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

Relativistic pendulum and invariant curves

Stefano Marò 1,

1. 

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

Received  January 2014 Revised  August 2014 Published  October 2014

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, 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), 1099-1111. doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[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), 463-471. doi: 10.1007/s10884-010-9172-3.  Google Scholar

[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), 2713-2719. doi: 10.1090/S0002-9939-2011-11101-8.  Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810.  Google Scholar

[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), 530-542. doi: 10.1016/j.jde.2008.11.013.  Google Scholar

[6]

A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane, Adv. Nonlinear Stud., 12 (2012), 395-408.  Google Scholar

[7]

M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 1, vol. 103 of Astérisque, Société Mathématique de France, Paris, 1983. Google Scholar

[8]

M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 2,, Astérisque, ().   Google Scholar

[9]

M. Kunze and R. Ortega, Twist mappings with non-periodic angles, in Stability and bifurcation theory for non-autonomous differential equations, vol. 2065 of Lecture Notes in Math., Springer, Berlin, (2013), 265-300. doi: 10.1007/978-3-642-32906-7_5.  Google Scholar

[10]

M. Levi, KAM theory for particles in periodic potentials, Ergodic Theory Dynam. Systems, 10 (1990), 777-785. doi: 10.1017/S0143385700005897.  Google Scholar

[11]

S. Marò, Coexistence of bounded and unbounded motions in a bouncing ball model, Nonlinearity, 26 (2013), 1439-1448. doi: 10.1088/0951-7715/26/5/1439.  Google Scholar

[12]

S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75.  Google Scholar

[13]

J. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[14]

J. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263. doi: 10.1090/S0894-0347-1991-1080112-5.  Google Scholar

[15]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.  Google Scholar

[16]

J. Moser, Selected Chapters in the Calculus of Variations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003, Lecture notes by Oliver Knill. doi: 10.1007/978-3-0348-8057-2.  Google Scholar

[17]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. (2), 53 (1996), 325-342. doi: 10.1112/jlms/53.2.325.  Google Scholar

[18]

R. Ortega, Invariant curves of mappings with averaged small twist, Adv. Nonlinear Stud., 1 (2001), 14-39.  Google Scholar

[19]

R. Ortega, Twist mappings, invariant curves and periodic differential equations, in Nonlinear analysis and its applications to differential equations (Lisbon, 1998), vol. 43 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, (2001), 85-112.  Google Scholar

[20]

P. Torres, Periodic oscillations of the relativistic pendulum with friction, Phys. Lett. A, 372 (2008), 6386-6387. doi: 10.1016/j.physleta.2008.08.060.  Google Scholar

[21]

J. You, Invariant tori and Lagrange stability of pendulum-type equations, J. Differential Equations, 85 (1990), 54-65. doi: 10.1016/0022-0396(90)90088-7.  Google Scholar

show all references

References:
[1]

J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111. doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[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), 463-471. doi: 10.1007/s10884-010-9172-3.  Google Scholar

[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), 2713-2719. doi: 10.1090/S0002-9939-2011-11101-8.  Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810.  Google Scholar

[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), 530-542. doi: 10.1016/j.jde.2008.11.013.  Google Scholar

[6]

A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane, Adv. Nonlinear Stud., 12 (2012), 395-408.  Google Scholar

[7]

M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 1, vol. 103 of Astérisque, Société Mathématique de France, Paris, 1983. Google Scholar

[8]

M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 2,, Astérisque, ().   Google Scholar

[9]

M. Kunze and R. Ortega, Twist mappings with non-periodic angles, in Stability and bifurcation theory for non-autonomous differential equations, vol. 2065 of Lecture Notes in Math., Springer, Berlin, (2013), 265-300. doi: 10.1007/978-3-642-32906-7_5.  Google Scholar

[10]

M. Levi, KAM theory for particles in periodic potentials, Ergodic Theory Dynam. Systems, 10 (1990), 777-785. doi: 10.1017/S0143385700005897.  Google Scholar

[11]

S. Marò, Coexistence of bounded and unbounded motions in a bouncing ball model, Nonlinearity, 26 (2013), 1439-1448. doi: 10.1088/0951-7715/26/5/1439.  Google Scholar

[12]

S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75.  Google Scholar

[13]

J. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[14]

J. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263. doi: 10.1090/S0894-0347-1991-1080112-5.  Google Scholar

[15]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.  Google Scholar

[16]

J. Moser, Selected Chapters in the Calculus of Variations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003, Lecture notes by Oliver Knill. doi: 10.1007/978-3-0348-8057-2.  Google Scholar

[17]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. (2), 53 (1996), 325-342. doi: 10.1112/jlms/53.2.325.  Google Scholar

[18]

R. Ortega, Invariant curves of mappings with averaged small twist, Adv. Nonlinear Stud., 1 (2001), 14-39.  Google Scholar

[19]

R. Ortega, Twist mappings, invariant curves and periodic differential equations, in Nonlinear analysis and its applications to differential equations (Lisbon, 1998), vol. 43 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, (2001), 85-112.  Google Scholar

[20]

P. Torres, Periodic oscillations of the relativistic pendulum with friction, Phys. Lett. A, 372 (2008), 6386-6387. doi: 10.1016/j.physleta.2008.08.060.  Google Scholar

[21]

J. You, Invariant tori and Lagrange stability of pendulum-type equations, J. Differential Equations, 85 (1990), 54-65. doi: 10.1016/0022-0396(90)90088-7.  Google Scholar

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