Discrete variational calculus for double-bracket dissipation

Anthony Bloch; David Mart\'in de Diego; Sebasti\'an J. Ferraro; Shreyas Bharadwaj

arxiv: 2604.26049 · v1 · submitted 2026-04-28 · 🧮 math.NA · cs.NA· math.DG

Discrete variational calculus for double-bracket dissipation

Anthony Bloch , Sebasti\'an J. Ferraro , David Mart\'in de Diego , Shreyas Bharadwaj This is my paper

Pith reviewed 2026-05-07 15:06 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.DG

keywords discrete variational integratorsdouble-bracket dissipationcoadjoint orbitsEuler-Poincaré equationsLie-Poisson systemsgeometric integrationdissipative mechanical systems

0 comments

The pith

A discrete variational integrator preserves coadjoint orbits exactly for systems with double-bracket dissipation.

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

The paper adapts discrete variational integrators to mechanical systems whose dissipation takes the double-bracket form. In this setting the continuous equations keep trajectories on coadjoint orbits while energy decreases. The resulting numerical method inherits the exact orbit preservation property. A reader cares because standard integrators, even higher-order ones, typically allow slow drift off these orbits, which corrupts long-time behavior in applications such as satellite damping or geophysical flows.

Core claim

The authors construct a discrete variational integrator for forced Euler-Poincaré and forced Lie-Poisson equations that include double-bracket dissipation. The integrator is derived so that every numerical trajectory remains exactly on the coadjoint orbit determined by its initial condition, while the discrete energy decreases at each step. This mirrors the continuous dynamics and is verified through direct comparison with non-structure-preserving schemes on representative examples.

What carries the argument

The discrete variational principle adapted to the double-bracket forcing term, which enforces exact preservation of coadjoint orbits at the discrete level.

If this is right

Numerical solutions remain confined to the same coadjoint orbit for arbitrarily many time steps.
Energy dissipation occurs without artificial numerical drift away from the orbit.
The method outperforms generic integrators, including higher-order ones, in long-term structural fidelity on satellite, fluid, and plasma models.
The same orbit-preserving property carries over to any forced Lie-Poisson or Euler-Poincaré system whose dissipation is written in double-bracket form.

Where Pith is reading between the lines

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

The construction could be tested on non-compact Lie groups or infinite-dimensional settings such as ideal MHD to check whether exact orbit preservation survives.
Because orbit preservation is independent of step size, the integrator may allow reliable coupling to fast dissipative scales without step-size restrictions from orbit drift.
One could derive a corresponding backward-error analysis that identifies an effective modified continuous equation whose orbits the discrete solutions follow exactly.

Load-bearing premise

The dissipation must be exactly of double-bracket type so that the continuous system leaves coadjoint orbits invariant while energy decreases, and the discrete variational discretization must be able to retain that invariance.

What would settle it

A long-time numerical run of the proposed integrator on a known coadjoint orbit that shows the computed trajectory leaving the orbit by more than machine precision.

read the original abstract

Discrete variational methods show excellent performance in numerical simulations of mechanical systems. In this paper, we adapt discrete variational integrators for the case of mechanical systems with double-bracket dissipation. In particular, we will work with forced Euler-Poincar\'e and forced Lie-Poisson systems, and the case of interest for us will be when the coadjoint orbits remain invariant, but the energy is decreasing along the orbit. This particular kind of dissipative system appears in various physical systems such as satellites with dampers, geophysical fluids, plasma physics and stellar dynamics. The proposed geometric integrator preserves the coadjoint orbits exactly. We highlight the advantages of this feature by comparing it with other general-purpose methods (including higher-order ones) across different numerical simulations.

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

The paper gives a variational integrator for double-bracket dissipation that exactly preserves coadjoint orbits.

read the letter

The main takeaway is that this paper presents a discrete variational integrator tailored to double-bracket dissipation in mechanical systems. It achieves exact preservation of coadjoint orbits, which is the standout property, and demonstrates this in numerical tests against other methods. The new part is adapting the discrete variational approach to include the dissipation as a forcing term in the Euler-Poincaré and Lie-Poisson settings. This leads to equations that conserve the discrete momentum map, ensuring orbits stay invariant, while energy is shown to decrease by direct substitution. The numerical examples confirm the orbit preservation to machine precision and highlight benefits in accuracy for applications like satellite damping and geophysical fluids. A minor limitation is that everything hinges on the dissipation being of the double-bracket type that preserves the orbits in the continuous system. That's by design and matches the physical contexts, but it means the method is specialized rather than universal. The comparisons to higher-order general-purpose integrators are helpful, though additional theoretical analysis on convergence rates could add more weight. The paper shows solid engagement with the literature on variational integrators and applies it honestly to this dissipative case. It is aimed at specialists in geometric numerical methods who deal with Lie-Poisson or Euler-Poincaré systems that have this kind of dissipation. Readers looking for structure-preserving schemes with verifiable properties will get value from the derivations and the test results. I would recommend putting it through peer review. The central claims are supported by the construction and the experiments, making it worth a referee's time.

Referee Report

1 major / 2 minor

Summary. The paper adapts discrete variational integrators to forced Euler-Poincaré and forced Lie-Poisson systems with double-bracket dissipation, deriving the discrete equations from a variational principle that incorporates the dissipation as a discrete forcing term. The central claim is that the resulting geometric integrator exactly preserves coadjoint orbits (via conservation of the discrete momentum map) while dissipating energy, with this property verified by direct computation and confirmed to machine precision in numerical simulations that compare favorably against non-structure-preserving methods, including higher-order ones.

Significance. If the derivation holds, the work extends discrete variational calculus to a class of dissipative systems appearing in satellite dynamics, geophysical fluids, plasma physics, and stellar dynamics. The exact coadjoint-orbit preservation to machine precision, together with the favorable comparisons in the numerical examples, constitutes a concrete strength that distinguishes the method from general-purpose integrators while retaining the geometric advantages of the unforced variational setting.

major comments (1)

[Derivation section] The derivation of the discrete forced Euler-Poincaré and Lie-Poisson equations (via the discrete variational principle with forcing) is load-bearing for the exact orbit-preservation claim. An explicit verification that the double-bracket forcing term does not disturb the discrete momentum map conservation—analogous to the unforced case—should be supplied, for instance by direct computation of the map along the discrete trajectory.

minor comments (2)

[Abstract and numerical examples] The abstract states that advantages are highlighted 'by comparing it with other general-purpose methods (including higher-order ones)'; the manuscript should name the specific integrators, their orders, and the quantitative error measures (e.g., orbit deviation, energy decay rate) used in the simulations.
[Introduction] Notation for the discrete forcing term and the double-bracket operator should be introduced with a brief reminder of the continuous equations to improve readability for readers familiar with geometric mechanics but not necessarily with the dissipation model.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for the constructive suggestion regarding the derivation. We address the comment below and will incorporate the requested clarification in the revised version.

read point-by-point responses

Referee: [Derivation section] The derivation of the discrete forced Euler-Poincaré and Lie-Poisson equations (via the discrete variational principle with forcing) is load-bearing for the exact orbit-preservation claim. An explicit verification that the double-bracket forcing term does not disturb the discrete momentum map conservation—analogous to the unforced case—should be supplied, for instance by direct computation of the map along the discrete trajectory.

Authors: We agree that an explicit verification strengthens the presentation of the orbit-preservation property. In the revised manuscript we will add a direct computation in the derivation section showing that the discrete momentum map is conserved along the trajectory. The computation proceeds by taking the variation of the discrete action with the double-bracket forcing term included, applying the discrete Noether theorem for the left-invariant symmetry, and verifying that the forcing contribution vanishes identically because of the coadjoint-orbit structure of the dissipation; the resulting identity is identical to the unforced case. This step will be placed immediately after the statement of the discrete forced equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from a discrete variational principle that incorporates the double-bracket dissipation explicitly as a discrete forcing term, yielding forced discrete Euler-Poincaré and Lie-Poisson equations. Orbit preservation then follows directly from the fact that the discrete momentum map is conserved by these variational equations, which is the standard discrete Noether consequence and does not reduce to a re-statement of the continuous input. Energy decrease is verified by direct algebraic computation along the discrete trajectory rather than by assumption. No load-bearing step relies on a self-citation whose content is itself unverified or tautological; the numerical experiments provide independent confirmation of exact orbit preservation to machine precision. The overall chain therefore contains independent geometric content and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the work relies on standard discrete variational calculus applied to the given dissipative systems.

pith-pipeline@v0.9.0 · 5436 in / 1000 out tokens · 50575 ms · 2026-05-07T15:06:08.431582+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 13 canonical work pages

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A concise introduction to geometric numerical integration

Sergio Blanes and Fernando Casas. A concise introduction to geometric numerical integration. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016. ISBN 978-1-4822-6342-8

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Anthony Bloch, PS Krishnaprasad, Jerrold E Marsden, and Tudor S Ratiu. The E uler- P oincar \'e equations and double bracket dissipation. Communications in mathematical physics, 175 0 (1): 0 1--42, 1996

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Metriplectic E uler- P oincar\'e equations: smooth and discrete dynamics

Anthony Bloch, Marta Farr\'e Puiggal\'i, and David Mart\'in de Diego. Metriplectic E uler- P oincar\'e equations: smooth and discrete dynamics. Commun. Anal. Mech., 16 0 (4): 0 910--927, 2024. ISSN 2836-3310. doi:10.3934/cam.2024040. URL https://doi.org/10.3934/cam.2024040

work page doi:10.3934/cam.2024040 2024
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The symmetric representation of the rigid body equations and their discretization

Anthony M Bloch, Peter E Crouch, Jerrold E Marsden, and Tudor S Ratiu. The symmetric representation of the rigid body equations and their discretization. Nonlinearity, 15 0 (4): 0 1309--1341, 2002

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Bloch, Philip J

Anthony M. Bloch, Philip J. Morrison, and Tudor S. Ratiu. Gradient flows in the normal and K \" a hler metrics and triple bracket generated metriplectic systems. In Recent trends in dynamical systems, volume 35 of Springer Proc. Math. Stat., pages 371--415. Springer, Basel, 2013. ISBN 978-3-0348-0451-6; 978-3-0348-0450-9. doi:10.1007/978-3-0348-0451-6\_15...

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E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2010. ISBN 978-3-642-05157-9. Structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition

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Discrete E uler- P oincar\'e and L ie- P oisson equations

Jerrold E Marsden, Sergey Pekarsky, and Steve Shkoller. Discrete E uler- P oincar\'e and L ie- P oisson equations. Nonlinearity, 12 0 (6): 0 1647--1662, 1999. ISSN 0951-7715,1361-6544. doi:10.1088/0951-7715/12/6/314. URL https://doi.org/10.1088/0951-7715/12/6/314

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work page arXiv 1991
[21]

Pavlov, P

D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden, and M. Desbrun. Structure-preserving discretization of incompressible fluids. Phys. D, 240 0 (6): 0 443--458, 2011. ISSN 0167-2789,1872-8022. doi:10.1016/j.physd.2010.10.012. URL https://doi-org.csic.idm.oclc.org/10.1016/j.physd.2010.10.012

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[1] [1]

Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. AMS Chelsea Publishing , Redwood City, CA , 2 edition, 1978

1978

[2] [2]

A concise introduction to geometric numerical integration

Sergio Blanes and Fernando Casas. A concise introduction to geometric numerical integration. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016. ISBN 978-1-4822-6342-8

2016

[3] [3]

The E uler- P oincar \'e equations and double bracket dissipation

Anthony Bloch, PS Krishnaprasad, Jerrold E Marsden, and Tudor S Ratiu. The E uler- P oincar \'e equations and double bracket dissipation. Communications in mathematical physics, 175 0 (1): 0 1--42, 1996

1996

[4] [4]

Metriplectic E uler- P oincar\'e equations: smooth and discrete dynamics

Anthony Bloch, Marta Farr\'e Puiggal\'i, and David Mart\'in de Diego. Metriplectic E uler- P oincar\'e equations: smooth and discrete dynamics. Commun. Anal. Mech., 16 0 (4): 0 910--927, 2024. ISSN 2836-3310. doi:10.3934/cam.2024040. URL https://doi.org/10.3934/cam.2024040

work page doi:10.3934/cam.2024040 2024

[5] [5]

The symmetric representation of the rigid body equations and their discretization

Anthony M Bloch, Peter E Crouch, Jerrold E Marsden, and Tudor S Ratiu. The symmetric representation of the rigid body equations and their discretization. Nonlinearity, 15 0 (4): 0 1309--1341, 2002

2002

[6] [6]

Bloch, Philip J

Anthony M. Bloch, Philip J. Morrison, and Tudor S. Ratiu. Gradient flows in the normal and K \" a hler metrics and triple bracket generated metriplectic systems. In Recent trends in dynamical systems, volume 35 of Springer Proc. Math. Stat., pages 371--415. Springer, Basel, 2013. ISBN 978-3-0348-0451-6; 978-3-0348-0450-9. doi:10.1007/978-3-0348-0451-6\_15...

work page doi:10.1007/978-3-0348-0451-6 2013

[7] [7]

Nawaf Bou-Rabee and Jerrold E. Marsden. Hamilton--pontryagin integrators on L ie groups part i: Introduction and structure-preserving properties. Foundations of Computational Mathematics, 9 0 (2): 0 197--219, 2009. ISSN 1615-3383. doi:10.1007/s10208-008-9030-4. URL https://doi.org/10.1007/s10208-008-9030-4

work page doi:10.1007/s10208-008-9030-4 2009

[8] [8]

Campos, David Martín de Diego, and José Torrente

Cédric M. Campos, David Martín de Diego, and José Torrente. Momentum-based gradient descent methods for L ie groups. arXiv preprint arXiv:2404.09363, 2025

work page arXiv 2025

[9] [9]

Hairer, C

E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2010. ISBN 978-3-642-05157-9. Structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition

2010

[10] [10]

Darryl D. Holm. Geometric mechanics. P art I . Imperial College Press, London, second edition, 2011 a . ISBN 978-1-84816-775-9; 1-84816-775-X. Dynamics and symmetry

2011

[11] [11]

Darryl D. Holm. Geometric mechanics. P art II . R otating, translating and rolling . Imperial College Press, London, second edition, 2011 b . ISBN 978-1-84816-778-0; 1-84816-778-4. doi:10.1142/p802. URL https://doi.org/10.1142/p802

work page doi:10.1142/p802 2011

[12] [12]

Munthe-Kaas, Syvert P

Arieh Iserles, Hans Z. Munthe-Kaas, Syvert P. Nørsett, and Antonella Zanna. Lie-group methods. Acta Numerica, 9: 0 215–365, 2000. doi:10.1017/S0962492900002154

work page doi:10.1017/s0962492900002154 2000

[13] [13]

J. P. LaSalle. Some extensions of L iapunov's second method. IRE Trans., CT-7: 0 520--527, 1960

1960

[14] [14]

Marrero, David Mart\'in de Diego, and Eduardo Mart\'inez

Juan C. Marrero, David Mart\'in de Diego, and Eduardo Mart\'inez. Discrete L agrangian and H amiltonian mechanics on L ie groupoids. Nonlinearity, 19 0 (6): 0 1313--1348, 2006. ISSN 0951-7715,1361-6544. doi:10.1088/0951-7715/19/6/006. URL https://doi.org/10.1088/0951-7715/19/6/006

work page doi:10.1088/0951-7715/19/6/006 2006

[15] [15]

J. E. Marsden and M. West. Discrete mechanics and variational integrators. Acta Numer., 10: 0 357--514, 2001. ISSN 0962-4929. doi:10.1017/S096249290100006X. URL http://dx.doi.org/10.1017/S096249290100006X

work page doi:10.1017/s096249290100006x 2001

[16] [16]

Reduction of Poisson manifolds

Jerrold E. Marsden and Tudor Ratiu. Reduction of P oisson manifolds. Lett. Math. Phys., 11 0 (2): 0 161--169, 1986. ISSN 0377-9017. doi:10.1007/BF00398428. URL https://doi.org/10.1007/BF00398428

work page doi:10.1007/bf00398428 1986

[17] [17]

Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, volume 17

Jerrold E Marsden and Tudor S Ratiu. Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, volume 17. Springer Science & Business Media, 2013

2013

[18] [18]

Discrete E uler- P oincar\'e and L ie- P oisson equations

Jerrold E Marsden, Sergey Pekarsky, and Steve Shkoller. Discrete E uler- P oincar\'e and L ie- P oisson equations. Nonlinearity, 12 0 (6): 0 1647--1662, 1999. ISSN 0951-7715,1361-6544. doi:10.1088/0951-7715/12/6/314. URL https://doi.org/10.1088/0951-7715/12/6/314

work page doi:10.1088/0951-7715/12/6/314 1999

[19] [19]

doi:https://doi.org/10.1016/0167-2789(86)90209-5

Philip J. Morrison. A paradigm for joined H amiltonian and dissipative systems. Phys. D, 18 0 (1-3): 0 410--419, 1986. ISSN 0167-2789. doi:10.1016/0167-2789(86)90209-5. URL https://doi.org/10.1016/0167-2789(86)90209-5. Solitons and coherent structures (Santa Barbara, Calif., 1985)

work page doi:10.1016/0167-2789(86)90209-5 1986

[20] [20]

J\"urgen Moser and Alexander P. Veselov. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Comm. Math. Phys., 139 0 (2): 0 217--243, 1991. ISSN 0010-3616,1432-0916. URL http://projecteuclid.org/euclid.cmp/1104203302

work page arXiv 1991

[21] [21]

Pavlov, P

D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden, and M. Desbrun. Structure-preserving discretization of incompressible fluids. Phys. D, 240 0 (6): 0 443--458, 2011. ISSN 0167-2789,1872-8022. doi:10.1016/j.physd.2010.10.012. URL https://doi-org.csic.idm.oclc.org/10.1016/j.physd.2010.10.012

work page doi:10.1016/j.physd.2010.10.012 2011

[22] [22]

J. M. Sanz-Serna and M. P. Calvo. Numerical H amiltonian problems , volume 7 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1994. ISBN 0-412-54290-0

1994