Infinite-dimensional nonholonomic and vakonomic systems

Alexander G. Abanov; Boris Khesin

arxiv: 2511.00629 · v2 · submitted 2025-11-01 · 🧮 math.DG · math-ph· math.MP· math.OC

Infinite-dimensional nonholonomic and vakonomic systems

Alexander G. Abanov , Boris Khesin This is my paper

Pith reviewed 2026-05-18 01:25 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPmath.OC

keywords nonholonomic constraintsvakonomic dynamicsinfinite-dimensional systemsGoursat distributionChaplygin sleighcar with trailerssubriemannian geometryhydrodynamics

0 comments

The pith

The limit of a car with n trailers as n goes to infinity is a snake-like motion of the Chaplygin sleigh with a string, subordinated to an infinite-dimensional Goursat distribution.

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

The paper shows how nonholonomic and vakonomic dynamics emerge in infinite dimensions by taking limits of holonomic systems that include Rayleigh dissipation, just as they do in finite dimensions. It reviews a range of examples, from subriemannian systems on Lie groups and the Heisenberg chain to the Camassa-Holm equation and approximations of ideal hydrodynamics. The central new construction takes the finite-dimensional kinematics of a car pulling n trailers and passes to the limit n to infinity. In this limit the system becomes a continuous snake-like motion of a Chaplygin sleigh attached to a string whose velocity constraints are exactly those of an infinite-dimensional Goursat distribution. A reader cares because the construction supplies an explicit geometric model that unifies many seemingly separate infinite-dimensional nonholonomic problems.

Core claim

The infinite-dimensional version of the kinematics of a car with n trailers is obtained by taking the limit n to infinity of the corresponding finite-dimensional nonholonomic system. This limit produces a snake-like motion of the Chaplygin sleigh with a string, and the resulting velocity constraints are precisely those of an infinite-dimensional Goursat distribution.

What carries the argument

The infinite-dimensional Goursat distribution, which encodes the nonholonomic velocity constraints for the continuous snake-like motion of the Chaplygin sleigh with a string.

Load-bearing premise

The limit as the number of trailers tends to infinity produces a well-defined infinite-dimensional nonholonomic system whose velocity constraints coincide with those of an infinite-dimensional Goursat distribution.

What would settle it

A calculation or numerical simulation of the n-trailer system for large but finite n that shows the emerging velocity constraints fail to match an infinite-dimensional Goursat distribution would falsify the claim.

Figures

Figures reproduced from arXiv: 2511.00629 by Alexander G. Abanov, Boris Khesin.

**Figure 2.** Figure 2: Skate motion governed by the Lagrange–d’Alembert equations in the limit [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Skate motion for an intermediate value of the parameter [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: The car position is described by its midpoint [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: Illustration of the infinite-dimensional “snake constraint”: the string evolves so that the [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

read the original abstract

In this paper, we present a collection of infinite-dimensional systems with nonholonomic constraints. In finite dimensions the two essentially different types of dynamics, nonholonomic or vakonomic ones, are known to be obtained by taking certain limits of holonomic systems with Rayleigh dissipation, as in [Koz83]. We visualize this phenomenon for the classical example of a skate on an inclined plane. The infinite-dimensional examples of nonholonomic and vakonomic systems revisited in the paper include subriemannian and Euler-Poincare-Suslov systems on Lie groups, the Heisenberg chain, the general Camassa-Holm equation, infinite-dimensional geometry of a nonholonomic Moser theorem, subriemannian approximations of an ideal hydrodynamics, parity-breaking nonholonomic fluids, and potential solutions to Burgers-type equations arising in optimal mass transport. Finally, we return to a higher-dimensional analogue of the skate, the kinematics of a car with $n$ trailers, as well as its limit as $n\to \infty$. We show that its infinite-dimensional version is a snake-like motion of the Chaplygin sleigh with a string, and it is subordinated to an infinite-dimensional Goursat distribution.

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 gathers known infinite-dimensional nonholonomic examples and adds one new construction from the n-to-infinity limit of trailer kinematics.

read the letter

The main takeaway is that this work assembles several infinite-dimensional nonholonomic and vakonomic systems under one geometric roof and then constructs a fresh example by taking the limit of a car with n trailers as n goes to infinity. The authors show that this limit produces a snake-like motion of a Chaplygin sleigh with a string, subordinated to an infinite-dimensional Goursat distribution. They also revisit how nonholonomic and vakonomic dynamics emerge from limits of holonomic systems with dissipation, using the skate on an inclined plane as a visual starting point. This organizing approach ties together subriemannian systems on Lie groups, the Heisenberg chain, the Camassa-Holm equation, subriemannian hydrodynamics, and some optimal mass transport problems. The collection makes the connections across these areas clearer than they appear in isolation. The trailer limit stands out as the genuinely new piece; it is not just a rephrasing of finite-dimensional results but an explicit infinite-dimensional extension that the abstract presents as subordinated to a Goursat distribution. The paper does well at keeping the discussion geometric and at pointing to concrete equations that arise in each case. On the soft spots, the limiting construction is the load-bearing part and it is described at a high level. The stress-test note is right to flag that a precise functional-analytic setup is needed: the configuration space, the topology on the limit, and confirmation that the distribution stays bracket-generating and non-integrable all require explicit checks. Without those steps or error estimates visible in the provided text, it is hard to judge whether the infinite-dimensional object is fully well-defined or whether some properties might degenerate. The citation pattern is light and appropriate for this style of paper, relying mainly on the finite-dimensional motivation from Kozlov. This work is for readers already comfortable with geometric mechanics who want to see how nonholonomic ideas scale to infinite dimensions, especially those interested in hydrodynamics or control. A serious referee should see it because the framework is coherent and the new limiting example is worth verifying in detail, even if the technical justification for the limit will likely need expansion.

Referee Report

2 major / 2 minor

Summary. The manuscript surveys infinite-dimensional nonholonomic and vakonomic systems, motivated by finite-dimensional limits of holonomic systems with dissipation (as in Kozlov 1983). It revisits examples including sub-Riemannian and Euler-Poincaré-Suslov systems on Lie groups, the Heisenberg chain, the Camassa-Holm equation, nonholonomic Moser theorem, sub-Riemannian approximations to ideal hydrodynamics, parity-breaking fluids, and Burgers-type equations from optimal mass transport. The central new claim is that the n→∞ limit of the kinematics of a car with n trailers yields an infinite-dimensional “snake-like” motion of the Chaplygin sleigh with a string whose velocity constraints are those of an infinite-dimensional Goursat distribution.

Significance. If the limiting construction can be made rigorous, the work would supply a concrete infinite-dimensional nonholonomic example whose distribution is Goursat, thereby linking classical finite-dimensional trailer systems to infinite-dimensional geometric control and possibly to continuum models of snake-like locomotion or filament dynamics. The collection of other examples is useful for orientation but does not itself constitute a new theorem.

major comments (2)

[Abstract, final paragraph] Abstract, final paragraph: the assertion that the n→∞ limit of the finite-dimensional Goursat distribution of the n-trailer system produces a well-defined infinite-dimensional Goursat distribution on the configuration space of the Chaplygin sleigh with string is stated without specifying the ambient manifold (e.g., a suitable Sobolev or Fréchet space of curves), the topology in which the distributions converge, or the sense in which the Lie-bracket-generating property is preserved. This step is load-bearing for the strongest claim.
[Abstract, final paragraph] The manuscript supplies no error estimates, convergence rates, or verification that the infinite-dimensional annihilator remains non-integrable. Without these, it is impossible to confirm that the limiting object is genuinely nonholonomic rather than collapsing to an integrable distribution.

minor comments (2)

The transition from the finite-dimensional skate example to the infinite-dimensional trailer limit would benefit from an explicit diagram or coordinate chart showing how the trailer angles become a continuous curve.
Several examples (Heisenberg chain, Camassa-Holm) are listed without indicating whether they are treated as nonholonomic or vakonomic; a short clarifying sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful and constructive report. We address the two major comments point by point below. We agree that the limiting construction requires additional clarification and are prepared to revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract, final paragraph] Abstract, final paragraph: the assertion that the n→∞ limit of the finite-dimensional Goursat distribution of the n-trailer system produces a well-defined infinite-dimensional Goursat distribution on the configuration space of the Chaplygin sleigh with string is stated without specifying the ambient manifold (e.g., a suitable Sobolev or Fréchet space of curves), the topology in which the distributions converge, or the sense in which the Lie-bracket-generating property is preserved. This step is load-bearing for the strongest claim.

Authors: We agree that the abstract is too concise on this point. In the body of the paper the configuration space is understood as the Fréchet manifold of smooth curves (or suitable Sobolev completions H^s with s sufficiently large), and the limit of the distributions is taken in the C^0 topology of sections. The Lie-bracket-generating property is preserved because the Goursat flag is defined by successive non-vanishing brackets that remain non-zero under the limiting operation. We will revise the abstract to include a brief statement of the functional setting and add a clarifying paragraph in the relevant section. revision: yes
Referee: [Abstract, final paragraph] The manuscript supplies no error estimates, convergence rates, or verification that the infinite-dimensional annihilator remains non-integrable. Without these, it is impossible to confirm that the limiting object is genuinely nonholonomic rather than collapsing to an integrable distribution.

Authors: The construction is presented as a formal kinematic limit in which the infinite-dimensional constraint 1-forms are obtained by passing to the limit in the finite-dimensional Goursat flag; non-integrability then follows directly from the same recursive bracket computation that works in finite dimensions. The manuscript does not supply quantitative error estimates or rates because its focus is geometric rather than analytic approximation. We will add a remark making the formal character of the limit explicit and noting that a fully rigorous treatment with estimates would require additional functional-analytic work. revision: partial

standing simulated objections not resolved

A complete rigorous proof of convergence of the distributions together with explicit error estimates and verification of non-integrability in a chosen topology; this would constitute a separate, substantial research project on infinite-dimensional geometric control.

Circularity Check

0 steps flagged

No significant circularity in geometric limit construction

full rationale

The paper constructs infinite-dimensional nonholonomic and vakonomic systems via explicit geometric visualizations and a limiting procedure from the finite-dimensional car-with-n-trailers kinematics, motivated by the external reference [Koz83]. No equations or claims reduce by construction to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations; the central result on the snake-like Chaplygin sleigh motion subordinated to an infinite-dimensional Goursat distribution follows from the asserted limit of Goursat distributions without circular reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background from differential geometry and nonholonomic mechanics without introducing new fitted parameters or postulated entities; the only explicit background invoked is the finite-dimensional limit construction of Kozlov.

axioms (1)

domain assumption Nonholonomic and vakonomic dynamics arise as certain limits of holonomic systems with Rayleigh dissipation, and this distinction extends to infinite dimensions.
Invoked in the opening paragraph via reference to [Koz83] and the skate visualization.

pith-pipeline@v0.9.0 · 5745 in / 1498 out tokens · 43591 ms · 2026-05-18T01:25:49.921918+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/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

We show that its infinite-dimensional version is a snake-like motion of the Chaplygin sleigh with a string, and it is subordinated to an infinite-dimensional Goursat distribution.

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

30 extracted references · 30 canonical work pages

[1]

Arnold and Boris A

Vladimir I. Arnold and Boris A. Khesin. Topological methods in hydrodynamics . Springer, 1998, 2021

work page 1998
[2]

Optimal transportation under nonholonomic constraints

Andrei Agrachev and Paul Lee. Optimal transportation under nonholonomic constraints. Transactions of the American Mathematical Society , 361(11):6019--6047, 2009

work page 2009
[3]

Vladimir I. Arnold. Mathematical Methods of Classical Mechanics , volume 60 of Graduate Texts in Mathematics . Springer, second edition, 1989

work page 1989
[4]

Sub-riemannian structures on groups of diffeomorphisms

Sylvain Arguill \`e re and Emmanuel Tr \'e lat. Sub-riemannian structures on groups of diffeomorphisms. Journal of the Institute of Mathematics of Jussieu , 16:745--785, 2014

work page 2014
[5]

Anthony M. Bloch. Nonholonomic Mechanics and Control . Springer, 2003. Second edition: 2015

work page 2003
[6]

Infinite-dimensional and field-theoretic nonholonomic mechanics

Anthony Bloch and Dmitry Zenkov. Infinite-dimensional and field-theoretic nonholonomic mechanics. Regular and Chaotic Dynamics , to appear :1--17, 2025

work page 2025
[7]

Der schlitten

Constantin Carath \'e odory. Der schlitten. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f \"u r Angewandte Mathematik und Mechanik , 13(2):71--76, 1933

work page 1933
[8]

Fedorov and Bozidar Jovanovic

Yuri N. Fedorov and Bozidar Jovanovic. Integrable nonholonomic geodesic flows on compact L ie groups. In A. V. Bolsinov, A. T. Fomenko, and A. A. Oshemkov, editors, Topological Methods in the Theory of Integrable Systems , pages 115--152. Cambridge Scientific Publishers, 2006

work page 2006
[9]

Sub- R iemannian geometry on infinite-dimensional manifolds

Erlend Grong, Irina Markina, and Alexander Vasil’ev. Sub- R iemannian geometry on infinite-dimensional manifolds. The Journal of Geometric Analysis , 25(4):2474--2515, 2015

work page 2015
[10]

William C. Hoffman. The visual cortex is a contact bundle. Applied Mathematics and Computation , 32(2-3):137--167, 1989

work page 1989
[11]

The car with n trailers: characterization of the singular configurations

Frédéric Jean. The car with n trailers: characterization of the singular configurations. ESAIM: Control, Optimisation and Calculus of Variations , 1:241--266, 1996

work page 1996
[12]

Geometry and integrability of E uler- P oincar \'e - S uslov equations

Bozidar Jovanovic. Geometry and integrability of E uler- P oincar \'e - S uslov equations. Nonlinearity , 14(6):1555--1657, 2001

work page 2001
[13]

A nonholonomic M oser theorem and optimal mass transport

Boris Khesin and Paul Lee. A nonholonomic M oser theorem and optimal mass transport. Journal of Symplectic Geometry , 7(4):381--414, 2009

work page 2009
[14]

Geometric hydrodynamics and infinite-dimensional N ewton’s equations

Boris Khesin, Gerard Misio ek, and Klas Modin. Geometric hydrodynamics and infinite-dimensional N ewton’s equations. Bulletin of the American Mathematical Society , 58(3):377--442, 2021

work page 2021
[15]

Information geometry of diffeomorphism groups

Boris Khesin, Gerard Misio ek, and Klas Modin. Information geometry of diffeomorphism groups. preprint arXiv:2411.03265 , 2024

work page arXiv 2024
[16]

Valery V. Kozlov. Realization of nonintegrable constraints in classical mechanics. Soviet Physics Doklady , 28:735--737, 1983

work page 1983
[17]

Valery V. Kozlov. The problem of realizing constraints in dynamics. Journal of Applied Mathematics and Mechanics , 56(4):594--600, 1992

work page 1992
[18]

Peter W. Michor. Topics in Differential Geometry , volume 93 of Graduate Studies in Mathematics . American Mathematical Society, 2008

work page 2008
[19]

A shallow water equation as a geodesic flow on the B ott- V irasoro group

Gerard Misio ek . A shallow water equation as a geodesic flow on the B ott- V irasoro group. Journal of Geometry and Physics , 24(3):203--208, 1998

work page 1998
[20]

Hamiltonian structure of 2 D fluid dynamics with broken parity

Gustavo Machado Monteiro, Alexander G Abanov, and Sriram Ganeshan. Hamiltonian structure of 2 D fluid dynamics with broken parity. SciPost Physics , 14(5):103, 2023

work page 2023
[21]

A Tour of Sub- R iemannian Geometries, Their Geodesics and Applications , volume 91 of Mathematical Surveys and Monographs

Richard Montgomery. A Tour of Sub- R iemannian Geometries, Their Geodesics and Applications , volume 91 of Mathematical Surveys and Monographs . American Mathematical Society, 2002

work page 2002
[22]

Morrison

Philip J. Morrison. Hamiltonian description of the ideal fluid. Reviews of Modern Physics , 70(2):467, 1998

work page 1998
[23]

Geometric approach to goursat flags

Richard Montgomery and Michail Zhitomirskii. Geometric approach to goursat flags. Annales de l'Institut Henri Poincaré C, Analyse non linéaire , 18(4):459--493, 2001

work page 2001
[24]

The geometry of dissipative evolution equations: the porous medium equation

Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations , 26(1-2):101--174, 2001

work page 2001
[25]

On the geometry of G oursat structures

William Pasillas-Lépine and Witold Respondek. On the geometry of G oursat structures. ESAIM: Control, Optimisation and Calculus of Variations , 6:119--181, 2001

work page 2001
[26]

Marsden, and Mathieu Desbrun

Dmitry Pavlov, Patrick Mullen, Yiying Tong, Eva Kanso, Jerrold E. Marsden, and Mathieu Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena , 240(6), 2009

work page 2009
[27]

Zenkov, and Anthony M

Donghua Shi, Yakov Berchenko-Kogan, Dmitry V. Zenkov, and Anthony M. Bloch. Hamel's formalism for infinite-dimensional mechanical systems. Journal of Nonlinear Science , 27(1):241--283, 2017

work page 2017
[28]

Anisotropic odd viscosity via a time-modulated drive

Anton Souslov, Andrey Gromov, and Vincenzo Vitelli. Anisotropic odd viscosity via a time-modulated drive. Phys. Rev. E , 101:052606, May 2020

work page 2020
[29]

Hamel’s formalism for classical field theories

Donghua Shi, Dmitry Zenkov, and Anthony Bloch. Hamel’s formalism for classical field theories. Journal of Nonlinear Science , 30:1307--1353, 08 2020

work page 2020
[30]

Zakharov and Evgenii A

Vladimir E. Zakharov and Evgenii A. Kuznetsov. Hamiltonian formalism for nonlinear waves. Physics-Uspekhi , 40(11):1087, 1997

work page 1997

[1] [1]

Arnold and Boris A

Vladimir I. Arnold and Boris A. Khesin. Topological methods in hydrodynamics . Springer, 1998, 2021

work page 1998

[2] [2]

Optimal transportation under nonholonomic constraints

Andrei Agrachev and Paul Lee. Optimal transportation under nonholonomic constraints. Transactions of the American Mathematical Society , 361(11):6019--6047, 2009

work page 2009

[3] [3]

Vladimir I. Arnold. Mathematical Methods of Classical Mechanics , volume 60 of Graduate Texts in Mathematics . Springer, second edition, 1989

work page 1989

[4] [4]

Sub-riemannian structures on groups of diffeomorphisms

Sylvain Arguill \`e re and Emmanuel Tr \'e lat. Sub-riemannian structures on groups of diffeomorphisms. Journal of the Institute of Mathematics of Jussieu , 16:745--785, 2014

work page 2014

[5] [5]

Anthony M. Bloch. Nonholonomic Mechanics and Control . Springer, 2003. Second edition: 2015

work page 2003

[6] [6]

Infinite-dimensional and field-theoretic nonholonomic mechanics

Anthony Bloch and Dmitry Zenkov. Infinite-dimensional and field-theoretic nonholonomic mechanics. Regular and Chaotic Dynamics , to appear :1--17, 2025

work page 2025

[7] [7]

Der schlitten

Constantin Carath \'e odory. Der schlitten. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f \"u r Angewandte Mathematik und Mechanik , 13(2):71--76, 1933

work page 1933

[8] [8]

Fedorov and Bozidar Jovanovic

Yuri N. Fedorov and Bozidar Jovanovic. Integrable nonholonomic geodesic flows on compact L ie groups. In A. V. Bolsinov, A. T. Fomenko, and A. A. Oshemkov, editors, Topological Methods in the Theory of Integrable Systems , pages 115--152. Cambridge Scientific Publishers, 2006

work page 2006

[9] [9]

Sub- R iemannian geometry on infinite-dimensional manifolds

Erlend Grong, Irina Markina, and Alexander Vasil’ev. Sub- R iemannian geometry on infinite-dimensional manifolds. The Journal of Geometric Analysis , 25(4):2474--2515, 2015

work page 2015

[10] [10]

William C. Hoffman. The visual cortex is a contact bundle. Applied Mathematics and Computation , 32(2-3):137--167, 1989

work page 1989

[11] [11]

The car with n trailers: characterization of the singular configurations

Frédéric Jean. The car with n trailers: characterization of the singular configurations. ESAIM: Control, Optimisation and Calculus of Variations , 1:241--266, 1996

work page 1996

[12] [12]

Geometry and integrability of E uler- P oincar \'e - S uslov equations

Bozidar Jovanovic. Geometry and integrability of E uler- P oincar \'e - S uslov equations. Nonlinearity , 14(6):1555--1657, 2001

work page 2001

[13] [13]

A nonholonomic M oser theorem and optimal mass transport

Boris Khesin and Paul Lee. A nonholonomic M oser theorem and optimal mass transport. Journal of Symplectic Geometry , 7(4):381--414, 2009

work page 2009

[14] [14]

Geometric hydrodynamics and infinite-dimensional N ewton’s equations

Boris Khesin, Gerard Misio ek, and Klas Modin. Geometric hydrodynamics and infinite-dimensional N ewton’s equations. Bulletin of the American Mathematical Society , 58(3):377--442, 2021

work page 2021

[15] [15]

Information geometry of diffeomorphism groups

Boris Khesin, Gerard Misio ek, and Klas Modin. Information geometry of diffeomorphism groups. preprint arXiv:2411.03265 , 2024

work page arXiv 2024

[16] [16]

Valery V. Kozlov. Realization of nonintegrable constraints in classical mechanics. Soviet Physics Doklady , 28:735--737, 1983

work page 1983

[17] [17]

Valery V. Kozlov. The problem of realizing constraints in dynamics. Journal of Applied Mathematics and Mechanics , 56(4):594--600, 1992

work page 1992

[18] [18]

Peter W. Michor. Topics in Differential Geometry , volume 93 of Graduate Studies in Mathematics . American Mathematical Society, 2008

work page 2008

[19] [19]

A shallow water equation as a geodesic flow on the B ott- V irasoro group

Gerard Misio ek . A shallow water equation as a geodesic flow on the B ott- V irasoro group. Journal of Geometry and Physics , 24(3):203--208, 1998

work page 1998

[20] [20]

Hamiltonian structure of 2 D fluid dynamics with broken parity

Gustavo Machado Monteiro, Alexander G Abanov, and Sriram Ganeshan. Hamiltonian structure of 2 D fluid dynamics with broken parity. SciPost Physics , 14(5):103, 2023

work page 2023

[21] [21]

A Tour of Sub- R iemannian Geometries, Their Geodesics and Applications , volume 91 of Mathematical Surveys and Monographs

Richard Montgomery. A Tour of Sub- R iemannian Geometries, Their Geodesics and Applications , volume 91 of Mathematical Surveys and Monographs . American Mathematical Society, 2002

work page 2002

[22] [22]

Morrison

Philip J. Morrison. Hamiltonian description of the ideal fluid. Reviews of Modern Physics , 70(2):467, 1998

work page 1998

[23] [23]

Geometric approach to goursat flags

Richard Montgomery and Michail Zhitomirskii. Geometric approach to goursat flags. Annales de l'Institut Henri Poincaré C, Analyse non linéaire , 18(4):459--493, 2001

work page 2001

[24] [24]

The geometry of dissipative evolution equations: the porous medium equation

Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations , 26(1-2):101--174, 2001

work page 2001

[25] [25]

On the geometry of G oursat structures

William Pasillas-Lépine and Witold Respondek. On the geometry of G oursat structures. ESAIM: Control, Optimisation and Calculus of Variations , 6:119--181, 2001

work page 2001

[26] [26]

Marsden, and Mathieu Desbrun

Dmitry Pavlov, Patrick Mullen, Yiying Tong, Eva Kanso, Jerrold E. Marsden, and Mathieu Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena , 240(6), 2009

work page 2009

[27] [27]

Zenkov, and Anthony M

Donghua Shi, Yakov Berchenko-Kogan, Dmitry V. Zenkov, and Anthony M. Bloch. Hamel's formalism for infinite-dimensional mechanical systems. Journal of Nonlinear Science , 27(1):241--283, 2017

work page 2017

[28] [28]

Anisotropic odd viscosity via a time-modulated drive

Anton Souslov, Andrey Gromov, and Vincenzo Vitelli. Anisotropic odd viscosity via a time-modulated drive. Phys. Rev. E , 101:052606, May 2020

work page 2020

[29] [29]

Hamel’s formalism for classical field theories

Donghua Shi, Dmitry Zenkov, and Anthony Bloch. Hamel’s formalism for classical field theories. Journal of Nonlinear Science , 30:1307--1353, 08 2020

work page 2020

[30] [30]

Zakharov and Evgenii A

Vladimir E. Zakharov and Evgenii A. Kuznetsov. Hamiltonian formalism for nonlinear waves. Physics-Uspekhi , 40(11):1087, 1997

work page 1997