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arxiv: 2501.12940 · v1 · submitted 2025-01-22 · 🧮 math-ph · cs.NA· math.DS· math.MP· math.NA

Euler--Poincar\'e reduction and the Kelvin--Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

Yusuke Ono , Simone Fiori , Linyu Peng This is my paper

Pith reviewed 2026-05-23 05:30 UTC · model grok-4.3

classification 🧮 math-ph cs.NAmath.DSmath.MPmath.NA
keywords discrete mechanicsEuler-Poincaré equationsKelvin-Noether theoremLie groupsadvected parametersunderwater vehiclesgeometric integration
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The pith

Discrete Euler-Poincaré reduction for Lie-group systems with advected parameters is achieved via a group difference map that extends the Kelvin-Noether theorem.

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

The paper constructs a discrete analogue of the Euler-Poincaré equations for Lagrangian mechanics on Lie groups that includes both advected parameters and extra dynamics. It defines a group difference map through the Cayley transform or the matrix exponential to produce discrete equations that retain the continuous Lie-algebra structure. The same construction yields a discrete Kelvin-Noether theorem that tracks the same conserved quantities as the continuous version. The method is illustrated on the dynamics of underwater vehicles, where numerical runs show the discrete flow stays on the Lie group and respects the geometric invariants over long intervals.

Core claim

The discrete Euler-Poincaré reduction is introduced for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics by employing the group difference map technique defined via the Cayley transform or the matrix exponential. This leads to discrete equations that mirror the continuous ones, and the Kelvin-Noether theorems are extended to account for the corresponding quantities in both continuous and discrete cases, as shown in the underwater vehicle example.

What carries the argument

The group difference map, defined using the Cayley transform or matrix exponential, which discretizes the continuous Euler-Poincaré equations while preserving the Lie-group structure and the dynamics of advected parameters.

If this is right

  • The discrete equations inherit the reduction structure of the continuous Euler-Poincaré system for advected parameters.
  • A discrete Kelvin-Noether quantity is conserved exactly along the discrete flow.
  • Numerical integration of underwater-vehicle dynamics stays on the Lie group and preserves the geometric invariants for long times.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same difference-map construction may supply structure-preserving integrators for other rigid-body systems in fluids.
  • If the map works for semidirect products, analogous discrete reductions could be written for more general mechanical systems with symmetry.

Load-bearing premise

The group difference map based on Cayley transform or matrix exponential can discretize the equations without breaking the Lie group structure or the form of the advected parameter dynamics.

What would settle it

A numerical run on the underwater-vehicle equations in which the discrete trajectory leaves the Lie group or the computed Kelvin-Noether quantity drifts by more than round-off over many steps would falsify the preservation claim.

Figures

Figures reproduced from arXiv: 2501.12940 by Linyu Peng, Simone Fiori, Yusuke Ono.

Figure 1
Figure 1. Figure 1: shows the relative error of energy over the time span of [0, 500] in a semi-logarithmic plot. The energy fluctuates within a certain range but tends to increase slightly over time. This is because of the Gauss–Newton method, which brings truncation errors. The time evolution of the Kelvin–Noether quantity, namely ez-component of the vector associated to πk given by (78) using the Lie algebra isomorphism (5… view at source ↗
Figure 2
Figure 2. Figure 2: Relative error of the Kelvin–Noether quantity, [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: 3-dimensional visualization of the vehicle’s trajectory over time span [0 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
read the original abstract

The Euler--Poincar\'e equations, firstly introduced by Henri Poincar\'e in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various settings, such as semidirect products, advected parameters, and field theory, and have been widely applied to mechanics and physics. In this paper, we introduce the discrete Euler--Poincar\'e reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics, utilizing the group difference map technique. Specifically, the group difference map is defined using either the Cayley transform or the matrix exponential. The continuous and discrete Kelvin--Noether theorems are extended accordingly, that account for Kelvin--Noether quantities of the corresponding continuous and discrete Euler--Poincar\'e equations. As an application, we show both continuous and discrete Euler--Poincar\'e formulations about the dynamics of underwater vehicles, followed by numerical simulations. Numerical results illustrate the scheme's ability to preserve geometric properties over extended time intervals, highlighting its potential for practical applications in the control and navigation of underwater vehicles, as well as in other domains.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a discrete Euler-Poincaré reduction for Lagrangian systems on Lie groups that incorporate advected parameters and additional dynamics. The reduction is performed via group difference maps (Cayley transform or matrix exponential). The continuous and discrete Kelvin-Noether theorems are extended to this setting, and the framework is applied to underwater-vehicle dynamics with accompanying numerical simulations that illustrate long-term geometric preservation.

Significance. If the central constructions are correct, the work supplies a variational integrator that exactly conserves a discrete Kelvin-Noether quantity for a class of systems with Lie-group symmetries and advected quantities. The explicit supply of the discrete Lagrangian, reduced equations, momentum map, and conservation proof, together with the underwater-vehicle example, strengthens the case for structure-preserving methods in long-term simulation of mechanical systems with symmetries.

minor comments (2)
  1. [Abstract] The abstract states that the group difference map is defined via Cayley transform or matrix exponential but does not indicate which is used in the underwater-vehicle example or why; a short statement in §4 or the numerical section would clarify reproducibility.
  2. Notation for the discrete momentum map and the advected-parameter update appears only after the main reduction theorem; moving a brief definition to the preliminaries would improve readability for readers unfamiliar with the continuous case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct variational construction

full rationale

The manuscript constructs the discrete Euler-Poincaré reduction and extended Kelvin-Noether theorem from the continuous equations via the group difference map (Cayley or exponential), supplying explicit discrete Lagrangians, reduced equations, momentum maps, and conservation proofs. No step reduces a claimed result to a fitted input, self-definition, or load-bearing self-citation chain; the work is a self-contained geometric discretization whose validity rests on the variational derivation itself rather than on any input quantity being renamed as output.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard Lie-group Lagrangian mechanics and the existence of suitable group difference maps; no new free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Lagrangian mechanics on Lie groups with symmetries and advected parameters admits Euler-Poincaré reduction
    Invoked as the starting point for both continuous and discrete extensions.
  • domain assumption Group difference maps (Cayley or exponential) provide a valid discretization that respects the group structure
    Central to the discrete reduction technique described in the abstract.

pith-pipeline@v0.9.0 · 5775 in / 1325 out tokens · 25044 ms · 2026-05-23T05:30:38.163268+00:00 · methodology

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Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    H. A. Ardakani. A variational principle for three-dimensional interactions between water waves and a floating rigid body with interior fluid motion. Journal of Fluid Mechanics , 866:630–659, 2019

    work page 2019

  2. [2]

    V. Arnold. Sur la g´ eom´ etrie diff´ erentielle des groupes de Lie de dimension infinie et ses applications ` a l’hydrodynamique des fluides parfaits.Annales de l’Institut Fourier , 16:319–361, 1966

    work page 1966

  3. [3]

    Bloch, P

    A. Bloch, P. Krishnaprasad, J. E. Marsden, and T. S. Ratiu. The Euler–Poincar´ e equations and double bracket dissipation. Communications in Mathematical Physics , 175:1–42, 1996. 17

    work page 1996

  4. [4]

    A. I. Bobenko and Y. B. Suris. Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top. Communications in Mathematical Physics , 204:147–188, 1999

    work page 1999

  5. [5]

    Bou-Rabee and J

    N. Bou-Rabee and J. E. Marsden. Hamilton–Pontryagin integrators on Lie groups part I: Introduc- tion and structure-preserving properties. Foundations of Computational Mathematics, 9:197–219, 2009

    work page 2009

  6. [6]

    N. M. Bou-Rabee. Hamilton–Pontryagin Integrators on Lie Groups . PhD thesis, California Institute of Technology, 2007

    work page 2007

  7. [7]

    Cendra, D

    H. Cendra, D. D. Holm, J. E. Marsden, and T. S. Ratiu. Lagrangian reduction, the Euler–Poincar´ e equations, and semidirect products. American Mathematical Society Translations, 186:1–25, 1998

    work page 1998

  8. [8]

    Cendra, J

    H. Cendra, J. E. Marsden, and T. S. Ratiu. Lagrangian Reduction by Stages . American Mathe- matical Society, Providence, 2001

    work page 2001

  9. [9]

    N. G. Chetayev. On the equations of Poincar´ e. Journal of Applied Mathematics and Mechanics , 5:253–262, 1941

    work page 1941

  10. [10]

    C. J. Cotter and D. D. Holm. On Noether’s theorem for the Euler–Poincar´ e equation on the diffeomorphism group with advected quantities. Foundations of Computational Mathematics , 13:457–477, 2013

    work page 2013

  11. [11]

    Cou´ eraud and F

    B. Cou´ eraud and F. Gay-Balmaz. Variational discretization of thermodynamical simple systems on Lie groups. Discrete and Continuous Dynamical Systems - Series S , 13:1075–1102, 2020

    work page 2020

  12. [12]

    Cristi, F

    R. Cristi, F. A. Papoulias, and A. J. Healey. Adaptive sliding mode control of autonomous underwater vehicles in the dive plane. IEEE Journal of Oceanic Engineering , 15:152–160, 1990

    work page 1990

  13. [13]

    Fels and P

    M. Fels and P. J. Olver. Moving coframes: II. Regularization and theoretical foundations. Acta Applicandae Mathematica, 55:127–208, 1999

    work page 1999

  14. [14]

    S. Fiori. Coordinate-free Lie-group-based modeling and simulation of a submersible vehicle. AIMS Mathematics, 9:10157–10184, 2024

    work page 2024

  15. [15]

    T. I. Fossen and O. E. Fjellstad. Nonlinear modelling of marine vehicles in 6 degrees of freedom. Mathematical Modelling of Systems , 1:17–27, 1995

    work page 1995

  16. [16]

    E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden, and M. Desbrun. Geometric, variational discretization of continuum theories. Physica D, 240:1724–1760, 2011

    work page 2011

  17. [17]

    Ge and J

    Z. Ge and J. E. Marsden. Lie–Poisson Hamilton–Jacobi theory and Lie–Poisson integrators. Physics Letters A , 133:134–139, 1988

    work page 1988

  18. [18]

    G. Hamel. Die Lagrange–Eulerschen gleichungen der mechanik. Z. Mathematik und Physik , 50:1–57, 1904

    work page 1904

  19. [19]

    G. Hamel. Theoretische Mechanik. Springer-Verlag, Berlin, 1949

    work page 1949

  20. [20]

    D. D. Holm, R. Hu, and O. D. Street. Lagrangian reduction and wave mean flow interaction. Physica D, 454:133847, 2023

    work page 2023

  21. [21]

    D. D. Holm, J. E. Marsden, and T. S. Ratiu. The Euler–Poincar´ e equations and semidirect products with applications to continuum theories. Advances in Mathematics, 137:1–81, 1998

    work page 1998

  22. [22]

    D. D. Holm, J. E. Marsden, and T. S. Ratiu. The Euler–Poincar´ e equations in geophysical fluid dynamics. In Large-scale Atmosphere-Ocean Dynamics. II. Geometric Methods and Models, pages 251–299. Cambridge University Press, 2002

    work page 2002

  23. [23]

    D. D. Holm, T. Schmah, C. Stoica, and D. C. P. Ellis. Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions . Oxford University Press, Oxford, 2009. 18

    work page 2009

  24. [24]

    D. D. Holm and C. Tronci. Euler–Poincar´ e formulation of hybrid plasma models.Communications in Mathematical Sciences, 10:191–222, 2012

    work page 2012

  25. [25]

    I. I. Hussein, M. Leok, A. K. Sanyal, and A. M. Bloch. A discrete variational integrator for optimal control problems on SO(3). In Proceedings of the 45th IEEE Conference on Decision and Control, pages 6636–6641, 2006

    work page 2006

  26. [26]

    Iserles, H

    A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna. Lie-group methods. Acta Numerica, 9:215–365, 2000

    work page 2000

  27. [27]

    C. Kane, J. E. Marsden, M. Ortiz, and M. West. Variational integrators and the Newmark algo- rithm for conservative and dissipative mechanical systems. International Journal for Numerical Methods in Engineering, 49:1295–1325, 2000

    work page 2000

  28. [28]

    M. B. Kobilarov and J. E. Marsden. Discrete geometric optimal control on Lie groups. IEEE Transactions on Robotics, 27:641–655, 2011

    work page 2011

  29. [29]

    T. Lee, M. Leok, and N. H. McClamroch. Optimal attitude control of a rigid body using geomet- rically exact computations on SO(3). Journal of Dynamical and Control Systems , 14:465–487, 2008

    work page 2008

  30. [30]

    M. Leok. An overview of Lie group variational integrators and their applications to optimal control. In International Conference on Scientific Computation and Differential Equations . The French National Institute for Research in Computer Science and Control, 2007

    work page 2007

  31. [31]

    N. E. Leonard. Stability of a bottom-heavy underwater vehicle. Automatica, 33:331–346, 1997

    work page 1997

  32. [32]

    E. L. Mansfield. A Practical Guide to the Invariant Calculus . Cambridge University Press, Cambridge, 2010

    work page 2010

  33. [33]

    E. L. Mansfield, A. Rojo-Echebur´ ua, P. E. Hydon, and L. Peng. Moving frames and Noether’s finite difference conservation laws I. Transactions of Mathematics and Its Applications , 3:tnz004, 2019

    work page 2019

  34. [34]

    J. E. Marsden, S. Pekarsky, and S. Shkoller. Discrete Euler–Poincar´ e and Lie–Poisson equations. Nonlinearity, 12:1647, 1999

    work page 1999

  35. [35]

    J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, New York, 2nd edition, 2013

    work page 2013

  36. [36]

    J. E. Marsden, T. S. Ratiu, and J. Scheurle. Reduction theory and the Lagrange–Routh equations. Journal of Mathematical Physics , 41:3379–3429, 2000

    work page 2000

  37. [37]

    J. E. Marsden and J. Scheurle. Lagrangian reduction and the double spherical pendulum. Zeitschrift f¨ ur Angewandte Mathematik und Physik ZAMP, 44:17–43, 1993

    work page 1993

  38. [38]

    J. E. Marsden and J. Scheurle. The reduced Euler–Lagrange equations. Fields Institute Commu- nications, 1:139–164, 1993

    work page 1993

  39. [39]

    J. E. Marsden and M. West. Discrete mechanics and variational integrators. Acta Numerica, 10:357–514, 2001

    work page 2001

  40. [40]

    Moser and A

    J. Moser and A. Veselov. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Communications in Mathematical Physics , 139:217–243, 1991

    work page 1991

  41. [41]

    E. Noether. Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse, 2:235–257, 1918

    work page 1918

  42. [42]

    Nordkvist and A

    N. Nordkvist and A. K. Sanyal. A Lie group variational integrator for rigid body motion in SE(3) with applications to underwater vehicle dynamics. In 49th IEEE Conference on Decision and Control, pages 5414–5419, 2010. 19

    work page 2010

  43. [43]

    P. J. Olver. Applications of Lie Groups to Differential Equations . Springer, New York, 2nd edition, 1993

    work page 1993

  44. [44]

    J. P. Panda, A. Mitra, and H. V. Warrior. A review on the hydrodynamic characteristics of autonomous underwater vehicles. Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment , 235:15–29, 2021

    work page 2021

  45. [45]

    Pavlov, P

    D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden, and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D, 240:443–458, 2011

    work page 2011

  46. [46]

    L. Peng. Symmetries, conservation laws, and Noether’s theorem for differential-difference equa- tions. Studies in Applied Mathematics , 139:457–502, 2017

    work page 2017

  47. [47]

    Peng and P

    L. Peng and P. E. Hydon. Transformations, symmetries and Noether theorems for differential- difference equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 478:20210944, 2022

    work page 2022

  48. [48]

    Poincar´ e

    H. Poincar´ e. Sur une forme nouvelle des ´ equations de la m´ ecanique. Comptes Rendus de l’Acad´ emie des Sciences de Paris, 132:369–371, 1901

    work page 1901

  49. [49]

    Squire, H

    J. Squire, H. Qin, W. M. Tang, and C. Chandre. The Hamiltonian structure and Euler–Poincar´ e formulation of the Vlasov–Maxwell and gyrokinetic systems. Physics of Plasmas , 20, 2013

    work page 2013

  50. [50]

    A. P. Veselov. Integrable discrete-time systems and difference operators. Functional Analysis and Its Applications, 22:83–93, 1988

    work page 1988

  51. [51]

    J. M. Wendlandt and J. E. Marsden. Mechanical integrators derived from a discrete variational principle. Physica D, 106:223–246, 1997. 20

    work page 1997