Abstract:
In this paper we study Chaplygin’s Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton–Poincaré–d’Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler–Poincaré–Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems.
\Bibitem{FerMesBlo09}
\by O. E. Fernandez, T. Mestdag, A. M. Bloch
\paper A generalization of Chaplygin’s Reducibility Theorem
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 6
\pages 635--655
\mathnet{http://mi.mathnet.ru/rcd1004}
\crossref{https://doi.org/10.1134/S1560354709060033}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2591865}
\zmath{https://zbmath.org/?q=an:1229.37087}
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This publication is cited in the following 26 articles:
Malika Belrhazi, Tom Mestdag, “Geodesic Extensions of Mechanical Systems with Nonholonomic Constraints”, J Nonlinear Sci, 35:2 (2025)
Federico Talamucci, “Nonlinear nonholonomic systems: a simple approach and various examples”, Meccanica, 59:3 (2024), 333
Oscar E. Fernandez, “Quantizing Chaplygin Hamiltonizable nonholonomic systems”, Sci Rep, 12:1 (2022)
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Oscar E. Fernandez, Mala L. Radhakrishnan, “The Quantum Mechanics of a Rolling Molecular “Nanocar””, Sci Rep, 8:1 (2018)
Paula Balseiro, “Hamiltonization of Solids of Revolution Through Reduction”, J Nonlinear Sci, 27:6 (2017), 2001
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Oscar E. Fernandez, “Poincaré transformations in nonholonomic mechanics”, Applied Mathematics Letters, 43 (2015), 96
A. V. Bolsinov, A. A. Kilin, A. O. Kazakov, “Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?”, J. Geom. Phys., 87 (2015), 61–75
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Paula Balseiro, Fields Institute Communications, 73, Geometry, Mechanics, and Dynamics, 2015, 1
A. V. Borisov, I. S. Mamaev, A. V. Tsiganov, “Non-holonomic dynamics and Poisson geometry”, Russian Math. Surveys, 69:3 (2014), 481–538
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of Nonholonomic Systems Consisting of a
Spherical Shell with a Moving Rigid Body Inside”, Regul. Chaotic Dyn., 19:2 (2014), 198–213
Oscar E. Fernandez, Anthony M. Bloch, Dmitry V. Zenkov, “The geometry and integrability of the Suslov problem”, Journal of Mathematical Physics, 55:11 (2014)
Paula Balseiro, “The Jacobiator of Nonholonomic Systems and the Geometry of Reduced Nonholonomic Brackets”, Arch Rational Mech Anal, 214:2 (2014), 453
A. V. Bolsinov, A. A. Kilin, A. O. Kazakov, “Topologicheskaya monodromiya v negolonomnykh sistemakh”, Nelineinaya dinam., 9:2 (2013), 203–227
Alberto G. Rojo, Anthony M. Bloch, “Optical mechanical analogy and Hamiltonization of a nonholonomic system”, Phys. Rev. E, 88:1 (2013)
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