Contact Hamiltonian Description of 1D Frictional Systems

Year 2021, Volume: 4 Issue: 2, 100 - 107, 30.06.2021

Furkan Semih DÜNDAR Gülhan AYAR

https://doi.org/10.33434/cams.937807

Abstract

In this paper, we consider contact Hamiltonian description of 1D frictional dynamics with no conserved force. Friction forces that are monomials of velocity, and sum of two monomials are considered. For that purpose, quite general forms of contact Hamiltonians are taken into account. We conjecture that it is impossible to give a contact Hamiltonian description dissipative systems where the friction force is not in the form considered in this paper.

Keywords

Contact geometry, Contact Hamiltonian mechanics, Friction force

Thanks

We would like to thank anonymous referees whose comments improved the paper.

References

• [1] A. McInerney, A. First Steps in Differential Geometry. Springer, New York, 2013.
• [2] S. Lie, Geometrie der Beru ̈hrungstransformationen (dargestellt von S. Lie und G. Scheffers), B. G. Teubner, Leipzig, 1896
• [3] J.W. Gibbs, Part 1,”Graphical methods in the thermodynamics of fluids” and Part 2, ”A method of geometrical representation of the thermodynamic properties of substances by means of surfaces”, Trans. Connecticut Acad., Part 1,309–342 and Part 2,382–404, 1873.
• [4] H. Geiges. Christiaan huygens and contact geometry, Nieuw Arch. Wiskd, 6(2) (2005), 117–123.
• [5] H. Geiges. A brief history of contact geometry and topology, Expositiones Math., 19(1) (2001), 25–53.
• [6] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1989.
• [7] A. Bravetti, H. Cruz, D. Tapias, Contact Hamiltonian mechanics, Ann. Phys., 376 (2017), 17–39.
• [8] Q. Liu, P.J. Torres, C. Wang. Contact hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Phys., 395 (2018), 26–44.
• [9] F.S. Dündar. Contact hamiltonian description of systems with exponentially decreasing force and friction that is quadratic in velocity, Fundam. J. Math. Appl., 3 (2020), 29–32.