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March  2014, 34(3): 1121-1130. doi: 10.3934/dcds.2014.34.1121

Integrability of nonholonomically coupled oscillators

Klas Modin 1, and  Olivier Verdier 2,

1. 

Department of Mathematical Sciences, Chalmers University of Technology, Sweden
2. 

Department of Mathematics, University of Bergen, Norway

Received  December 2012 Revised  February 2013 Published  August 2013

We study a family of nonholonomic mechanical systems. These systems consist of harmonic oscillators coupled through nonholonomic constraints. The family includes the contact oscillator, which has been used as a test problem for numerical methods for nonholonomic mechanics. The systems under study constitute simple models for continuously variable transmission gearboxes.
    The main result is that each system in the family is integrable reversible with respect to the canonical reversibility map on the cotangent bundle. By using reversible Kolmogorov--Arnold--Moser theory, we then establish preservation of invariant tori for reversible perturbations. This result explains previous numerical observations, that some discretisations of the contact oscillator have favourable structure preserving properties.
Keywords: geometric integration., Nonholonomic mechanics, KAM theory, Continuously variable transmission, reversible integrability, Lagrange D'Alembert.
Mathematics Subject Classification: Primary: 70H08, 70F25, 37J60; Secondary: 65P99, 37M9.
Citation: Klas Modin, Olivier Verdier. Integrability of nonholonomically coupled oscillators. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1121-1130. doi: 10.3934/dcds.2014.34.1121
References:
[1]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer-Verlag, New York, second edition, 1989.

[2]

V. I. Arnold, V. Kozlov, A. I. Neishtadt and E. Khukhro, "Mathematical Aspects of Classical and Celestial Mechanics," Springer-Verlag, Berlin, third edition, 2006. doi: 10.1007/978-3-540-48926-9.

[3]

V. I. Arnold, "Ordinary Differential Equations," Springer-Verlag, Berlin, 2006.

[4]

A. M. Bloch, "Nonholonomic Mechanics and Control," Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[5]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in non-holonomic systems, Dynamical Systems, 24 (2009), 187-222. doi: 10.1080/14689360802609344.

[6]

J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems," Springer-Verlag, Berlin, 2002. doi: 10.1007/b84020.

[7]

M. de León, J. C. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics, J. Geom. Mech., 2 (2010), 159-198. doi: 10.3934/jgm.2010.2.159.

[8]

S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928. doi: 10.1088/0951-7715/21/8/009.

[9]

S. J. Ferraro, D. Iglesias-Ponte and D. Martín de Diego, Numerical and geometric aspects of the nonholonomic Shake and Rattle methods, Discrete Contin. Dyn. Syst., (Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl.), (2009), 220-229.

[10]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer-Verlag, Berlin, second edition, 2006. doi: 10.1007/3-540-30666-8.

[11]

D. Iglesias-Ponte, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, J. Phys. A, 41 (2008), 015205, 14. doi: 10.1088/1751-8113/41/1/015205.

[12]

M. Kobilarov, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics, Bol. Soc. Esp. Mat. Apl. S$\vec{\e}$MA, 50 (2010), 61-81.

[13]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61-84. doi: 10.3934/dcdss.2010.3.61.

[14]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regul. Chaotic Dyn., 7 (2002), 161-176. doi: 10.1070/RD2002v007n02ABEH000203.

[15]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, New York, second edition, 1999. doi: 10.1007/978-0-387-21792-5.

[16]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1.

[17]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geom. Phys., 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.

[18]

M. B. Sevryuk, KAM-stable Hamiltonians, J. Dynam. Control Systems, 1 (1995), 351-366. doi: 10.1007/BF02269374.

[19]

M. B. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.

[20]

Z. Shang, KAM theorem of symplectic algorithms for Hamiltonian systems, Numer. Math., 83 (1999), 477-496. doi: 10.1007/s002110050460.

[21]

Z. Shang, A note on the KAM theorem for symplectic mappings, J. Dynam. Differential Equations, 12 (2000), 357-383. doi: 10.1023/A:1009068425415.

show all references

References:
[1]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer-Verlag, New York, second edition, 1989.

[2]

V. I. Arnold, V. Kozlov, A. I. Neishtadt and E. Khukhro, "Mathematical Aspects of Classical and Celestial Mechanics," Springer-Verlag, Berlin, third edition, 2006. doi: 10.1007/978-3-540-48926-9.

[3]

V. I. Arnold, "Ordinary Differential Equations," Springer-Verlag, Berlin, 2006.

[4]

A. M. Bloch, "Nonholonomic Mechanics and Control," Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[5]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in non-holonomic systems, Dynamical Systems, 24 (2009), 187-222. doi: 10.1080/14689360802609344.

[6]

J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems," Springer-Verlag, Berlin, 2002. doi: 10.1007/b84020.

[7]

M. de León, J. C. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics, J. Geom. Mech., 2 (2010), 159-198. doi: 10.3934/jgm.2010.2.159.

[8]

S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928. doi: 10.1088/0951-7715/21/8/009.

[9]

S. J. Ferraro, D. Iglesias-Ponte and D. Martín de Diego, Numerical and geometric aspects of the nonholonomic Shake and Rattle methods, Discrete Contin. Dyn. Syst., (Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl.), (2009), 220-229.

[10]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer-Verlag, Berlin, second edition, 2006. doi: 10.1007/3-540-30666-8.

[11]

D. Iglesias-Ponte, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, J. Phys. A, 41 (2008), 015205, 14. doi: 10.1088/1751-8113/41/1/015205.

[12]

M. Kobilarov, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics, Bol. Soc. Esp. Mat. Apl. S$\vec{\e}$MA, 50 (2010), 61-81.

[13]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61-84. doi: 10.3934/dcdss.2010.3.61.

[14]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regul. Chaotic Dyn., 7 (2002), 161-176. doi: 10.1070/RD2002v007n02ABEH000203.

[15]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, New York, second edition, 1999. doi: 10.1007/978-0-387-21792-5.

[16]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1.

[17]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geom. Phys., 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.

[18]

M. B. Sevryuk, KAM-stable Hamiltonians, J. Dynam. Control Systems, 1 (1995), 351-366. doi: 10.1007/BF02269374.

[19]

M. B. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.

[20]

Z. Shang, KAM theorem of symplectic algorithms for Hamiltonian systems, Numer. Math., 83 (1999), 477-496. doi: 10.1007/s002110050460.

[21]

Z. Shang, A note on the KAM theorem for symplectic mappings, J. Dynam. Differential Equations, 12 (2000), 357-383. doi: 10.1023/A:1009068425415.

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