Relativistic pendulum and invariant curves

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March 2015, 35(3): 1139-1162. doi: 10.3934/dcds.2015.35.1139

Relativistic pendulum and invariant curves

1.	Dipartimento di Matematica - Università di Torino, Via Carlo Alberto, 10, 10123 Torino, Italy

Received January 2014 Revised August 2014 Published October 2014

Abstract
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We apply KAM theory to the equation of the forced relativistic pendulum to prove that all the solutions have bounded momentum. Subsequently, we detect the existence of quasiperiodic solutions in a generalized sense. This is achieved using a modified version of the Aubry-Mather theory for compositions of twist maps.

Keywords: Poincaré map, Aubry-Mather theory., invarinat curves, Relativisitc pendulum, KAM theory.

Mathematics Subject Classification: Primary: 70H08, 70K43; Secondary: 34C15, 37C5.

Citation: Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139

References:

[1]	J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099. doi: 10.1088/0951-7715/9/5/003.
[2]	C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians,, J. Dynam. Differential Equations, 22 (2010), 463. doi: 10.1007/s10884-010-9172-3.
[3]	C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum,, Proc. Amer. Math. Soc., 140 (2012), 2713. doi: 10.1090/S0002-9939-2011-11101-8.
[4]	H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.
[5]	J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator,, J. Differential Equations, 247 (2009), 530. doi: 10.1016/j.jde.2008.11.013.
[6]	A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane,, Adv. Nonlinear Stud., 12 (2012), 395.
[7]	M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 1, vol. 103 of Astérisque,, Société Mathématique de France, (1983).
[8]	M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 2,, Astérisque, ().
[9]	M. Kunze and R. Ortega, Twist mappings with non-periodic angles,, in Stability and bifurcation theory for non-autonomous differential equations, (2013), 265. doi: 10.1007/978-3-642-32906-7_5.
[10]	M. Levi, KAM theory for particles in periodic potentials,, Ergodic Theory Dynam. Systems, 10 (1990), 777. doi: 10.1017/S0143385700005897.
[11]	S. Marò, Coexistence of bounded and unbounded motions in a bouncing ball model,, Nonlinearity, 26 (2013), 1439. doi: 10.1088/0951-7715/26/5/1439.
[12]	S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics,, Topol. Methods Nonlinear Anal., 42 (2013), 51.
[13]	J. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus,, Topology, 21 (1982), 457. doi: 10.1016/0040-9383(82)90023-4.
[14]	J. Mather, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207. doi: 10.1090/S0894-0347-1991-1080112-5.
[15]	J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1.
[16]	J. Moser, Selected Chapters in the Calculus of Variations,, Lectures in Mathematics ETH Zürich, (2003). doi: 10.1007/978-3-0348-8057-2.
[17]	R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc. (2), 53 (1996), 325. doi: 10.1112/jlms/53.2.325.
[18]	R. Ortega, Invariant curves of mappings with averaged small twist,, Adv. Nonlinear Stud., 1 (2001), 14.
[19]	R. Ortega, Twist mappings, invariant curves and periodic differential equations,, in Nonlinear analysis and its applications to differential equations (Lisbon, (2001), 85.
[20]	P. Torres, Periodic oscillations of the relativistic pendulum with friction,, Phys. Lett. A, 372 (2008), 6386. doi: 10.1016/j.physleta.2008.08.060.
[21]	J. You, Invariant tori and Lagrange stability of pendulum-type equations,, J. Differential Equations, 85 (1990), 54. doi: 10.1016/0022-0396(90)90088-7.

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References:

[1]	J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099. doi: 10.1088/0951-7715/9/5/003.
[2]	C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians,, J. Dynam. Differential Equations, 22 (2010), 463. doi: 10.1007/s10884-010-9172-3.
[3]	C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum,, Proc. Amer. Math. Soc., 140 (2012), 2713. doi: 10.1090/S0002-9939-2011-11101-8.
[4]	H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.
[5]	J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator,, J. Differential Equations, 247 (2009), 530. doi: 10.1016/j.jde.2008.11.013.
[6]	A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane,, Adv. Nonlinear Stud., 12 (2012), 395.
[7]	M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 1, vol. 103 of Astérisque,, Société Mathématique de France, (1983).
[8]	M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 2,, Astérisque, ().
[9]	M. Kunze and R. Ortega, Twist mappings with non-periodic angles,, in Stability and bifurcation theory for non-autonomous differential equations, (2013), 265. doi: 10.1007/978-3-642-32906-7_5.
[10]	M. Levi, KAM theory for particles in periodic potentials,, Ergodic Theory Dynam. Systems, 10 (1990), 777. doi: 10.1017/S0143385700005897.
[11]	S. Marò, Coexistence of bounded and unbounded motions in a bouncing ball model,, Nonlinearity, 26 (2013), 1439. doi: 10.1088/0951-7715/26/5/1439.
[12]	S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics,, Topol. Methods Nonlinear Anal., 42 (2013), 51.
[13]	J. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus,, Topology, 21 (1982), 457. doi: 10.1016/0040-9383(82)90023-4.
[14]	J. Mather, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207. doi: 10.1090/S0894-0347-1991-1080112-5.
[15]	J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1.
[16]	J. Moser, Selected Chapters in the Calculus of Variations,, Lectures in Mathematics ETH Zürich, (2003). doi: 10.1007/978-3-0348-8057-2.
[17]	R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc. (2), 53 (1996), 325. doi: 10.1112/jlms/53.2.325.
[18]	R. Ortega, Invariant curves of mappings with averaged small twist,, Adv. Nonlinear Stud., 1 (2001), 14.
[19]	R. Ortega, Twist mappings, invariant curves and periodic differential equations,, in Nonlinear analysis and its applications to differential equations (Lisbon, (2001), 85.
[20]	P. Torres, Periodic oscillations of the relativistic pendulum with friction,, Phys. Lett. A, 372 (2008), 6386. doi: 10.1016/j.physleta.2008.08.060.
[21]	J. You, Invariant tori and Lagrange stability of pendulum-type equations,, J. Differential Equations, 85 (1990), 54. doi: 10.1016/0022-0396(90)90088-7.

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