Contact Hamiltonian Description of Systems with Exponentially Decreasing Force and Friction that is Quadratic in Velocity

Abstract

We have given a simple contact Hamiltonian description of a system with exponentially vanishing (or zero) potential under a friction term that is quadratic in velocity. We have given two applications: to cavity solitons and to a free body under air friction.

Keywords

• Contact geometry,Contact Hamiltonian,Friction

References

1. [1] H. Geiges, A brief history of contact geometry and topology, Expo. Math., 19(1) (2001), 25–53.
2. [2] H. Geiges, Christiaan huygens and contact geometry, (2005) arXiv:math/0501255.
3. [3] A. Bravetti, H. Cruz, D. Tapias, Contact Hamiltonian mechanics, Ann. Phys.-New York, 376 (2017), 17–39.
4. [4] Q. Liu, P. J. Torres, C. Wang, Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Phys.-New York, 395 (2018), 26–44.
5. [5] D. Sloan, Dynamical similarity. Phys. Rev. D, 97(12) (2018), 123541.
6. [6] E. Anderson, J. Barbour, B. Foster, N. O Murchadha, Scale-invariant gravity: Geometrodynamics. Classical Quant. Grav., 20 (2003), 1571–1604.
7. [7] E. Anderson, J. Barbour, B. Z. Foster, B. Kelleher, N. O. Murchadha, The physical gravitational degrees of freedom, Classical Quant. Grav., 22 (2005), 1795–1802.
8. [8] J. Barbour, N. O Murchadha, Classical and Quantum Gravity on Conformal Superspace, (1999), arXiv:gr-qc/9911071.

9. [9] F. Mercati, A Shape Dynamics Tutorial, (2014), arXiv:1409.0105.
10. [10] S. R. Anbardan, C. Rimoldi, R. Kheradmand, G. Tissoni, F. Prati, Exponentially decaying interaction potential of cavity solitons, Phys. Rev. E, 97(3) (2018), 032208.
11. [11] L. A. Lugiato, F. Prati, M. Brambilla, L. Columbo, S. Barland, G. Tissoni, K. M. Aghdami, R. Kheradmand, H. Tajalli, H. Vahed, Cavity solitons, In Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics, Springer, 2013, 395–404.