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Regular and Chaotic Dynamics, 2005, Volume 10, Issue 3, Pages 267–284
DOI: https://doi.org/10.1070/RD2005v010n03ABEH000315 (Mi rcd710) 		 
This article is cited in 24 scientific papers (total in 24 papers)

150th anniversary of H. Poincaré

Periodic flows, rank-two Poisson structures, and nonholonomic mechanics

F. Fassò, A. Giacobbe, N. Sansonetto

Dipartimento di Matematica Pura e Applicata, Università di Padova, Via G. Belzoni 7, 35131 Padova, Italy
Citations (24)
DOI: https://doi.org/10.1070/RD2005v010n03ABEH000315
Abstract: It has been recently observed that certain (reduced) nonholonomic systems are Hamiltonian with respect to a rank-two Poisson structure. We link the existence of these structures to a dynamical property of the (reduced) system: its periodicity, with positive period depending continuously on the initial data. Moreover, we show that there are in fact infinitely many such Poisson structures and we classify them. We illustrate the situation on the sample case of a heavy ball rolling on a surface of revolution.
Keywords: Poisson structures, non-holonomic systems, periodic flows.
Received: 22.04.2005
Accepted: 15.05.2005
Bibliographic databases:
Document Type: Article
MSC: 53D17, 37J60
Language: English
Citation: F. Fassò, A. Giacobbe, N. Sansonetto, “Periodic flows, rank-two Poisson structures, and nonholonomic mechanics”, Regul. Chaotic Dyn., 10:3 (2005), 267–284
Citation in format AMSBIB
\Bibitem{FasGiaSan05}
\by F.~Fass\`o, A. Giacobbe, N.~Sansonetto
\paper Periodic flows, rank-two Poisson structures, and nonholonomic mechanics
\jour Regul. Chaotic Dyn.
\yr 2005
\vol 10
\issue 3
\pages 267--284
\mathnet{http://mi.mathnet.ru/rcd710}
\crossref{https://doi.org/10.1070/RD2005v010n03ABEH000315}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2155187}
\zmath{https://zbmath.org/?q=an:1077.37043}
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  • https://www.mathnet.ru/eng/rcd710
  • https://www.mathnet.ru/eng/rcd/v10/i3/p267
    • This publication is cited in the following 24 articles:
    1. 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)  crossref
    2. Paula Balseiro, Maria Eugenia Garcia, Cora Inés Tori, Marcela Zuccalli, “Momentum map reduction for nonholonomic systems”, Nonlinearity, 36:10 (2023), 5401  crossref
    3. Francesco Fassò, Nicola Sansonetto, “On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art”, Regul. Chaotic Dyn., 27:4 (2022), 409–423  mathnet  crossref  mathscinet
    4. Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruiz-Pantaleón, Eduardo Velasco-Barreras, “Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems”, SIGMA, 18 (2022), 038, 29 pp.  mathnet  crossref  mathscinet
    5. 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)  crossref
    6. Francesco Fassò, Encyclopedia of Complexity and Systems Science Series, Perturbation Theory, 2022, 307  crossref
    7. Paula Balseiro, Nicola Sansonetto, “First Integrals and Symmetries of Nonholonomic Systems”, Arch Rational Mech Anal, 244:2 (2022), 343  crossref
    8. Francesco Fassò, Encyclopedia of Complexity and Systems Science, 2022, 1  crossref
    9. Balseiro P. Yapu L.P., “Conserved Quantities and Hamiltonization of Nonholonomic Systems”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 38:1 (2021), 23–60  crossref  mathscinet  isi  scopus
    10. Fasso F. Garcia-Naranjo L.C. Montaldi J., “Integrability and Dynamics of the N-Dimensional Symmetric Veselova TOP”, J. Nonlinear Sci., 29:3 (2019), 1205–1246  crossref  mathscinet  zmath  isi  scopus
    11. Garcia-Naranjo L.C., “Generalisation of Chaplygin'S Reducing Multiplier Theorem With An Application to Multi-Dimensional Nonholonomic Dynamics”, J. Phys. A-Math. Theor., 52:20 (2019), 205203  crossref  mathscinet  isi  scopus
    12. Benito Hernández-Bermejo, “Congruence method for global Darboux reduction of finite-dimensional Poisson systems”, Journal of Mathematical Physics, 59:6 (2018)  crossref
    13. Luis C. García-Naranjo, James Montaldi, “Gauge Momenta as Casimir Functions of Nonholonomic Systems”, Arch Rational Mech Anal, 228:2 (2018), 563  crossref
    14. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.  mathnet  crossref
    15. Francesco Fassò, Nicola Sansonetto, “Conservation of 'Moving' Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces”, J Nonlinear Sci, 26:2 (2016), 519  crossref
    16. I. A. Bizyaev, A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and bifurcations in nonholonomic mechanics”, 25, no. 10, 2015, 15300–21  mathnet  mathnet  crossref  isi  scopus
    17. A. V. Borisov, I. S. Mamaev, A. V. Tsiganov, “Non-holonomic dynamics and Poisson geometry”, Russian Math. Surveys, 69:3 (2014), 481–538  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Andrey Tsiganov, “Poisson structures for two nonholonomic systems with partially reduced symmetries”, Journal of Geometric Mechanics, 6:3 (2014), 417  crossref
    19. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Ierarkhiya dinamiki pri kachenii tverdogo tela bez proskalzyvaniya i vercheniya po ploskosti i sfere”, Nelineinaya dinam., 9:2 (2013), 141–202  mathnet
    20. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere”, Regul. Chaotic Dyn., 18:3 (2013), 277–328  mathnet  crossref  mathscinet  zmath
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