Abstract
In this paper, we continue the construction of variational integrators adapted to contact geometry started in Vermeeren et al. (J Phys A 52(44):445206, 2019), in particular, we introduce a discrete Herglotz Principle and the corresponding discrete Herglotz Equations for a discrete Lagrangian in the contact setting. This allows us to develop convenient numerical integrators for contact Lagrangian systems that are conformally contact by construction. The existence of an exact Lagrangian function is also discussed.
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Anahory Simoes, A., Marrero, J.C. and de Diego, D.M.: Exact discrete Lagrangian mechanics for nonholonomic mechanics (2020). arxiv:2003.11362
Barbero Liñán, M., Cendra, H., García Toraño, E., de Diego, D.M.: Morse families and Dirac systems. J. Geom. Mech. 11(4), 487–510 (2019)
Blanes, S., Casas, F.: A Concise Introduction to Geometric Numerical Integration. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL (2016)
Bloch, A.: Nonholonomic Mechanics and Control. Springer, Interdisciplinary Applied Mathematics 24 (2015)
Bravetti, A.: Contact Hamiltonian dynamics: the concept and its use. Entropy 19(12), 535 (2017)
Bravetti, A.: Contact geometry and thermodynamics. Int. J. Geom. Methods Mod. Phys. 16(supp01), 1940003 (2018)
Bravetti, A., Seri, M., Vermeeren, M., Zadra, F.: Numerical integration in Celestial Mechanics: a case for contact geometry. Celestial Mech. Dyn. Astronom. 132(1), 1–29 (2020)
Ciaglia, F.M., Cruz, H., Marmo, G.: Contact manifolds and dissipation, classical and quantum. Ann. Phys. 398, 159–179 (2018)
de León, M., de Diego, D.M.: On the geometry of non-holonomic Lagrangian systems. J. Math. Phys. 37, 3389–3414 (1996)
de León, M., Lainz Valcázar, M.: Contact Hamiltonian systems. J. Math. Phys. 60(10), 102–902 (2019a)
de León, M., Lainz Valcázar, M.: Singular Lagrangians and precontact Hamiltonian systems. Int. J. Geom. Methods Modern Phys. 16(10), 1950158 (2019b)
de León, M., Valcázar, M.L.: Infinitesimal symmetries in contact Hamiltonian systems. J. Geom. Phys. 153, 103651 (2020)
Ferraro, S., de León, M., Marrero, J.C., Martín de Diego, D., Vaquero, M.: On the geometry of the Hamilton–Jacobi equation and generating functions. Arch. Ration. Mech. Anal. 226(1), 243–302 (2017)
Gaset, J., Gràcia, X., Muñoz-Lecanda, M.C., Rivas, X., Román-Roy, N.: New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries. Int. J. Geom. Methods Mod. Phys. 17(6), 2050090 (2020)
Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer, Heidelberg. Structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition (2010)
Hartman, P. (2002). Ordinary differential equations, volume 38 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e:34002)], With a foreword by Peter Bates
Herglotz, G.: Beruhrungstransformationen. In: Lectures at the University of Gottingen, Gottingen (1930)
Libermann, P., Marle, C.-M.: Symplectic geometry and analytical mechanics, volume 35 of Mathematics and its Applications. D. Reidel Publishing Co., Dordrecht. Translated from the French by Bertram Eugene Schwarzbach (1987)
Marrero, J., de Diego, D.M., Martínez, E.: On the exact discrete lagrangian function for variational integrators: theory and applications. arxiv:1608.01586v1 [math.dg] (2016)
Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)
Patrick, G.W., Cuell, C.: Error analysis of variational integrators of unconstrained Lagrangian systems. Numer. Math. 113(2), 243–264 (2009)
Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems, Volume 7 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London (1994)
Vermeeren, M., Bravetti, A., Seri, M.: Contact variational integrators. J. Phys. A 52(44), 445206 (2019)
Acknowledgements
Manuel Lainz wishes to thank MICINN and ICMAT for a FPI-Severo Ochoa predoctoral contract PRE2018-083203. The authors are supported by Ministerio de Ciencia e Innovación (Spain) under Grants PID2019-106715GB-C21, MTM2016-76702-P and “Severo Ochoa Programme for Centres of Excellence” in R&D (SEV-2015-0554). A. Simoes is supported by the FCT (Portugal) research fellowship SFRH/BD/129882/2017 partially funded by the European Union (ESF). M. Lainz wishes to thank MICINN and ICMAT for a FPI-Severo Ochoa predoctoral contract PRE2018-083203. The authors also want to thank the referees for the careful reading and useful feedback that contributed to improve the quality of the paper.
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Communicated by Arash Yavari.
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Anahory Simoes, A., Martín de Diego, D., Lainz Valcázar, M. et al. On the Geometry of Discrete Contact Mechanics. J Nonlinear Sci 31, 53 (2021). https://doi.org/10.1007/s00332-021-09708-2
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DOI: https://doi.org/10.1007/s00332-021-09708-2