Contact mechanics
Goal: Understand the geometric formulation of contact mechanics, its physics applications and theoretical problems. Understand the geometric theory of contact structures and contact transformations.
Supervisor: Katarjina Grabowska
References:
Basics and geometric formulation
1. A. Bravetti, H. Cruz, D. Tapias, Contact Hamiltonian mechanics, Ann. Phys. 376 (2017) 17-39.
2. M. de Leon, M. L. Valcazar, Contact Hamiltonian systems, J. Math. Phys. 60 (2019) 102902.
3. A. A. Simoes, A. A., M. de Leon, M.L. Valcazar, M. L., D. M. de Diego, Contact geometry for simple thermodynamical systems with friction, Proc. Royal Soc. A 476 (2020) 2241.
4. M. L. Valcázar, Contact Hamiltonian Systems, PhD thesis.
5. X. R. Guijarro, Geometrical aspects of contact mechanical systems and field theories, PhD thesis.
6. M. L. de Leon, A. M. Muniz-Brea, The Hamilton-Jacobi theory for contact Hamiltonian systems, Sigma Mathematics 9 (2021) 1993.
7. O. Esen, M. L. Valcázar, M. León, J. C. Marrero, Contact Dynamics: Legendrian and Lagrangian Submanifolds, Sigma Mathematics 9 (2021) 21.
8. K. Grabowska, J. Grabowski, A geometric approach to contactc Hamiltonians and contact Hamilton-Jacobi theory, J. Phys. A: Mathematical and Theoretical 55 (2022) 43.
9. K. Grabowska, J. Grabowski, Contact geometric mechanics: the Tulczyjew triples, arXiv:2209.03154 [math.SG].
10. K. Grabowska, J. Grabowski, Reductions: precontact versus presymplectic, Annali di Mat. Pura ed Appl. 202 (2023) 2803-2839.
11. K. Grabowska, J. Grabowski, Contactifications: a Lagrangian description of compact Hamiltonian systems, J. Phys. A 57 (2024) 395204.
Contact dynamics
12. J. Montaldi, Equilibria and bifurcations in contact dynamics, Geometric Mechanics 1(2023) 1.
13. A, Bravetti, M. de Leon, J. C. Marrero, E. Padron, Invariant measures for contact Hamiltonian systems, J. Phys. A 53 (2020) 455205.
The Herglotz variational principle
14. P. Cannarsa, W. Cheng, K. Wang, J. Yan, Herglotz’ Generalized Variational Principle and Contact Type Hamilton-Jacobi Equations, in Trends in Control Theory and Partial differential equations, Springer, 2019.
15. Q. Liu, P. J. Torres, C. Wang, Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Phys. 395 (2018) 26-44.
16. K. Wang, L. Wang. J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures et Appliquées 123 (219) 167-200.
Symmetries and contact integrability
17. M. de Leon, M. L. Valcazar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geom. Phys, 153 (2020) 103651.
18. L. Colombo, M. de León, M. Lainz, A. López-Gordón, Liouville-Arnold theorem for contact Hamiltonian systems, arXiv:2302.12061 [math.SG].
19. M. L. de Leon, M. Lainz, A. Lopez-Gordon, X. Rivas, : Hamilton–Jacobi theory and integrability for autonomous and non-autonomous contact systems, J. Geom. Phys, 187 (2023) 104787.
20. Khecin, Tabachnikov, Contact complete integrability, Regular and Chaotic Dynamics 15 (2010) 504-520.
21. Ignoul, Introduction to contact complete integrability, arXiv:2305.03553 [math.DS].
22.Jovanovic and Jovanovic, Contact flows and integrable systems, J. Geom. Phys 87 (2015) 217-232.
23. Boyer, Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on S2 × S3, SIGMA 7 (2011) 058.
Contact topology
24. H. Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics 109, Cambridge U.P., 2008.
Note. References on jet bundles, geometric PDEs and the calculus of variations can be found here.