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

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.
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. 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. 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. 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. 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. 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. 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, (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, (2013), 265. 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. doi: 10.1017/S0143385700005897. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. 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. Google Scholar

[16]

J. Moser, Selected Chapters in the Calculus of Variations,, Lectures in Mathematics ETH Zürich, (2003). 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. 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. Google Scholar

[19]

R. Ortega, Twist mappings, invariant curves and periodic differential equations,, in Nonlinear analysis and its applications to differential equations (Lisbon, (2001), 85. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. 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. 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. 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. 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. 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. 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, (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, (2013), 265. 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. doi: 10.1017/S0143385700005897. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. 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. Google Scholar

[16]

J. Moser, Selected Chapters in the Calculus of Variations,, Lectures in Mathematics ETH Zürich, (2003). 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. 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. Google Scholar

[19]

R. Ortega, Twist mappings, invariant curves and periodic differential equations,, in Nonlinear analysis and its applications to differential equations (Lisbon, (2001), 85. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379

[2]

Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135

[3]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807

[4]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683

[5]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413

[6]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155

[7]

Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823

[8]

Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103

[9]

Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983

[10]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018

[11]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[12]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[13]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[14]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693

[15]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545

[16]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839

[17]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345

[18]

Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017

[19]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757

[20]

Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]