### Journal of Geometric Mechanics

2023, Issue 1: 1-26. doi: 10.3934/jgm.2023001
Research article

# Lagrangian–Hamiltonian formalism for cocontact systems

• Received: 29 May 2022 Revised: 25 July 2022 Accepted: 29 July 2022 Published: 19 October 2022
• Primary: 37J55; Secondary: 70H03, 70H05, 53D10, 70G45, 53Z05, 70H45

• In this paper we present a unified Lagrangian–Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.

Citation: Xavier Rivas, Daniel Torres. Lagrangian–Hamiltonian formalism for cocontact systems[J]. Journal of Geometric Mechanics, 2023, 15(1): 1-26. doi: 10.3934/jgm.2023001

### Related Papers:

• In this paper we present a unified Lagrangian–Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.

 [1] R. Skinner, R. Rusk, Generalized Hamiltonian dynamics I: Formulation on $\mathrm{T}^\ast Q\oplus \mathrm{T} Q$, J. Math. Phys., 24 (1983), 2589–2594. https://doi.org/10.1063/1.525654 doi: 10.1063/1.525654 [2] K. Kamimura, Singular Lagrangian and constrained Hamiltonian systems, generalized canonical formalism, Nuovo Cim. B, 68 (1982), 33–54. https://doi.org/10.1007%2FBF02888859 [3] M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda, N. Román-Roy, Unified formalism for non-autonomous mechanical systems, J. Math. Phys., 49 (2008), 062902. https://doi.org/10.1063/1.2929668 doi: 10.1063/1.2929668 [4] J. Cortés, S. Martínez, F. Cantrijn, Skinner–Rusk approach to time-dependent mechanics, Phys. Lett., 300 (2002), 250–258. https://doi.org/10.1016/S0375-9601(02)00777-6 doi: 10.1016/S0375-9601(02)00777-6 [5] X. Gràcia, R. Martín, Geometric aspects of time-dependent singular differential equations, Int. J. Geom. Methods Mod. Phys., 2 (2005), 597–618. https://doi.org/10.1142/S0219887805000697 doi: 10.1142/S0219887805000697 [6] J. Cortés, M. de León, D. Martín de Diego, S. Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison solutions, SIAM J. Control Optim., 41 (2002), 1389–1412. https://doi.org/10.1137/S036301290036817X doi: 10.1137/S036301290036817X [7] X. Gràcia, J. M. Pons, N. Román-Roy, Higher-order Lagrangian systems: Geometric structures, dynamics and constraints, J. Math. Phys., 32 (1991), 2744–2763. https://doi.org/10.1063/1.529066 doi: 10.1063/1.529066 [8] X. Gràcia, J. M. Pons, N. Román-Roy, Higher-order conditions for singular Lagrangian systems, J. Phys. A: Math. Gen., 25 (1992), 1981–2004. https://doi.org/10.1088/0305-4470/25/7/037 doi: 10.1088/0305-4470/25/7/037 [9] P. D. Prieto-Martínez, N. Román-Roy, Lagrangian–Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A: Math. Theor., 44 (2011), 385203. https://doi.org/10.1088/1751-8113/44/38/385203 doi: 10.1088/1751-8113/44/38/385203 [10] P. D. Prieto-Martínez, N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems, J. Math. Phys., 53 (2012), 032901. https://doi.org/10.1063/1.3692326 doi: 10.1063/1.3692326 [11] M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda, N. Román-Roy, Skinner–Rusk unified formalism for optimal control systems and applications, J. Phys. A: Math. Theor., 40 (2007), 12071–12093. https://doi.org/10.1088/1751-8113/40/40/005 doi: 10.1088/1751-8113/40/40/005 [12] L. Colombo, D. Martín de Diego, M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach, J. Math. Phys., 51 (2010), 083519. https://doi.org/10.1063/1.3456158 doi: 10.1063/1.3456158 [13] C. M. Campos, M. de León, D. Martín de Diego, J. Vankerschaver, Unambiguous formalism for higher order Lagrangian field theories, J. Phys. A: Math. Theor., 42 (2009), 475207. https://doi.org/10.1088/1751-8113/42/47/475207 doi: 10.1088/1751-8113/42/47/475207 [14] M. de León, J. C. Marrero, D. Martín de Diego, A new geometrical setting for classical field theories, in Classical and Quantum Integrability, Inst. of Math., Polish Acad. Sci., Warsawa: Banach Center Pub., 59 (2003), 189–209. https://doi.org/10.4064/bc59-0-10 [15] A. Echeverría-Enríquez, C. López, J. Marín-Solano, M. C. Muñoz-Lecanda, N. Román-Roy, Lagrangian–Hamiltonian unified formalism for field theory, J. Math. Phys., 45 (2004), 360–385. https://doi.org/10.1063/1.1628384 doi: 10.1063/1.1628384 [16] A. M. Rey, N. Román-Roy, M. Salgado, Günther formalism ($k$-symplectic formalism) in classical field theory: Skinner–Rusk approach and the evolution operator, J. Math. Phys., 46 (2005), 052901. https://doi.org/10.1063/1.1876872 doi: 10.1063/1.1876872 [17] A. M. Rey, N. Román-Roy, M. Salgado, S. Vilariño, $k$-cosymplectic classical field theories: Tulczyjew and Skinner–Rusk formulations, Math. Phys. Anal. Geom., 15 (2012), 85–119. https://doi.org/10.1007/s11040-012-9104-z doi: 10.1007/s11040-012-9104-z [18] L. Vitagliano, The Lagrangian–Hamiltonian formalism for higher order field theories," J. Geom. Phys., 60 (2010), 857–873. https://doi.org/10.1016/j.geomphys.2010.02.003 [19] M. de León, J. Gaset, M. Lainz-Valcázar, X. Rivas, N. Román-Roy, Unified Lagrangian-Hamiltonian formalism for contact systems, Fortschr. Phys., 68 (2020), 2000045. https://doi.org/10.1002/prop.202000045 doi: 10.1002/prop.202000045 [20] X. Gràcia, X. Rivas, N. Román-Roy, Skinner–Rusk formalism for $k$-contact systems, J. Geom. Phys., 172 (2022), 104429. https://doi.org/10.1016/j.geomphys.2021.104429 doi: 10.1016/j.geomphys.2021.104429 [21] A. Banyaga, D. F. Houenou, A brief introduction to symplectic and contact manifolds, vol. 15. Singapore: World Scientific Publishing Co. Pte. Ltd., 2016. https://doi.org/10.1142/9667 [22] H. Geiges, An Introduction to Contact Topology, vol. 109 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2008. https://doi.org/10.1017/CBO9780511611438 [23] A. L. Kholodenko, Applications of Contact Geometry and Topology in Physics. World Scientific, 2013. https://doi.org/10.1142/8514 [24] R. Abraham, J. E. Marsden, Foundations of mechanics, vol. 364 of AMS Chelsea publishing. New York: Benjamin/Cummings Pub. Co., 2nd ed., 1978. https://doi.org/10.1090/chel/364 [25] M. de León, C. Sardón, Cosymplectic and contact structures to resolve time-dependent and dissipative Hamiltonian systems, J. Phys. A: Math. Theor., 50 (2017), 255205. https://doi.org/10.1088/1751-8121/aa711d doi: 10.1088/1751-8121/aa711d [26] P. Libermann, C. M. Marle, Symplectic Geometry and Analytical Mechanics. Reidel, Dordretch: Springer Netherlands, oct 1987. http://doi.org/10.1007/978-94-009-3807-6 [27] B. Cappelletti-Montano, A. De Nicola, I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys., 25 (2013), 1343002. https://doi.org/10.1142/S0129055X13430022 doi: 10.1142/S0129055X13430022 [28] A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 10 (2017), 535. https://doi.org/10.3390/e19100535 doi: 10.3390/e19100535 [29] M. de León, M. Lainz-Valcázar, Contact Hamiltonian systems, J. Math. Phys., 60 (2019), 102902. https://doi.org/10.1063/1.5096475 doi: 10.1063/1.5096475 [30] J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas, N. Román-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050090. https://doi.org/10.1142/S0219887820500905 doi: 10.1142/S0219887820500905 [31] Q. Liu, P. J. Torres, C. Wang, Contact Hamiltonian dynamics: variational principles, invariants, completeness and periodic behaviour, Ann. Phys., 395 (2018), 26–44. https://doi.org/10.1016/j.aop.2018.04.035 doi: 10.1016/j.aop.2018.04.035 [32] M. Visinescu, Contact Hamiltonian systems and complete integrability, in AIP Conference Proceedings, 1916 (2017), 020002. https://doi.org/10.1063/1.5017422 [33] F. M. Ciaglia, H. Cruz, G. Marmo, Contact manifolds and dissipation, classical and quantum, Ann. Phys., 398 (2018), 159–179. https://doi.org/10.1016/j.aop.2018.09.012 doi: 10.1016/j.aop.2018.09.012 [34] S. Goto, Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics, J. Math. Phys., 57 (2016), 102702. https://doi.org/10.1063/1.4964751 doi: 10.1063/1.4964751 [35] H. Ramirez, B. Maschke, D. Sbarbaro, Partial stabilization of input-output contact systems on a Legendre submanifold, IEEE Trans. Autom. Control, 62 (2017), 1431–1437. https://doi.org/10.1109/TAC.2016.2572403 doi: 10.1109/TAC.2016.2572403 [36] A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Methods Mod. Phys., 16 (2018), 1940003. https://doi.org/10.1142/S0219887819400036 doi: 10.1142/S0219887819400036 [37] A. A. Simoes, M. de León, M. Lainz-Valcázar, D. Martín de Diego, Contact geometry for simple thermodynamical systems with friction, Proc. R. Soc. A., 476 (2020), 20200244. https://doi.org/10.1098/rspa.2020.0244 doi: 10.1098/rspa.2020.0244 [38] A. Bravetti, M. de León, J. C. Marrero, E. Padrón, Invariant measures for contact Hamiltonian systems: symplectic sandwiches with contact bread, J. Phys. A: Math. Theor., 53 (2020), 455205. https://doi.org/10.1088/1751-8121/abbaaa doi: 10.1088/1751-8121/abbaaa [39] M. de León, V. M. Jiménez, M. Lainz-Valcázar, Contact Hamiltonian and Lagrangian systems with nonholonomic constraints, J. Geom. Mech., 13 (2021), 25–53. https://doi.org/10.3934/jgm.2021001 doi: 10.3934/jgm.2021001 [40] M. de León, M. Lainz-Valcázar, M. C. Muñoz-Lecanda, The Herglotz Principle and Vakonomic Dynamics, in Geometric Science of Information (F. Nielsen and F. Barbaresco, eds.), Lecture Notes in Computer Science, (Cham), Springer International Publishing, 12829 (2021), 183–190. https://doi.org/10.1007/978-3-030-80209-7_21 [41] M. de León, M. Lainz-Valcázar, M. C. Muñoz-Lecanda, N. Román-Roy, Constrained Lagrangian dissipative contact dynamics, J. Math. Phys., 62 (2021), 122902. https://doi.org/10.1063/5.0071236 doi: 10.1063/5.0071236 [42] O. Esen, M. Lainz-Valcázar, M. de León, J. C. Marrero, Contact Dynamics versus Legendrian and Lagrangian Submanifolds, Mathematics, 9 (2021), 2704. https://doi.org/10.3390/math9212704 doi: 10.3390/math9212704 [43] H. J. Sussmann, Geometry and optimal control. Mathematical control theory, New York, NY: Springer, 1999. https://doi.org/10.1007/978-1-4612-1416-8_5 [44] J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas, N. Román-Roy, A contact geometry framework for field theories with dissipation, Ann. Phys., 414 (2020), 168092. https://doi.org/10.1016/j.aop.2020.168092 doi: 10.1016/j.aop.2020.168092 [45] J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas, N. Román-Roy, A $k$-contact Lagrangian formulation for nonconservative field theories, Rep. Math. Phys., 87 (2021), 347–368. https://doi.org/10.1016/S0034-4877(21)00041-0 doi: 10.1016/S0034-4877(21)00041-0 [46] M. de León, J. Gaset, M. C. Muñoz-Lecanda, X. Rivas, Time-dependent contact mechanics, Monatsh. Math., 2022. https://doi.org/10.1007/s00605-022-01767-1 [47] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, Springer, New York, NY, 42 (1938). https://doi.org/10.1007/978-1-4612-1140-2 [48] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics. Springer, New York, NY, 2 (2003). https://doi.org/10.1007/b97481 [49] M. C. Muñoz-Lecanda, N. Román-Roy, Lagrangian theory for presymplectic systems, Ann. Inst. H. Poincaré, 57 (1992), 27–45.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

0.737 1.4

Article outline

Figures(2)

• On This Site