Abstract
The nonholonomic dynamics can be described by the so-called nonholonomic bracket in the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket in the cotangent bundle of the configuration manifold. This bracket was defined in [2, 10], although there was already some particular and less direct definition. On the other hand, another bracket, also called noholonomic, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems.
M. de León, M. Lainz and A. López-Gordón—Acknowledge financial support from Grants PID2019-106715GB-C21 and CEX2019-000904-S funded by MCIN/AEI/ 10.13039/501100011033. Asier López-Gordón—would also like to thank MCIN for the predoctoral contract PRE2020-093814. J. C. Marrero—Ackowledges financial support from the Spanish Ministry of Science and Innovation and European Union (Feder) Grant PGC2018-098265-B-C32.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bates, L., Sniatycki, J.: Nonholonomic reduction. Rep. Math. Phys. 32, 99–115 (1992)
Cantrijn, F., de León, M., de Diego, D.M.: On almost-Poisson structures in nonholonomic mechanics. Nonlinearity. 12(3), 721–737 (1999)
Cantrijn, F., de León, M., Marrero, J.C., de Diego, D.M.: On almost-Poisson structures in nonholonomic mechanics II. The time-dependent framework. Nonlinearity. 13(4), 1379–1409 (2000)
de León, M., Martín de Diego, D.: On the geometry of non-holonomic Lagrangian systems. J. Math. Phys. 37(7), 3389–3414 (1996)
de León, M., Marrero, J.C., de Diego, D.M.: Linear almost Poisson structures and Hamilton-Jacobi equation. Appl. Nonholon. Mech. J. Geom. Mech. 2(2), 159–198 (2010)
de León, M., Rodrigues, P.R.: Methods of differential geometry in analytical mechanics. North-Holland Mathematics Studies, 158. North-Holland Publishing Co., pp. x+483. Amsterdam (1989). ISBN: 0-444-88017-8
Eden, R.J.: The Hamiltonian dynamics of non-holonomic systems. Proc. Roy. Soc. London Ser. A 205, 564–583 (1951)
Eden, R.J.: The quantum mechanics of non-holonomic systems. Proc. Roy. Soc. London Ser. A 205, 583–595 (1951)
Grabowski, J., de León, M., Marrero, J.C., de Diego, D.M.: Nonholonomic constraints: a new viewpoint. J. Math. Phys. 50(1), 17, 013520 (2009)
Ibort, A., de Leon, M., Marrero, J.C., de Diego, D.M.: Dirac brackets in constrained dynamics. Fortschr. Phys. 47(5), 459–492 (1999)
van der Schaft, A.J., Maschke, B.M.: On the Hamiltonian formulation of nonholonomic mechanical systems. Rep. Math. Phys. 34, 225–33 (1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
de León, M., Lainz, M., López-Gordón, A., Marrero, J.C. (2023). Nonholonomic Brackets: Eden Revisited. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-031-38299-4_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-38298-7
Online ISBN: 978-3-031-38299-4
eBook Packages: Computer ScienceComputer Science (R0)