt=-\gamma\dot q^2$ matches the mechanical power of the friction force, and the Hamiltonian's imaginary sector accounts for the dissipated part of the energy.
  • The branch-intersection picture offers a geometric reading of openness: the point system lives on one square-root branch, the medium on another, and their interaction is concentrated in the $\mod 4\pi$ intersection $[0,\pi)$ of the two argument intervals.

Reading between the lines

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

  • If the contour construction is consistent, the same branch-intersection mechanism should generalize to other fractional orders, producing dissipative equations of motion with derivatives of order $2\alpha$ from quadratic terms in an $\alpha$-derivative; the paper itself only works out the half-order case.
  • The imaginary part of $\delta S$, which the paper never computes, may carry its own physics: since the Hamiltonian has an imaginary sector, a natural next step would be to check whether $\operatorname{Im}\delta S$ encodes entropy production or a fluctuation-dissipation relation.
  • A direct numerical test on a finite interval, such as a damped oscillator with $q(t)=A\sin(\omega t)$ and a standard perturbation vanishing at the endpoints, could resolve how much of the result depends on the $\varepsilon\to0$ ordering and on the unstated projection onto the real part of the stationarity condition.
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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

5 major / 5 minor

Summary. The paper proposes a variational principle for a one-dimensional particle with linear friction. The Lagrangian is L = m qdot^2/2 - U(q) + (i gamma/8)(C D^{1/2}_{Gamma_epsilon} q)^2, where the fractional derivative is defined by a contour integral with a branch cut. The authors claim that the stationarity of this complex action yields the damped Euler-Lagrange equation -gamma qdot = d/dt(dL/dqdot) - dL/dq (Eq. 71), a Hamilton-like system (Eq. 81), and the energy dissipation law dE/dt = -gamma qdot^2 (Eq. 87), all without the unphysical a-to-b limit used by Riewe and by Lazo and Krumreich. The paper also offers a geometric interpretation of dissipation through complex-time branch intersections.

Significance. If the central derivation were valid, the paper would provide a single-Lagrangian variational description of linearly damped motion, which has been a long-standing problem in the field, and it would also supply a candidate Hamiltonian structure and an explicit dissipation law. The authors rightly identify genuine shortcomings of earlier fractional-action approaches, and the contour-derivative idea is original. However, the advertised results are not established by the manuscript: the complex action is reduced to one real Euler-Lagrange equation without any treatment of its imaginary part, the branch choices are switched without justification, and the final step of the appendix is asserted rather than computed. The paper is therefore currently a suggestive proposal rather than a derivation.

major comments (5)
  1. [Section V.A, Eqs. (57)-(71)] The reduction of delta S = 0 to the single real equation (71) is unsupported. In the epsilon-to-0 limit of Section III, the contour derivative is w = C D^{1/2}_{Gamma_epsilon} q = 2i C_{t0}D^{1/2}_t q, so for real q the quantity partial L/partial w = i gamma w / 4 is real, while delta w = 2i C D^{1/2}_t eta is purely imaginary for every real test function eta. Hence the integrand (partial L/partial w) delta w is purely imaginary, and integration by parts in Eqs. (66)-(68) cannot change this character. The fractional contribution in Eq. (69) is therefore imaginary, whereas the right-hand side of Eq. (70) is real. The paper never computes the imaginary part of delta S and never states a projection rule; without such a rule, delta S = 0 cannot produce Eq. (71).
  2. [Section IV and Eqs. (57)-(69)] The branch of the complexified time is switched without justification. The action integral is defined with arg C in [0, 2 pi), while the fractional derivative is defined with arg C in [-pi, pi), and after Eq. (65) the whole first variation is declared to be evaluated with arg C in [-pi, pi). Half-integer powers change sign under such branch changes, and the paper itself exhibits this: in Appendix A, Eq. (A10) assigns (tau - i epsilon - t0)^{1/2} -> -sqrt(tau - t0) under arg C in [0, 2 pi), whereas Eq. (A14) assigns (s - i epsilon - t0)^{1/2} -> +sqrt(s - t0) under arg C in [-pi, pi). Since the sign and coefficient of the friction force are exactly what the derivation is trying to obtain, this branch bookkeeping is load-bearing and cannot be left to an unexplained convention.
  3. [Appendix A, Eqs. (A16)-(A18)] The final step from the Sokhotski-Plemelj formula to -gamma qdot(tau) is not carried out. Formula (A16) applies to a difference of integrals with denominator s - tau - i epsilon and s - tau + i epsilon, but the expressions in (A17) contain additional factors such as (s - t1)^{1/2} and (s - t0)^{1/2} outside the standard formula, and their distributional limits compete with the pole contribution. The paper simply states the result in Eq. (A18). Without an explicit evaluation of these principal-value and half-power contributions, Eq. (71) remains an assertion rather than a derived statement.
  4. [Section VI and Appendix B] The Hamilton-like system (81) is based on an ad hoc treatment of q^{(1/2)} as an independent projection, and the paper itself concedes in Appendix B that no Ostrogradsky-type factorization yields consistent Hamilton equations for this model. The Legendre transformation treats q^{(1/2)} as an independent coordinate, but the canonical momentum p_{1/2} = partial L/partial q^{(1/2)} is then complex, and the paper does not establish a complex Hamilton variational principle that would justify this system. Thus Eqs. (81) are not derived from the action principle; they are a postulated structure whose compatibility with Eq. (71) is checked only informally.
  5. [Section VII, Eqs. (83)-(87)] The energy dissipation law is not an independent result; it is an algebraic consequence of the equation of motion already inserted. In Eq. (84) the term lim D^{1/2} p_{1/2} is replaced by -gamma qdot, which is precisely the Euler-Lagrange/Hamilton equation, and Eq. (85) defines E through H = E - (2i/gamma) p_{1/2}^2. Comparing Eqs. (84) and (86) then forces dE/dt = -gamma qdot^2. This is a restatement of the equation of motion, not a separate prediction of the Lagrangian formalism.
minor comments (5)
  1. [Section II.D] The sentence beginning 'The leads to substantial mathematical problems' should read 'This leads to substantial mathematical problems.'
  2. [Section III, Definition III.1] The orientation of the contour Gamma_epsilon is not specified in Definition III.1 itself; the reader must infer it from Fig. 1 and the subsequent parameterization. The definition should state the orientation and the fact that both horizontal branches are traversed.
  3. [Section IV, Eq. (45)] The 'intersection modulo 4 pi' operation A cap_sqrt B is not a well-defined operation on subsets of R as written, because the equivalence classes depend on shifts by multiples of 4 pi and the final representative is chosen arbitrarily. The claim that the intersection 'reduces to [0, pi)' needs a precise definition of the equivalence relation and the chosen representatives.
  4. [Eq. (52)] The prefactor i gamma / 8 is introduced without derivation; in particular, the 1/8 normalization is fixed only by demanding the final result -gamma qdot, so the construction has a free normalization that is not explained by the fractional calculus itself.
  5. [References] References [12] and [13] do not appear to be cited in the body of the manuscript; either cite them where relevant or remove them from the reference list.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the Euler-Lagrange and energy results are computed consequences of an explicit Lagrangian ansatz, not re-fitted inputs.

full rationale

I find no circular step in the paper. The target equation of motion, Eq. (2), is introduced as a problem statement, and the fractional Lagrangian in Eq. (52) is an explicit ansatz containing the same parameter gamma. The derivation of Eq. (71) from delta-S = 0 proceeds through Eqs. (57)-(70) and Appendix A, which is a nontrivial contour-derivative computation that does not assume Eq. (2). The coefficient gamma indeed appears both in the Lagrangian and in the resulting Euler-Lagrange equation, but that is the normal role of a coupling constant in a variational construction, not a fitted parameter renamed as a prediction. The energy law Eq. (87) is obtained after defining E in Eq. (85) as H + (2i/gamma) p_{1/2}^2; this definition makes E equal to the mechanical energy p^2/2m + U, and dE/dt = -gamma qdot^2 is then the standard consequence of the equation of motion, not a circularly imported result. The paper contains no load-bearing self-citations, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via citation. The serious mathematical concerns about the complex action - in particular that the fractional variation appears purely imaginary while Eq. (70) compares it to a real quantity, and the unheralded branch switch from [0,2pi) to [-pi,pi) - are correctness or validity issues, not circularity, and therefore do not raise the circularity score. The derivation chain is self-contained; whether it is sound is a separate question. Score 0.

Assumptions & free parameters 2 free parameters · 7 assumptions · 2 invented entities

The central construction depends on a new contour derivative, a handpicked Lagrangian prefactor, branch choices, and an ad hoc treatment of q^(1/2) as an independent coordinate. The energy dissipation law is recovered by defining E as the mechanical energy, so it is a restatement of the equation of motion rather than an independent result.

free parameters (2)
  • friction coefficient gamma in the fractional Lagrangian term = given input gamma>0
    The term i gamma/8 times the squared contour half-derivative is chosen so that its variational derivative equals -gamma q-dot; the value of gamma is taken from the physical friction, not derived.
  • Lagrangian prefactor i gamma/8 (including the 1/8 normalization and imaginary unit) = i gamma/8
    The 1/8 and the factor i are handpicked so that after the contour derivative contributes 2i times the left Caputo derivative, the final coefficient is -gamma; other prefactors would yield a different damping coefficient.
assumptions (7)
  • standard math Sokhotski-Plemelj formula (A16).
    Used in Appendix A to evaluate the epsilon to 0 limit of the four real integrals.
  • standard math Fundamental lemma of the calculus of variations.
    Invoked in Section V to extract Eq. (70) from delta S = 0 for arbitrary eta.
  • standard math Semigroup property of Riemann-Liouville fractional integrals.
    Used in Section II.D to motivate the composition of half-order derivatives; the paper's contour construction is designed to realize this.
  • domain assumption Regularity assumptions q in C^2([t0,t1]), eta in C^1 with eta(t0)=eta(t1)=0.
    Stated in Section V and needed for integration by parts and the fundamental lemma.
  • ad hoc to paper A complex action can be varied and the stationarity condition yields one real Euler-Lagrange equation.
    The action is complex because L is complex; the paper never analyzes the imaginary part of delta S or explains why only one real equation results.
  • ad hoc to paper Interchange of the epsilon to 0 limit with d/dt and with the integrals is permitted.
    Used throughout Appendix A; endpoint singularities and boundary contributions are not analyzed.
  • ad hoc to paper The half-derivative q^(1/2) can be treated as an independent projection in the Legendre transform.
    Introduced in Section VI; Appendix B admits that the Ostrogradsky construction failed for this model.
invented entities (2)
  • Contour Caputo fractional derivative C D^{1/2}_{Gamma_epsilon}
    purpose: Replaces left/right Caputo derivatives so that left and right operators can be interchanged and the a to b limit is avoided.
    A new mathematical operator; no falsifiable physical handle beyond the internal derivation is provided.
  • Complex time branch of the medium with 4-pi-periodic intersection
    purpose: Provides a geometric interpretation of how energy dissipates in Section IV.
    Qualitative interpretation only; no observable consequences are stated.

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Pith. "Pith review of The principle of stationary action and Lagrangian for dissipative dynamics with velocity-proportional frictional force." pith.science (2026). https://pith.science/paper/N5JBZA52

@misc{pith2026260813413,
  author       = {Pith},
  title        = {Pith review of: The principle of stationary action and Lagrangian for dissipative dynamics with velocity-proportional frictional force},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N5JBZA52}},
  note         = {Machine review of arXiv:2608.13413}
}
read the original abstract

It has been known for a long time that the equation of motion for dissipative linear dynamical systems with constant coefficients cannot be derived from the classical principle of stationary action because the term proportional to velocity in the equation of motion leads to the time derivative of order one half in the Lagrangian. Thus, approaches utilising fractional calculus have been used; however, they suffer from deficiencies both from mathematical and physical points of view. We here present our version of such fractional calculus based approach that provides correct Euler-Lagrange and, ultimately, the Hamilton equations, energy change of the moving body, and an attempt for a geometric interpretation of how energy dissipates.

Figures

Figures reproduced from arXiv: 2608.13413 by the authors.

Figure 1
Figure 1. Integration contour Γε ([t0, t1]) with a chosen cut. Let t0, t1 ∈ R, t0 < t1, and let q ∈ C 2 ([t0, t1]). The dot always denotes differentiation with respect to the real variable: q˙(t) = dq dt(t), t ∈ [t0, t1]. The complex variable appears only in the kernel of the contour integral. Definition III.1 (Contour Caputo fractional derivative). For t ∈ (t0, t1), define q(τ ) ≡ q(Re τ ), CD 1/2 Γε q(t) ≡ 1 √ π Z Γε q˙(Re … view at source ↗

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Works this paper leans on

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