arxiv: 2605.00483 · v1 · submitted 2026-05-01 · 🧮 math.DG

Recognition: unknown

Hamiltonian semisprays on Lie algebroids

Misael Avenda\~no Camacho , Jhonny Kama Mamani , Eduardo Velasco Barreras

Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:56 UTC · model grok-4.3

classification 🧮 math.DG

keywords Lie algebroidsHamiltonian semispraysPoisson bracketsregular Lagrangianssymplectic geometryprolongationcohomologysecond-order dynamics

0 comments

The pith

A family of Poisson brackets on a Lie algebroid makes the energy Hamiltonian vector field a semispray for any regular Lagrangian.

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

The authors address the existence of Hamiltonian semisprays in the context of Lie algebroids, generalizing a problem previously considered for tangent bundles. They show that starting from a regular Lagrangian on the algebroid, it is possible to define a family of Poisson brackets on the algebroid. For each such bracket, the Hamiltonian vector field of the energy function turns out to be a semispray. This construction uses the symplectic geometry on the prolongation of the Lie algebroid and a cohomological study of its vertical subbundle, providing a setup for second-order dynamics in this generalized geometric setting.

Core claim

Given a Lie algebroid and a regular Lagrangian, we construct a family of Poisson brackets on the algebroid such that the Hamiltonian vector field associated with the corresponding energy function is a semispray. Our approach is based on the symplectic geometry of the prolongation of a Lie algebroid and a cohomological analysis of its vertical subbundle. The results provide a geometric framework for second-order Hamiltonian dynamics on Lie algebroids, extending some known facts in the classical tangent bundle case and revealing new interactions between Poisson geometry and algebroid structures.

What carries the argument

The symplectic structure on the prolongation of the Lie algebroid combined with cohomological analysis of the vertical subbundle, which allows construction of the family of Poisson brackets making the energy Hamiltonian a semispray.

Load-bearing premise

The Lagrangian must be regular for the two-form defined on the prolongation to be symplectic, enabling the cohomological construction of the Poisson brackets without obstructions.

What would settle it

A specific Lie algebroid equipped with a regular Lagrangian for which no choice of Poisson bracket makes the energy Hamiltonian vector field a semispray would falsify the claim.

read the original abstract

We study the existence of Hamiltonian semisprays on Lie algebroids. This work is motivated by a problem studied by Vaisman for tangent bundles, and we extend this question to the setting of arbitrary Lie algebroids and provide a general solution. More precisely, given a Lie algebroid and a regular Lagrangian, we construct a family of Poisson brackets on the algebroid such that the Hamiltonian vector field associated with the corresponding energy function is a semispray. Our approach is based on the symplectic geometry of the prolongation of a Lie algebroid and a cohomological analysis of its vertical subbundle. The results provide a geometric framework for second-order Hamiltonian dynamics on Lie algebroids, extending some known facts in the classical tangent bundle case and revealing new interactions between Poisson geometry and algebroid structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

It extends Vaisman's tangent-bundle Hamiltonian semisprays to arbitrary Lie algebroids via prolongation symplectic geometry and vertical cohomology, with the construction holding up on its own terms.

read the letter

This paper extends Vaisman's construction of Hamiltonian semisprays from tangent bundles to general Lie algebroids. Given a regular Lagrangian, it builds a family of Poisson brackets on the algebroid such that the associated Hamiltonian vector field satisfies the semispray condition via the anchor projection. The approach relies on the standard prolongation of the Lie algebroid, equips it with a symplectic form induced by the Lagrangian, and applies a cohomological argument on the vertical subbundle to obtain the Poisson bivector. The stress-test note confirms no extra global obstructions appear beyond the stated regularity of the Lagrangian, so the argument stays internally consistent. What stands out is that the construction is a direct lift using well-defined objects rather than a rephrasing of prior results. The paper organizes the existence statement cleanly and shows how the tools interact without introducing fitted quantities or self-referential steps. One soft spot is the output family of Poisson structures: the existence is established, but the paper does not examine whether a preferred member exists or how the different brackets behave in concrete examples on non-trivial algebroids. The mention of new interactions between Poisson geometry and algebroid structures is noted but left without specific computations to illustrate the payoff. Readers already working in geometric mechanics or control on Lie algebroids will get the most from having the general case written down. It is not a broad advance but it fills a natural gap with standard methods. The work shows clear engagement with the literature and the math checks out on its terms, so it deserves a serious referee. I recommend sending it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that, given any Lie algebroid and a regular Lagrangian on it, there exists a family of Poisson brackets on the total space of the algebroid such that the Hamiltonian vector field of the associated energy function is a semispray (i.e., its anchor projection reproduces the identity section). The construction proceeds by equipping the prolongation Lie algebroid with a symplectic structure induced by the regular Lagrangian and then using a cohomological argument on the vertical subbundle to produce the required Poisson bivector. This extends Vaisman's earlier results for tangent bundles to the general Lie algebroid setting and supplies a geometric framework for second-order Hamiltonian dynamics on algebroids.

Significance. If the central construction holds, the paper supplies a clean existence result that places second-order dynamics on Lie algebroids inside a Poisson-geometric setting. The use of the standard prolongation and the symplectic form induced by a regular Lagrangian is a strength; the cohomological step on the vertical bundle is a natural extension of classical techniques and yields a whole family of Poisson structures rather than a single one. This framework is likely to be useful for further work on geometric mechanics, reduction, and integrability questions in the algebroid category.

minor comments (2)

[Abstract] The abstract states that a 'family' of Poisson brackets is constructed but does not indicate the dimension or parametrization of this family; a brief sentence clarifying this point would help readers gauge the generality of the result.
[§2] Notation for the prolongation Lie algebroid and its vertical subbundle is introduced without an accompanying diagram or explicit local-coordinate expressions; adding a short notational table or local-frame description in §2 would improve readability for readers less familiar with the prolongation construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough reading of the manuscript and for the positive evaluation. We are pleased that the central construction and its relation to Vaisman’s work on tangent bundles were found to be clear and of interest.

Circularity Check

0 steps flagged

No circularity: direct construction from standard symplectic and cohomological tools on Lie algebroid prolongations

full rationale

The derivation proceeds by equipping the prolongation of the Lie algebroid with the symplectic form induced by a regular Lagrangian, then using a standard cohomological argument on the vertical subbundle to produce the required Poisson bivector family. All objects invoked (prolongation Lie algebroid, induced symplectic structure, vertical cohomology) are independently defined in the literature on Lie algebroids and do not reduce to the target semispray condition by construction or by any self-citation chain. The abstract and skeptic analysis confirm the argument is self-contained against external benchmarks in symplectic geometry; no fitted parameters are renamed as predictions, no uniqueness theorems are imported from the authors' prior work, and no ansatz is smuggled via citation. This is the expected honest non-finding for a paper whose central claim is an explicit existence construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard Lie algebroid theory and symplectic geometry; no free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (2)

standard math Standard properties of Lie algebroids, their prolongations, and vertical subbundles
The construction invokes established differential-geometric structures on Lie algebroids.
domain assumption Existence of a regular Lagrangian on the given Lie algebroid
The result is stated for regular Lagrangians.

pith-pipeline@v0.9.0 · 5446 in / 1227 out tokens · 77549 ms · 2026-05-09T18:56:19.467031+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 2 canonical work pages

[1]

I., Vogtmann, K., Weinstein, A

Arnold, V. I., Vogtmann, K., Weinstein, A. (1989). Mathematical methods of classical mechan- ics (Vol. 60, p. 271). New York: Springer

1989
[2]

Marsden J

Abraham, R. Marsden J. E. (2008). Foundations of Mechanics AMS Chelsea publishing. AMS Chelsea Pub./American Mathematical Society. ISBN: 9780821844380

2008
[3]

(2016).Inverse problem for La- grangian systems on Lie algebroids and applications to reduction by symmetries

Barbero-Liñán, M., Farré Puiggalí, M., Martín de Diego, D. (2016).Inverse problem for La- grangian systems on Lie algebroids and applications to reduction by symmetries. Monatshefte für Mathematik,180(4), 665–691

2016
[4]

Brahic,On the infinitesimal gauge symmetries of closed forms, J

O. Brahic,On the infinitesimal gauge symmetries of closed forms, J. Geom. Mech.3(2011), no. 3, 277–312. DOI: 10.3934/jgm.2011.3.277

work page doi:10.3934/jgm.2011.3.277 2011
[5]

F., da Costa, J

Carinena, J. F., da Costa, J. M. N., Santos, P. (2007). Reduction of Lagrangian mechanics on Lie algebroids. Journal of Geometry and Physics, 57(3), 977-990

2007
[6]

C., Martín de Diego, D., Martínez, E

Cortés, J., de León, M., Marrero, J. C., Martín de Diego, D., Martínez, E. (2008).Nonholo- nomic Lagrangian systems on Lie algebroids. Discrete Contin. Dyn. Syst.,24(2), 213–271

2008
[7]

C., Martín de Diego, D., Martínez, E

Cortés, J., de León, M., Marrero, J. C., Martín de Diego, D., Martínez, E. (2006).A survey of Lagrangian mechanics and control on Lie algebroids and groupoids. International Journal of Geometric Methods in Modern Physics,3(3), 509–558. 23

2006
[8]

Coste, P

A. Coste, P. Dazord, and A. Weinstein,Groupe symplectique, Publ. Dép. Math. Nouvelle Sér. A, Univ. Claude-Bernard Lyon I2/A(1987), 1–62

1987
[9]

C., Martínez, E

de Diego, D., Marrero, J. C., Martínez, E. (2006).Discrete Lagrangian and Hamiltonian me- chanics on Lie groupoids. Nonlinearity 19 1313

2006
[10]

(2009).k-symplectic formalism on Lie algebroids

De León, M., Martín de Diego, D., Salgado, M., Vilariño, S. (2009).k-symplectic formalism on Lie algebroids. Journal of Physics A: Mathematical and Theoretical,42, 385209 (31pp)

2009
[11]

C., Martínez, E

de León, M., Marrero, J. C., Martínez, E. (2005).Lagrangian submanifolds and dynamics on Lie algebroids. Journal of Physics A: Mathematical and General,38(24), R241–R308

2005
[12]

(2006).Geometrical mechanics on algebroids

Grabowska, K., Urbański, P., Grabowski, J. (2006).Geometrical mechanics on algebroids. Int. J. Geom. Methods Mod. Phys.,3(03), 559-575

2006
[13]

C., Martín de Diego, D., Martínez, E., Padrón, E

Iglesias, D., Marrero, J. C., Martín de Diego, D., Martínez, E., Padrón, E. (2007).Reduction of symplectic Lie algebroids by a Lie subalgebroid and a symmetry Lie group. SIGMA,3, 049, 28 pp

2007
[14]

Klaasse, R.L.(2017).Geometric structures and Lie algebroids.Ph.D.thesis, UtrechtUniversity

2017
[15]

Lichnerowicz,Les variétés de Poisson et leurs algèbres de Lie associées, J

A. Lichnerowicz,Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12(1977), 253–300

1977
[16]

K. C. H. Mackenzie,General Theory of Lie Groupoids and Lie Algebroids, London Mathe- matical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005, xxxviii+501 pp

2005
[17]

K. C. H. Mackenzie and P. Xu,Lie bialgebroids and Poisson groupoids, Duke Mathematical Journal, 73(2):415–452, 1994

1994
[18]

E., Ratiu, T

Marsden, J. E., Ratiu, T. (1994). Introduction to symmetry and mechanics. Texts in Applied Mathematics, 17

1994
[19]

E., Ratiu, T., Weinstein, A

Marsden, J. E., Ratiu, T., Weinstein, A. (1984). Semidirect products and reduction in mechan- ics. Trans. Amer. Math. Soc., 281(1), 147-177

1984
[20]

Martınez, E., Mestdag, T., Sarlet, W. (2002). Lie algebroid structures and Lagrangian systems on affine bundles. Journal of Geometry and Physics, 44(1), 70-95

2002
[21]

Martínez,Lagrangian Mechanics on Lie Algebroids, Acta Appl

E. Martínez,Lagrangian Mechanics on Lie Algebroids, Acta Appl. Math.67(2001), 295–320

2001
[22]

(2018).The geometry of E-manifolds

Miranda, E., Scott, G. (2018).The geometry of E-manifolds. Revista Matemática Iberoameri- cana,35(5), 1207–1246

2018
[23]

(2021).On contact and symplectic Lie algebroids

Nazari, E., Heydari, A. (2021).On contact and symplectic Lie algebroids. Iranian Journal of Mathematical Sciences and Informatics,16(1), 35–53

2021
[24]

Popescu,Geometrical structures on Lie algebroids, Publ

L. Popescu,Geometrical structures on Lie algebroids, Publ. Math. Debrecen72(1-2) (2008), 95–109. 24

2008
[25]

Popescu, L.(2017).Symmetries of second order differential equations on Lie algebroids.Journal of Geometry and Physics,117, 84–98

2017
[26]

Rankin, S.(2024).Symplectic Hodge theory on Lie algebroids.arXivpreprintarXiv:2411.06012

work page arXiv 2024
[27]

L., Kertész, D

Szilasi, J., Lovas, R. L., Kertész, D. C. (2013). Connections, sprays and Finsler structures. World Scientific Publishing Company

2013
[28]

Stefan,Accessible sets, orbits, and foliations with singularities, Proc

P. Stefan,Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc.29 (1974), 699–713

1974
[29]

H. J. Sussmann,Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc.180(1973), 171–188

1973
[30]

(1995).Second order Hamiltonian vector fields on tangent bundles

Vaisman, I. (1995).Second order Hamiltonian vector fields on tangent bundles. Differential Geometry and its Applications, 5(2), 153-170

1995
[31]

Weinstein,Lagrangian mechanics and groupoids, Fields Inst

A. Weinstein,Lagrangian mechanics and groupoids, Fields Inst. Comm.7(1996), 207–231. 25

1996