On the geometry of discrete contact mechanics

In this paper, we continue the construction of variational integrators adapted to contact geometry started in \cite{VBS}, 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.

[1]  H. Cendra,et al.  Morse families and Dirac systems , 2018, Journal of Geometric Mechanics.

[2]  Marcello Seri,et al.  Contact variational integrators , 2019, Journal of Physics A: Mathematical and Theoretical.

[3]  Fernando Casas,et al.  A Concise Introduction to Geometric Numerical Integration , 2016 .

[4]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[5]  D. D. Diego,et al.  On the exact discrete Lagrangian function for variational integrators: theory and applications , 2016, 1608.01586.

[6]  Charles-Michel Marle,et al.  Symplectic geometry and analytical mechanics , 1987 .

[7]  M. Muñoz-Lecanda,et al.  New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries , 2019, 1907.02947.

[8]  Marcello Seri,et al.  Numerical integration in Celestial Mechanics: a case for contact geometry , 2019, ArXiv.

[9]  Manuel Lainz Valc'azar,et al.  Singular Lagrangians and precontact Hamiltonian systems , 2019, International Journal of Geometric Methods in Modern Physics.

[10]  George W. Patrick,et al.  Error analysis of variational integrators of unconstrained Lagrangian systems , 2008, Numerische Mathematik.

[11]  Sebastián J. Ferraro,et al.  On the Geometry of the Hamilton–Jacobi Equation and Generating Functions , 2016, 1606.00847.

[12]  Alessandro Bravetti,et al.  Contact Hamiltonian Dynamics: The Concept and Its Use , 2017, Entropy.

[13]  David Martín de Diego,et al.  On the geometry of non‐holonomic Lagrangian systems , 1996 .

[14]  E. Hairer,et al.  Structure-Preserving Algorithms for Ordinary Differential Equations , 2006 .

[15]  Manuel Lainz Valc'azar,et al.  Contact Hamiltonian systems , 2018, Journal of Mathematical Physics.

[16]  A. Bravetti.  Contact geometry and thermodynamics , 2019, International Journal of Geometric Methods in Modern Physics.

[17]  Alexandre Anahory Simoes,et al.  Exact discrete Lagrangian mechanics for nonholonomic mechanics , 2020, ArXiv.

[18]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .