On a geometric description of time dependent singular Lagrangians with applications to biological systems

A Ghose-Choudhury; Partha Guha; Sudip Garai

REVIEW 3 major objections 5 minor 26 references

On a geometric description of time dependent singular Lagrangians with applications to biological systems

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that the Kill the Winner model and its reductions admit a singular Lagrangian built from a Jacobi last multiplier, and that a Hamiltonian description can be supplied by Dirac brackets.

desk verdict The Lagrangian-construction half is solid and the m-species KtW Lagrangian is new, but the Dirac-bracket calculation gives ˙x=˙y=0 in every example, so the central Hamiltonian claim fails internally. read the letter →

arxiv 1908.09647 v2 pith:2V7ACFB4 submitted 2019-08-20 q-bio.PE nlin.SI

Sudip Garai , A Ghose-Choudhury , Partha Guha This is my paper · ORCID

classification q-bio.PEnlin.SI MSC 37J0570H4592D25 PACS 45.20.Jj11.30.Qc03.65.Ca11.30.Er

keywords singularLagrangiansJacobilastmultiplierDiracbracketsKilltheWinnermodelLotka-VolterracosymplecticgeometryconstrainedHamiltoniandynamicspopulation

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 tries to establish that the Kill the Winner model of microbial diversity, along with several classical ecological reductions, admits a singular Lagrangian description built from a Jacobi last multiplier. The authors show that the model equations possess a multiplier $M$ that converts the system into a closed two-form, from which a Lagrangian linear in the velocities is constructed. Because the Hessian vanishes, the Legendre transform is unavailable, and the paper proposes a Hamiltonian formulation using Dirac brackets on a constrained phase space. If correct, the construction would put a stochastic-coevolution ecology model into the same geometric framework used for constrained mechanical systems and predator-prey models.

What carries the argument

The carrying object is the Jacobi last multiplier $M(t,x,y)$, the solution of $\partial_t M + \partial_x(Mf) + \partial_y(Mg) = 0$ for a planar system $(\dot x,\dot y)=(f,g)$. It turns the Cartan two-form $M(dx-f\,dt)\wedge(dy-g\,dt)$ into a closed two-form, which is the geometric seed of the Lagrangian: with $F = -\int M\,dy$ and $G = \int M\,dx$, the velocity-linear Lagrangian $L = F\dot x + G\dot y - U$ is singular because its Hessian in the velocities vanishes. The Hamiltonian is then recovered not by Legendre transformation but by Dirac brackets, with the constraint matrix $C_{rs} = \{\phi_r,\phi_s\}$ built from the primary and secondary constraints; the Dirac bracket $\{f,g\}_D = \{f,g\} - \{f,\phi_r\}[C^{rs}]^{-1}\{\phi_s,g\}$ is claimed to generate the phase-space flow. The time-dependent setting is handled by cosymplectic geometry, with the Poincaré-Cartan form and its exterior derivative replacing the symplectic structure.

What would settle it

Evaluate the paper's Dirac-bracket equations for the Lotka-Volterra system without competition, equations (3.51)-(3.52), at any point where $a x - b x y \neq 0$; the bracket gives $\dot x = 0$, whereas the original model requires $\dot x = a x - b x y$. A single such evaluation settles whether the constrained flow reproduces the model, and the same check applies to the displayed $m=1$ KtW result $\dot x=0$, $\dot y=0$.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the KtW system $\dot x = a_1x - b_1x^2 - c_1xy$, $\dot y = a_2xy - b_2y$ has a Jacobi last multiplier $M = e^{\gamma t} y^{\sigma}/x$, where $\gamma = b_1b_2/a_2$ and $\sigma = b_1/a_2 - 1$, and that this multiplier generates a singular Lagrangian $L = \sum_k [F_k\dot x_k + G_k\dot y_k] - U$. The momenta are velocity-independent and define primary constraints; requiring their persistence in time produces secondary constraints, and the paper forms a $4m \times 4m$ constraint matrix $C$ whose inverse enters the Dirac bracket. Explicit Dirac brackets are written for the KtW model, Lotka-Volterra with and without competition, Kermack-McKendrick, and a Gierer-Meinhardt reduction. The intended upshot is a Hamiltonian phase-space description of ecological systems whose Lagrangians are time-dependent and singular.

Load-bearing premise

The calculation assumes that the extra equations found by demanding the constraints stay true over time are independent restrictions on the system; if instead those equations only fix the undetermined multipliers and place no restriction on the populations, the Dirac-bracket description collapses.

Editorial extensions

If this is right

The KtW equations would become a Hamiltonian system in the constrained sense, so phase-space methods for stability and conserved quantities could be applied to a model originally posed stochastically.
The Jacobi last multiplier supplies a natural density on phase space, giving a geometric handle on coexistence and boom-bust cycles.
All listed reductions would inherit the same singular-Lagrangian and Dirac-bracket framework, unifying prey-predator, host-parasite, and pattern-formation models under one formalism.
The cosymplectic formulation would allow time-dependent singular Lagrangians in biology to be treated with the same tools developed for time-dependent mechanical systems.

Reading between the lines

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

Editorial inference: the validity of the construction can be checked by substituting the Dirac-bracket equations back into the original model; in the paper's own $m=1$ KtW calculation the bracket flow is displayed as $\dot x=0$, $\dot y=0$, which would not reproduce the original equations.
Editorial inference: because the secondary constraints contain the undetermined Lagrange multipliers, the $4m \times 4m$ constraint matrix may mix genuine restrictions with equations that only fix the multipliers; eliminating the multipliers first could yield a smaller, different Dirac bracket.
Editorial inference: the same Jacobi-last-multiplier route could be tried on other ecology models; a concrete test is whether the multiplier equation has a solution for the given vector field, and if so whether the resulting Dirac flow matches the original rates.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Bottom line: the Lagrangian-construction half is fine, and the m-species KtW Lagrangian in (3.28) looks new. The Hamiltonian half collapses. The Dirac-bracket calculation in every worked example ends with ˙x=0 and ˙y=0, so the paper's central claim—that Dirac brackets supply a Hamiltonian description of KtW—fails on its own equations.

What is good: the Jacobi last multiplier for the single-species KtW system checks out, and the route from JLM to a singular Lagrangian via (3.5)-(3.6) is standard and applied carefully. The general m-species KtW Lagrangian is a genuine extension of the authors' earlier lower-dimensional examples, and the potential U is derived rather than fitted. The Gierer-Meinhardt section is explicit that a Lagrangian exists only after setting b=0, though that restriction is substantive and should have been flagged earlier.

The soft spot is load-bearing, not cosmetic. In §3.1, after building the 4×4 constraint matrix C and its inverse, the authors obtain ˙x=0 and ˙y=0 (3.34-3.35) for the m=1 KtW case. The same zero-velocity result appears for Lotka-Volterra with and without competition and for Kermack-McKendrick. A Hamiltonian formulation that forces both population variables to be constant does not reproduce (3.7)-(3.8); it contradicts them. The cause is visible in the secondary constraints (3.31)-(3.32): φ3 and φ4 contain the undetermined multipliers λ and μ, so they are consistency conditions that fix the multipliers, not independent second-class constraints on phase space. Once the multipliers are substituted, the assumed constraint structure—and the invertibility of C—has no basis. This is not a detail; the paper's stated result rests on it.

I would not cite this in its current form. If the Dirac-bracket sections were removed and the paper reframed as a Lagrangian construction via JLM for KtW and its reductions, there might be a modest publishable note. As is, the central claim is internally inconsistent. I would still send it to a serious referee rather than desk-reject—the failure is instructive and needs expert confirmation—but the likely outcome is rejection unless the constraint analysis is redone.

Best

Referee Report

3 major / 5 minor

Summary. The paper proposes a geometric framework, based on cosymplectic geometry, the Jacobi last multiplier (JLM), and Dirac brackets, to obtain a Lagrangian and Hamiltonian description of the "Kill the Winner" (KtW) model and several reductions: Lotka-Volterra with and without competition, Kermack-McKendrick, and Gierer-Meinhardt. The authors derive the JLM from the continuity equation (2.10), construct a singular Lagrangian linear in the velocities (3.5)-(3.6), identify primary constraints from the velocity-independent momenta, define secondary constraints, form a 4m x 4m constraint matrix C, and compute Dirac-bracket equations of motion. Explicit results are presented for the m=1 KtW case and for the Lotka-Volterra reductions.

Significance. The JLM computation and the associated Lagrangian construction are explicit, self-contained, and algebraically checkable; the paper does not fit the multiplier to data but derives it from the continuity equation. However, the central claim of the abstract—that a Hamiltonian description via Dirac brackets is achieved—is contradicted by the paper's own equations: in every worked example the Dirac-bracket time evolution gives dx/dt=0 and dy/dt=0 (e.g., (3.34)-(3.35), (3.51)-(3.52)), which does not reproduce the biological dynamics (3.7)-(3.8). This is an internal inconsistency, not a disagreement with an external consensus. The geometric and JLM construction is of some interest, but the advertised Hamiltonian description is not obtained.

major comments (3)

[Sec. 3.1, Eqs. (3.34)-(3.35)] The Dirac-bracket dynamics for the m=1 KtW model yield dx/dt=0 and dy/dt=0, whereas the original system (3.7)-(3.8) is nonzero on the constraint surface (e.g., dx/dt = a1 x - b1 x^2 - c1 x y). A Hamiltonian description that assigns zero velocity to every phase-space coordinate does not reproduce the model dynamics, so the abstract's assertion of a Hamiltonian description is falsified by the paper's own computation. The same failure occurs in Sec. 3.1.1 (LV with competition: dx/dt=0, dy/dt=0) and Sec. 3.1.2 (LV without competition: Eqs. (3.51)-(3.52)).
[Sec. 3.1, Eqs. (3.31)-(3.32)] The secondary constraints are defined as expressions that contain the undetermined Lagrange multipliers lambda_k and mu_k. In the Dirac-Bergmann algorithm, consistency conditions of the form d(phi)/dt=0 serve to determine the multipliers; they do not generically impose additional independent restrictions on phase space. By treating these multiplier-dependent expressions as independent second-class constraints and building the 4m x 4m matrix C in (3.33) from them, the paper constructs an invalid constraint algebra. This is explicitly visible in the LV-without-competition case: the text states that phi3 and phi4 in (3.47) fix lambda2 and lambda1, yet the C matrix in (3.48) is built from those same expressions. The Dirac brackets based on this C are therefore not a valid constrained Hamiltonian dynamics.
[Sec. 3.2] The Gierer-Meinhardt treatment is limited by the paper's own statement that "unless the parameter b = 0, we cannot find a Lagrangian"; hence the claimed application to the Gierer-Meinhardt model covers only a special case with the source term set to zero, not the standard model (3.63)-(3.64) with b != 0.

minor comments (5)

[Sec. 3.1, after Eq. (3.28)] The Hamiltonian expression contains the term "d_k log y_k," which appears to be a typo; the potential U in (3.27) has "d_i log x_i," and the two should match.
[Sec. 3.1, Eqs. (3.31)-(3.32)] The index k is used both as a free index and as a summation index in the same line; the sums should run over a separate index j.
[Sec. 3.1.2, Eq. (3.51)] The formula for dp_x/dt includes a factor (1 - 4/(x^2 y^2)) whose appearance is not justified by the preceding bracket calculation and appears dimensionally inconsistent; the determinant xi in (3.48)-(3.49) should be re-checked.
[Sec. 3.2] For the Gierer-Meinhardt model, the paper does not actually provide the final equations of motion obtained from the Dirac brackets, despite promising "explicit results" in the abstract; the section ends with the inverse of C only.
[Sec. 3.1, Eqs. (3.31)-(3.32)] The paper calls the constraints phi3, phi4 in (3.31)-(3.32) "second class" before establishing nonsingularity of the constraint matrix; these are secondary constraints, and the second-class property should be verified after eliminating the multipliers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Lagrangian is constructed from the ODEs via the Jacobi last multiplier, and the Dirac-bracket contradiction is a correctness failure rather than a reduction of outputs to inputs.

full rationale

The paper's derivation chain is self-contained rather than circular. In Section 2, the Jacobi last multiplier is defined as the function solving the continuity equation (2.10), and in Section 3 the coefficients F, G, and V of the proposed linear-in-velocity Lagrangian are obtained by quadrature from that multiplier and from the original right-hand sides f and g (Eqs. 3.3–3.6). Thus the Lagrangian is constructively built to satisfy the Euler–Lagrange equations; this is an inverse-problem reformulation, not a prediction extracted from an independently fitted parameter. The Dirac-bracket calculation is also an explicit computation from the primary and secondary constraints. The fact that the paper's own equations (3.34)–(3.35) yield dx/dt = 0 and dy/dt = 0, instead of the KtW equations (3.7)–(3.8), is an internal correctness failure of the claimed Hamiltonian description, not a circularity: the zero velocities do not reproduce the inputs by construction, they contradict them. The self-citations [5,19,20,22] recall the JLM-to-Lagrangian procedure and identify special cases, but the paper re-derives the formulas it actually uses and does not import the target result from those references. No fitted quantity is relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; all constants are inputs from the ecological models. The axioms are the standard JLM/Dirac machinery plus restrictive assumptions in the multi-species and Gierer-Meinhardt cases. No new physical entities are introduced.

assumptions (5)

domain assumption The system admits a Jacobi last multiplier of the assumed exponential monomial form M = e^{gamma t} x^alpha y^beta.
Used in Section 3.1 to construct the Lagrangian; the form is verified by substitution into (2.10), but the form itself is assumed.
domain assumption The Lagrangian is assumed linear in the velocities, L = F dot x + G dot y - V, making it singular.
Equation (3.1) is the starting point for the entire construction; it restricts the class of systems considered.
ad hoc to paper The construction assumes G_x = -F_y = mu in (3.3) and, for the multi-species case, A = D = 0 with B diagonal in (3.18)-(3.20).
These assumptions are needed to decouple the Euler-Lagrange equations and are not derived from the biological model.
ad hoc to paper For the Gierer-Meinhardt model, the parameter b is set to zero to obtain a Lagrangian.
Section 3.2 states that unless b=0, a Lagrangian cannot be found; this changes the original model (3.63)-(3.64).
standard math Standard Dirac-Bergmann constraint algorithm and Poisson geometry are assumed.
The Dirac bracket definition in Section 2.2 and the constraint algorithm are standard background results.

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Cite this review

Pith. "Pith review of On a geometric description of time dependent singular Lagrangians with applications to biological systems." pith.science (2026). https://pith.science/paper/2V7ACFB4

@misc{pith2026190809647,
  author       = {Pith},
  title        = {Pith review of: On a geometric description of time dependent singular Lagrangians with applications to biological systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2V7ACFB4}},
  note         = {Machine review of arXiv:1908.09647}
}

read the original abstract

We consider certain analytical features of a stochastic model that can explain among other things competition among species and simultaneous predation on the competing species from a geometric perspective which allows for a systematic description of models admitting singular Lagrangians. The model equations are shown to admit a Jacobi Last Multiplier which in turn allows for the construction of a Lagrangian. The Lagrangian is of singular nature so that construction of the Hamiltonian via a Legendre transformation is not possible. A Hamiltonian description of the model therefore requires the introduction of Dirac brackets. Explicit results are presented for the "Kill the winner" model and its reductions.

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Chi Xue and Nigel Goldenfeld, Phy. Rev. Lett. 119 (2017), 2681 01

work page 2017
[2]

Trubatch and A

S.L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics , J. Theor. Biol. 48 (1974), 299-324

work page 1974
[3]

Goriely, Integrability and nonintegrability of dynamical systems

A. Goriely, Integrability and nonintegrability of dynamical systems . Advanced Series in Nonlinear Dynamics, 19. World Scientiﬁc Publishing Co., Inc., River Edge , NJ, 2001. xviii+415 pp

work page 2001
[4]

Paine, The development of Lagrangians for biological models , Bull

G.H. Paine, The development of Lagrangians for biological models , Bull. Math. Biol. 44 (1982) 749-760

work page 1982
[5]

Nonlinear Math

M.C.Nucci and K M Tamizhmani Lagrangians for biological systems , J. Nonlinear Math. Phys (19) (3) 1250021 (2012)

work page 2012
[6]

P. A. M. Dirac, Generalized Hamiltonian Dynamics , Canadian J. Math 2 (1950) 129-148

work page 1950
[7]

M. J. Gotay, J. M. Nester, and G. Hinds, em Presymplectic mani-f olds and the Dirac- Bergmann theory of constraints, J. Math. Phys 19 (1978) 2388- 2399. 22

work page 1978
[8]

Esen and P

O. Esen and P. Guha, On time-dependent Hamiltonian realizations of planar and n on- planar systems , J. Geom. Phys. 127 ( 2018 ) 32-45

work page 2018

Show all 26 references

[9]

de Leon and D

M. de Leon and D. Martin de Diego, A constraint algorithm for singular Lagrangians subjected to nonholonomic constraints , J. Math. Phys. 38 (1997) 3055-3062

1997
[10]

de Leon and C

M. de Leon and C. Sardon, Cosymplectic and contact structures for time-dependent an d dissipative Hamiltonian systems , J. Phys A: Mathematical and Theoretical 50 (2017) 255205

2017
[11]

D. J. Saunders, The Geometry of Jet Bundles , LMS vol 142, Cambridge University Press, Cambridge, 1989

1989
[12]

Ibort, M

A. Ibort, M. de Leon, J. C. Marrero, and D. Martin de Diego, Dirac brackets in constrained dynamics, Fortschritte der Physik 47 (1999) 459-492

1999
[13]

Cari˜ nena and J

J.F. Cari˜ nena and J. Fernandez-Nu˜ nez,Geometric Theory of Time-Dependent Singular Lagrangians, Fortschritte der Physik 41, Issue 6 (1993) 517–552

1993
[14]

Cari˜ nena, J

J.F. Cari˜ nena, J. Fernandez-Nu˜ nez and M. Ra˜ nada,Singular Lagrangians aﬃne in ve- locities, J. Phys. A: Math. Gen. 36 (2003) 3789

2003
[15]

M.Crampin, Jet Bundle Techniques in Analytical Mechanics , (Gruppo Nazionale di Fisica Matematica) vol 47 (1995)

1995
[16]

Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies

E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1988

1988
[17]

M.C.Nucci and P.G.L.Leach, Jacobi’s last multiplier and Lagrangians for multidimen- sional systems.arXiv:0709.3231v1

arXiv
[18]

Nonlinear Math

M.C.Nucci, Jacobi last multiplier and Lie symmetries: a novel applicat ion of an old relationship, J. Nonlinear Math. Phys. 12 (2005) 284–304

2005
[19]

Ghose Choudhury, P

A. Ghose Choudhury, P. Guha and B. Khanra, On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of diﬀerential equa tions of the Painlev´ e-Gambier classiﬁcation, J. Math. Anal. Appl. 360 (2009), no. 2, 651–664

2009
[20]

Ghose Choudhury and P

A. Ghose Choudhury and P. Guha, Application of Jacobi’s last multiplier for construction of Hamiltonians of certain biological systems , Cent. Eur. J. Phys. • 10(2) • 2012 • 398-404

2012
[21]

Ruan, Diﬀusion-driven instability in the Gierer-Meinhardt mode l of morphogenesis , Natural Resourse Modeling 11, no

S. Ruan, Diﬀusion-driven instability in the Gierer-Meinhardt mode l of morphogenesis , Natural Resourse Modeling 11, no. 2, summer 1998

1998
[22]

Guha and A

P. Guha and A. Ghose Choudhury, Singular Lagrangian, Hamiltonization and Jacobi Last Multiplier for Certain Biological Systems , Eur. Phys. J. Special Topics 2013, Volume 222, Issue 3-4, 615-624. 23

2013
[23]

S Smale, On the diﬀerential equations of species in competition , J. Math. Biol (2016), 3 (2) 5-7

2016
[24]

Gierer and H

A. Gierer and H. Meinhardt, A theory of biological pattern formation . Kybernetik 12 (1972) 30-39

1972
[25]

Granero-Porati and A

M.I. Granero-Porati and A. Porati, Temporal organization in a morphogenesis ﬁeld , J. Math. Biol 20 (1984) 153-157

1984
[26]

Turing, The chemical basis of morphogenesis

A.M. Turing, The chemical basis of morphogenesis . Phil. Trans. R. Soc. Lond. B237, 37-72 (1952). 24

1952

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Chi Xue and Nigel Goldenfeld, Phy. Rev. Lett. 119 (2017), 2681 01

work page 2017

[2] [2]

Trubatch and A

S.L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics , J. Theor. Biol. 48 (1974), 299-324

work page 1974

[3] [3]

Goriely, Integrability and nonintegrability of dynamical systems

A. Goriely, Integrability and nonintegrability of dynamical systems . Advanced Series in Nonlinear Dynamics, 19. World Scientiﬁc Publishing Co., Inc., River Edge , NJ, 2001. xviii+415 pp

work page 2001

[4] [4]

Paine, The development of Lagrangians for biological models , Bull

G.H. Paine, The development of Lagrangians for biological models , Bull. Math. Biol. 44 (1982) 749-760

work page 1982

[5] [5]

Nonlinear Math

M.C.Nucci and K M Tamizhmani Lagrangians for biological systems , J. Nonlinear Math. Phys (19) (3) 1250021 (2012)

work page 2012

[6] [6]

P. A. M. Dirac, Generalized Hamiltonian Dynamics , Canadian J. Math 2 (1950) 129-148

work page 1950

[7] [7]

M. J. Gotay, J. M. Nester, and G. Hinds, em Presymplectic mani-f olds and the Dirac- Bergmann theory of constraints, J. Math. Phys 19 (1978) 2388- 2399. 22

work page 1978

[8] [8]

Esen and P

O. Esen and P. Guha, On time-dependent Hamiltonian realizations of planar and n on- planar systems , J. Geom. Phys. 127 ( 2018 ) 32-45

work page 2018