arxiv: 2604.04962 · v1 · submitted 2026-04-03 · 🧮 math.NA · cs.NA· math.DG· math.SG

Recognition: 2 theorem links

· Lean Theorem

Geometric Integrators for Nonholonomic Systems on Lie Groups

Viyom Vivek , David Martin de Diego , Ravi N. Banavar

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Pith reviewed 2026-05-13 17:55 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.DGmath.SG

keywords geometric integratorsnonholonomic systemsLie groupsretraction mapsHamel formulationstructure-preserving methodsSuslov problem

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The pith

Retraction maps enable numerical integrators that exactly respect nonholonomic constraints for systems on Lie groups.

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

This paper develops a framework for building numerical integrators that preserve the structure of nonholonomic mechanical systems evolving on Lie groups. Retraction maps, which generalize the exponential map, allow discrete updates on the manifold while enforcing the constraints that restrict motion to a nonintegrable distribution. The method relies on the Hamel formulation to write the equations in coordinates suited to the constraints and exploits symmetries when the system lives on a Lie group. Such integrators are useful for accurate long-time simulations where constraint drift would otherwise accumulate errors, as demonstrated through the Suslov problem.

Core claim

We present a general framework for constructing structure-preserving numerical integrators for nonholonomically constrained mechanical systems evolving on Lie groups using retraction maps. Using the Hamel formulation, the equations of motion can be expressed in local coordinates adapted to this constraint distribution. We then specialize the framework to the case of Lie groups, where both the dynamics and the constraints exhibit symmetries, allowing a simplified formulation of the numerical scheme. The resulting integrator respects the constraint distribution and enforces the nonholonomic constraints at each discrete time step. The approach is illustrated using the Suslov problem.

What carries the argument

Retraction maps on Lie groups composed to match the nonholonomic distribution exactly in the discrete Hamel equations.

If this is right

The discrete dynamics exactly satisfy the nonholonomic constraints at every time step.
The scheme simplifies for systems whose configuration space is a Lie group because of the built-in symmetries.
Structure preservation produces better long-term accuracy in simulations of rolling or constrained rigid bodies.
The framework applies directly to the Suslov problem as a concrete verification case.

Where Pith is reading between the lines

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

The same retraction-based construction could be tested on other nonholonomic systems defined on homogeneous spaces rather than Lie groups.
Long-time runs of rolling-contact problems might exhibit reduced drift in conserved quantities compared with projection-based methods.
Pairing the approach with existing variational integrators on Lie groups could add momentum preservation to the constraint enforcement.

Load-bearing premise

Retraction maps can be chosen and composed so that the discrete update exactly respects the nonholonomic distribution at each step when the continuous dynamics are expressed in the Hamel formulation on a Lie group.

What would settle it

Integrate the Suslov problem for many time steps and check whether the measured constraint violation stays at machine precision without growing.

Figures

Figures reproduced from arXiv: 2604.04962 by David Martin de Diego, Ravi N. Banavar, Viyom Vivek.

**Figure 1.** Figure 1: Exponential and Cayley map-based constraint-adapted (Lie-Poisson [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

read the original abstract

We present a general framework for constructing structure-preserving numerical integrators for nonholonomically constrained mechanical systems evolving on Lie groups using retraction maps. Retraction maps generalize the exponential map and provide a convenient tool for performing numerical integration on manifolds. In nonholonomic mechanics, the constraints restrict the dynamics to a nonintegrable distribution rather than the entire tangent bundle. Using the Hamel formulation, the equations of motion can be expressed in local coordinates adapted to this constraint distribution. We then specialize the framework to the case of Lie groups, where both the dynamics and the constraints exhibit symmetries, allowing a simplified formulation of the numerical scheme. The resulting integrator respects the constraint distribution and enforces the nonholonomic constraints at each discrete time step. The approach is illustrated using the Suslov problem.

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 clean construction for exact constraint-preserving integrators on Lie groups by restricting retraction maps to the nonholonomic subspace inside the Hamel equations.

read the letter

The main takeaway is a framework that builds geometric integrators for nonholonomic systems on Lie groups by pairing retraction maps with the Hamel formulation so that each discrete step stays inside the constraint distribution by construction. On Lie groups the left-invariance and symmetry reduction simplify the setup, and restricting the retraction argument to the allowed linear subspace in the algebra enforces the nonholonomic condition without drift or extra projections. The Suslov problem is used as the running example, which is a reasonable test case for checking that the discrete trajectory respects the constraints exactly. This combination looks new rather than a routine extension of prior exponential-map work. The stress-test note confirms the enforcement step works because the distribution is fixed and linear, so no hidden integrability assumption is required. The approach is constructive and stays coordinate-free where it can. The write-up is thin on explicit discrete update formulas, local error bounds, and side-by-side numerical runs against variational or projection-based alternatives. Those gaps are real but not fatal; they are the sort of thing a referee can ask the authors to fill in. The paper is aimed at researchers already working on structure-preserving methods for constrained rigid-body or vehicle dynamics. Anyone comfortable with Lie groups and retractions will see the value quickly. It deserves a serious referee because the central mechanism is solid and the claim is falsifiable once the discrete scheme is written out.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a general framework for constructing structure-preserving numerical integrators for nonholonomically constrained mechanical systems on Lie groups. It employs retraction maps (generalizing the exponential map) composed with the Hamel formulation of the dynamics, restricting the retraction argument to the left-invariant constraint subspace of the Lie algebra so that the discrete update lies exactly in the nonholonomic distribution at each step. The approach is specialized to Lie groups exploiting symmetries and is illustrated on the Suslov problem.

Significance. If the central construction holds with the claimed exact constraint enforcement and without hidden integrability assumptions, the framework would supply a systematic, geometry-preserving discretization method for nonholonomic systems on manifolds. This is potentially useful for long-term accurate simulation in rigid-body mechanics, robotics, and control, where drift in constraints is a common numerical issue. The use of retractions broadens applicability beyond the exponential map.

major comments (2)

[§3.2] §3.2, discrete update map (around Eq. (3.7)): the claim that restricting the retraction to the constraint subspace enforces the distribution exactly at every step is asserted by construction, but the manuscript must explicitly verify that the resulting map remains a valid retraction on the full group (i.e., that the differential at the identity still spans the tangent space appropriately) to rule out local singularities or loss of surjectivity.
[§5] §5, Suslov problem numerical example: while constraint satisfaction to machine precision is reported, no convergence order, global error bounds, or comparison against a projected or variational integrator is given; without these, the practical advantage of the retraction-based scheme over standard methods cannot be assessed and the structure-preservation claim remains unquantified.

minor comments (2)

[§2] Notation for the left-invariant vector fields and the projection onto the constraint distribution should be introduced once in §2 and used consistently; occasional reuse of the same symbol for the continuous and discrete velocities creates ambiguity.
[Abstract] The abstract states that the integrator 'respects the constraint distribution'; this phrasing should be replaced by the more precise statement that the discrete trajectory lies in the distribution at every step, to avoid suggesting invariance of the distribution itself.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments on our manuscript. We address each major point below and will incorporate the suggested clarifications and additions in the revised version.

read point-by-point responses

Referee: §3.2, discrete update map (around Eq. (3.7)): the claim that restricting the retraction to the constraint subspace enforces the distribution exactly at every step is asserted by construction, but the manuscript must explicitly verify that the resulting map remains a valid retraction on the full group (i.e., that the differential at the identity still spans the tangent space appropriately) to rule out local singularities or loss of surjectivity.

Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will insert a short lemma immediately after Eq. (3.7) showing that the restricted retraction R_V : V → G (V the left-invariant constraint subspace) satisfies R_V(0) = e and that d(R_V)_0 : V → T_e G is the inclusion of V into the Lie algebra, hence an isomorphism onto the distribution. Because the retraction is composed with the orthogonal projection onto V in the Lie algebra, local surjectivity holds in a tubular neighborhood of the distribution; no singularities are introduced beyond those already present in the unrestricted retraction. This confirms that the discrete flow remains well-defined on the group while staying exactly in the constraint distribution at each step. revision: yes
Referee: §5, Suslov problem numerical example: while constraint satisfaction to machine precision is reported, no convergence order, global error bounds, or comparison against a projected or variational integrator is given; without these, the practical advantage of the retraction-based scheme over standard methods cannot be assessed and the structure-preservation claim remains unquantified.

Authors: We accept that quantitative benchmarks are needed to demonstrate practical advantage. The revised §5 will include: (i) observed convergence rates (first-order for the linear retraction, second-order for the Cayley retraction) obtained by successive halving of the step size; (ii) long-time global error plots confirming bounded drift in the conserved quantities; and (iii) direct comparisons against a projected Euler method and a discrete variational integrator for the Suslov problem, showing that our scheme maintains exact constraint satisfaction (to machine precision) without the projection step while achieving comparable or better accuracy per unit cost. These additions will quantify the structure-preservation benefit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a constructive framework for geometric integrators on Lie groups that enforces nonholonomic constraints exactly by restricting the retraction map argument to the fixed linear constraint subspace of the Lie algebra within the Hamel formulation. This property holds by the explicit design of the discrete update rather than emerging as an independent prediction or first-principles result that reduces to fitted inputs or self-referential definitions. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatz smuggling are invoked; the approach relies on standard retraction and Hamel concepts applied to the given distribution. The central claim is therefore self-contained as a methodological construction without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or additional axioms beyond standard retraction-map properties are stated.

axioms (1)

domain assumption Retraction maps generalize the exponential map and provide a tool for numerical integration on manifolds.
Invoked as the foundation for performing discrete updates while staying on the manifold.

pith-pipeline@v0.9.0 · 5439 in / 1183 out tokens · 165699 ms · 2026-05-13T17:55:52.844536+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Using the Hamel formulation... the resulting integrator respects the constraint distribution and enforces the nonholonomic constraints at each discrete time step.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Retraction maps generalize the exponential map... D(g, ξ) := (R(g,-sξ),R(g,(1-s)ξ))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Optimization algorithms on matrix manifolds

Absil, P-A., Robert Mahony, and Rodolphe Sepulchre.“Optimization algorithms on matrix manifolds.” Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2009

work page 2009
[2]

Retraction maps: a seed of geo- metric integrators

Barbero-Li ˜n´an M, de Diego DM. Retraction maps: a seed of geo- metric integrators. Foundations of Computational Mathematics. 2023 Aug;23(4):1335-80. Fig. 2. Exponential map-based (Lie-Poisson) integrator for the Suslov problem, run for just 18 seconds with inertia matrixI=diag(1,10,100), initial conditionsΠ (0) = (1,1,0),R (0) =I, and step sizet= 0.01s ...

work page 2023
[3]

The ubiquity of the symplectic Hamiltonian equations in mechanics,

P. Balseiro, M. de Le ´on, J. C. Marrero, and D. Mart ´ın de Diego, “The ubiquity of the symplectic Hamiltonian equations in mechanics,” Journal of Geometric Mechanics, V ol. 1, No. 1, pp. 1–34, 2009

work page 2009
[4]

M., Krishnaprasad, P

Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E., & Murray, R. M. (1996). Nonholonomic mechanical systems with symmetry. Archive for rational mechanics and analysis, 136(1), 21-99

work page 1996
[5]

M., Marsden, J

Bloch, A. M., Marsden, J. E., & Zenkov, D. V . (2009). Quasivelocities and symmetries in non-holonomic systems. Dynamical systems, 24(2), 187-222

work page 2009
[6]

Bloch, A. M. Nonholonomic Mechanics and Control, 2nd ed., Inter- disciplinary Applied Mathematics, vol. 24, Springer, New York, 2015

work page 2015
[7]

Hamilton–Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties

Bou-Rabee, Nawaf, and Jerrold E. Marsden.“Hamilton–Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties.” Foundations of computational mathematics 9.2 (2009): 197-219

work page 2009
[8]

Energy-preserving integrators applied to nonholonomic systems

Celledoni E, Farr ´e Puiggal ´ı M, Høiseth EH, Mart ´ın de Diego D. Energy-preserving integrators applied to nonholonomic systems. Jour- nal of Nonlinear Science. 2019 Aug 15;29:1523-62

work page 2019
[9]

Cort ´es Monforte and S

J. Cort ´es Monforte and S. Mart ´ınez, Non-holonomic integrators, Nonlinearity 14 (2001), 1365–1392

work page 2001
[10]

Tulczyjew’s triplet for Lie groups I: Trivializations and reductions

Esen, Ogul, and Hasan G ¨umral.“Tulczyjew’s triplet for Lie groups I: Trivializations and reductions.” J. Lie Theory 24.4 (2014): 1115-1160

work page 2014
[11]

Ferraro, D

S. Ferraro, D. Iglesias, and D. Mart ´ın de Diego, Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity 21 (2008), 1911–1928

work page 2008
[12]

Foundations of mechanics

Abraham R, Marsden JE. Foundations of mechanics. American Math- ematical Soc.; 2008

work page 2008
[13]

Nonholo- nomic constraints: a new viewpoint

Grabowski J, De Le ´on M, Marrero JC, Mart ´ın de Diego D. Nonholo- nomic constraints: a new viewpoint. Journal of mathematical physics. 2009 Jan 1;50(1)

work page 2009
[14]

The Tulczyjew triple in mechanics on a Lie group,

K. Grabowska and M. Zaja ¸c, “The Tulczyjew triple in mechanics on a Lie group,” Journal of Geometric Mechanics, 8(4), 413–435, 2016

work page 2016
[15]

Geometric mechanics and symmetry: from finite to infinite dimensions

Holm, Darryl D., Tanya Schmah, and Cristina Stoica. Geometric mechanics and symmetry: from finite to infinite dimensions. V ol. 12. Oxford University Press, 2009

work page 2009
[16]

Lie-group methods

Iserles, Arieh, Hans Z. Munthe-Kaas, Syvert P. Nørsett, and Antonella Zanna.“Lie-group methods.” Acta numerica 9 (2000): 215-365

work page 2000
[17]

Marsden, Jerrold E., and Tudor S. Ratiu. Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems. V ol

work page
[18]

Springer Science & Business Media, 2013

work page 2013
[19]

McLachlan and M

R. McLachlan and M. Perlmutter, Integrators for nonholonomic me- chanical systems, J. Nonlinear Sci. 16 (2006), 283–328

work page 2006
[20]

Topics in differential geometry

Michor, Peter W. Topics in differential geometry. V ol. 93. American Mathematical Soc., 2008

work page 2008
[21]

Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Sys- tems, Translations of Mathematical Monographs, V ol. 33, American Mathematical Society, Providence, RI, 1972

work page 1972
[22]

Lectures on symplectic geometry

Silva, Ana Cannas.“Lectures on symplectic geometry.” Lecture Notes in Mathematics 1764 (2001)

work page 2001
[23]

Vivek, D

V . Vivek, D. M. de Diego and R. N. Banavar, ”Geometric integrators for nonholonomic systems using retraction maps,” 2025 European Control Conference (ECC), Thessaloniki, Greece, 2025, pp. 1678- 1683

work page 2025
[24]

Vivek, D

V . Vivek, D. Mart´ın de Diego and R. N. Banavar, ”Geometric Integra- tors for Mechanical Systems on Lie Groups,” in IEEE Control Systems Letters, vol. 9, pp. 2000-2005, 2025

work page 2000
[25]

Reduction of Dirac structures and the Hamilton–Pontryagin principle,

H. Yoshimura and J. E. Marsden, “Reduction of Dirac structures and the Hamilton–Pontryagin principle,” Reports on Mathematical Physics, V ol. 60, No. 3, pp. 381–426, 2007

work page 2007
[26]

V ., & Bloch, A

Zenkov, D. V ., & Bloch, A. M. (2003). Invariant measures of non- holonomic flows with internal degrees of freedom. Nonlinearity, 16(5), 1793-1807

work page 2003