×

Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior. (English) Zbl 1394.37099

Summary: This paper describes the connections between the notions of Hamiltonian system, contact Hamiltonian system and nonholonomic system from the perspective of differential equations and dynamical systems. It shows that action minimizing curves of nonholonomic system satisfy the dissipative Lagrange system, which is equivalent to the contact Hamiltonian system under some generic conditions. As the initial research of contact Hamiltonian dynamics in this direction, we investigate the dynamics of contact Hamiltonian systems in some special cases including invariants, completeness of phase flows and periodic behavior.

MSC:

37J55 Contact systems
37J60 Nonholonomic dynamical systems
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnold, V. I., Mathematical Methods of Classical Mechanics, 349-370, (1999), Springer-Verlag
[2] Andrianov, S., Math. Comput. Simul., 57, 3-5, 139-145, (2001) · Zbl 0984.65131
[3] Boyer, C. P.; Galicki, K., J. Geom. Phys., 35, 4, 288-298, (2000) · Zbl 0984.53032
[4] Grabowski, J., J. Geom. Phys., 68, 7, 27-58, (2013) · Zbl 1280.53070
[5] Contreras, G.; Iturriaga, R., Global Minimizers of Autonomous Lagrangians, (1999), IMPA Rio de Janeiro · Zbl 0957.37065
[6] Liu, Q.; Li, X.; Qian, D., J. Differential Equations, 261, 10, 5784-5802, (2016) · Zbl 1409.70013
[7] Fathi, A., Weak KAM Theorem in Lagrangian Dynamics Vol. 58, (2011), Cambridge Univ. Press
[8] Wang, K.; Wang, L.; Yan, J., Nonlinearity, 30, 2, 492, (2016)
[9] Bravetti, A.; Lopez-Monsalvo, C. S.; Nettel, F., Ann. Physics, 361, 377-400, (2014) · Zbl 1360.80003
[10] Goto, S., J. Math. Phys., 57, 10, 167-181, (2015)
[11] Grmela, M., Entropy, 16, 3, 1652-1686, (2014)
[12] Eberard, D.; Maschke, B.; Van Der Schaft, A., Rep. Math. Phys., 60, 2, 175-198, (2007) · Zbl 1210.80001
[13] Bravetti, A.; Cruz, H.; Tapias, D., Ann. Physics, 376, 17-39, (2017) · Zbl 1364.37138
[14] Cannarsa, P.; Sinestrari, C., Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Vol. 58, (2004), Springer Science & Business Media · Zbl 1095.49003
[15] Boyer, C. P., Symmetry Integrability Geom. Methods Appl., 7, 1-22, (2011) · Zbl 1242.53112
[16] Cendra, H.; Marsden, J. E.; Ratiu, T. S., (Enguist, B.; Schmid, W., Mathematics Unlimited-2001 and Beyond, (2001), Springer- Verlag New York) · Zbl 0955.00011
[17] Monforte, J. C., Geometric, Control and Numerical Aspects of Nonholonomic Systems, (2002), Springer New York · Zbl 1009.70001
[18] Brunt, B. V., Calculus of Variations, (2006), Springer New York
[19] Mazzucchelli, M., Critical Point Theory for Lagrangian Systems, 80, (2012), Springer Basel · Zbl 1246.37003
[20] P. Carnnarsa, W. Cheng, J. Yan, Herglotz generalized variational principle and contact type Hamilton-Jacobi equations, 2017, Preprint.; P. Carnnarsa, W. Cheng, J. Yan, Herglotz generalized variational principle and contact type Hamilton-Jacobi equations, 2017, Preprint.
[21] Su, X.; Wang, L.; Yan, J., Discrete Contin. Dyn. Syst., 36, 11, 6487-6522, (2016) · Zbl 1352.37171
[22] Davini, A.; Fathi, A.; Iturriaga, R.; Zavidovique, M., Invent. Math., 206, 1, 29-55, (2016) · Zbl 1362.35094
[23] Liu, Q.; Wang, K.; Wang, L.; Yan, J., J. Differential Equations, 261, 10, 5289-5305, (2016) · Zbl 1352.37157
[24] Lewis Jr., H. R., J. Math. Phys., 9, 11, 1976-1986, (1968) · Zbl 0167.52501
[25] Pinney, E., Proc. Amer. Math. Soc., 1, 5, 681, (1950) · Zbl 0038.24303
[26] Zhang, M., Adv. Nonlinear Stud., 6, 1, 57-67, (2006) · Zbl 1107.34037
[27] Leach, P.; Andriopoulos, K., Appl. Anal. Discrete Math., 146-157, (2008) · Zbl 1199.34006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.