Abstract:
We present a systematic geometric construction of reduced almost Poisson brackets for nonholonomic systems on Lie groups with invariant kinetic energy metric and constraints. Our construction is of geometric interest in itself and is useful in the Hamiltonization of some classical examples of nonholonomic mechanical systems.
Keywords:
nonholonomic systems, almost Poisson brackets, hamiltonization, geometric reduction.
Citation:
L. C. García-Naranjo, “Reduction of Almost Poisson Brackets for Nonholonomic Systems on Lie Groups”, Regul. Chaotic Dyn., 12:4 (2007), 365–388
\Bibitem{Gar07}
\by L.~C.~Garc{\'\i}a-Naranjo
\paper Reduction of Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
\jour Regul. Chaotic Dyn.
\yr 2007
\vol 12
\issue 4
\pages 365--388
\mathnet{http://mi.mathnet.ru/rcd629}
\crossref{https://doi.org/10.1134/S1560354707040028}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2350330}
\zmath{https://zbmath.org/?q=an:1229.37088}
Linking options:
https://www.mathnet.ru/eng/rcd629
https://www.mathnet.ru/eng/rcd/v12/i4/p365
This publication is cited in the following 14 articles:
Luis C. García-Naranjo, Juan C. Marrero, David Martín de Diego, Paolo E. Petit Valdés, “Almost-Poisson Brackets for Nonholonomic Systems with Gyroscopic Terms and Hamiltonisation”, J Nonlinear Sci, 34:6 (2024)
Marco Dalla Via, Francesco Fassò, Nicola Sansonetto, “On the Dynamics of a Heavy Symmetric Ball that Rolls Without Sliding on a Uniformly Rotating Surface of Revolution”, J Nonlinear Sci, 32:6 (2022)
Dong Eui Chang, Matthew Perlmutter, “Feedback Integrators for Nonholonomic Mechanical Systems”, J Nonlinear Sci, 29:3 (2019), 1165
Luis C. García-Naranjo, James Montaldi, “Gauge Momenta as Casimir Functions of Nonholonomic Systems”, Arch Rational Mech Anal, 228:2 (2018), 563
Francesco Fassò, Luis C García-Naranjo, Nicola Sansonetto, “Moving energies as first integrals of nonholonomic systems with affine constraints”, Nonlinearity, 31:3 (2018), 755
Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179
A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Geometrisation of Chaplygin's reducing multiplier theorem”, Nonlinearity, 28:7 (2015), 2307–2318
A. V. Borisov, I. S. Mamaev, “Simmetrii i reduktsiya v negolonomnoi mekhanike”, Nelineinaya dinam., 11:4 (2015), 763–823
Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604
Luis C. García-Naranjo, Joris Vankerschaver, “Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation”, Journal of Geometry and Physics, 73 (2013), 56
Paula Balseiro, Luis C. García-Naranjo, “Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems”, Arch Rational Mech Anal, 205:1 (2012), 267
O. E. Fernandez, T. Mestdag, A. M. Bloch, “A generalization of Chaplygin’s Reducibility Theorem”, Regul. Chaotic Dyn., 14:6 (2009), 635–655
YongXin Guo, Chang Liu, ShiXing Liu, Peng Chang, “Decomposition of almost Poisson structure of non-self-adjoint dynamical systems”, Sci. China Ser. E-Technol. Sci., 52:3 (2009), 761
A. V. Borisov, I. S. Mamaev, “Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems”, Regul. Chaotic Dyn., 13:5 (2008), 443–490
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