Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 018, 14 pages      arXiv:1510.08314
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries

Paula Balseiro a and Nicola Sansonetto b
a) Universidade Federal Fluminense, Instituto de Matemática, Rua Mario Santos Braga S/N, 24020-140, Niteroi, Rio de Janeiro, Brazil
b) Università degli Studi di Padova, Dipartimento di Matematica, via Trieste 64, 35121 Padova, Italy

Received October 29, 2015, in final form February 12, 2016; Published online February 21, 2016

We study the existence of first integrals in nonholonomic systems with symmetry. First we define the concept of $\mathcal{M}$-cotangent lift of a vector field on a manifold $Q$ in order to unify the works [Balseiro P., Arch. Ration. Mech. Anal. 214 (2014), 453-501, arXiv:1301.1091], [Fassò F., Ramos A., Sansonetto N., Regul. Chaotic Dyn. 12 (2007), 579-588], and [Fassò F., Giacobbe A., Sansonetto N., Rep. Math. Phys. 62 (2008), 345-367]. Second, we study gauge symmetries and gauge momenta, in the cases in which there are the symmetries that satisfy the so-called vertical symmetry condition. Under such condition we can predict the number of linearly independent first integrals (that are gauge momenta). We illustrate the theory with two examples.

Key words: nonholonomic systems; Lie group symmetries; first integrals; gauge symmetries and gauge momenta.

pdf (463 kb)   tex (25 kb)


  1. Balseiro P., The Jacobiator of nonholonomic systems and the geometry of reduced nonholonomic brackets, Arch. Ration. Mech. Anal. 214 (2014), 453-501, arXiv:1301.1091.
  2. Balseiro P., Fernandez O.E., Reduction of nonholonomic systems in two stages and Hamiltonization, Nonlinearity 28 (2015), 2873-2912, arXiv:1409.0456.
  3. Balseiro P., García-Naranjo L.C., Gauge transformations, twisted Poisson brackets and Hamiltonization of nonholonomic systems, Arch. Ration. Mech. Anal. 205 (2012), 267-310, arXiv:1104.0880.
  4. Balseiro P., Sansonetto N., Conserved quantities and Hamiltonization, work in progress.
  5. Bates L., Graumann H., MacDonnell C., Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys. 37 (1996), 295-308.
  6. Bates L., Śniatycki J., Nonholonomic reduction, Rep. Math. Phys. 32 (1993), 99-115.
  7. Benenti S., Meccanica dei sistemi anolonomi, in Complementi alle Lezioni di Meccanica Razionale di T. Levi-Civita e U. Amaldi, Editors E.N.M. Cirillo, G. Maschio, T. Ruggeri, G. Saccomandi, CompoMat, 2012, 213-257.
  8. Bloch A.M., Nonholonomic mechanics and control, Interdisciplinary Applied Mathematics, Vol. 24, Springer-Verlag, New York, 2003.
  9. Bloch A.M., Krishnaprasad P.S., Marsden J.E., Murray R.M., Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal. 136 (1996), 21-99.
  10. Borisov A.V., Mamaev I.S., Chaplygin's ball rolling problem is Hamiltonian, Math. Notes 70 (2001), 720-723.
  11. Borisov A.V., Mamaev I.S., Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Regul. Chaotic Dyn. 13 (2008), 443-490.
  12. Borisov A.V., Mamaev I.S., Kilin A.A., Rolling of a ball on a surface. New integrals and hierarchy of dynamics, Regul. Chaotic Dyn. 7 (2002), 201-219, nlin.SI/0303024.
  13. Cantrijn F., de León M., Marrero J.C., de Diego D.M., Reduction of nonholonomic mechanical systems with symmetries, Rep. Math. Phys. 42 (1998), 25-45.
  14. Cortés Monforte J., Geometric, control and numerical aspects of nonholonomic systems, Lecture Notes in Math., Vol. 1793, Springer-Verlag, Berlin, 2002.
  15. Crampin M., Mestdag T., The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem, Int. J. Geom. Methods Mod. Phys. 8 (2011), 897-923, arXiv:1101.3153.
  16. Crampin M., Pirani F.A.E., Applicable differential geometry, London Mathematical Society Lecture Note Series, Vol. 59, Cambridge University Press, Cambridge, 1986.
  17. Cushman R., Duistermaat H., Śniatycki J., Geometry of nonholonomically constrained systems, Advanced Series in Nonlinear Dynamics, Vol. 26, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.
  18. Fassò F., Giacobbe A., Sansonetto N., Gauge conservation laws and the momentum equation in nonholonomic mechanics, Rep. Math. Phys. 62 (2008), 345-367.
  19. Fassò F., Giacobbe A., Sansonetto N., On the number of weakly Noetherian constants of motion of nonholonomic systems, J. Geom. Mech. 1 (2009), 389-416.
  20. Fassò F., Giacobbe A., Sansonetto N., Linear weakly Noetherian constants of motion are horizontal gauge momenta, J. Geom. Mech. 4 (2012), 129-136.
  21. Fassò F., Ramos A., Sansonetto N., The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions, Regul. Chaotic Dyn. 12 (2007), 579-588.
  22. Fassò F., Sansonetto N., Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces, J. Nonlinear Sci., to appear, arXiv:1503.06661.
  23. García-Naranjo L., Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), 37-60, arXiv:0808.0854.
  24. Giachetta G., First integrals of non-holonomic systems and their generators, J. Phys. A: Math. Gen. 33 (2000), 5369-5389.
  25. Ibort A., de Leon M., Marrero J.C., Martin de Diego D., Dirac brackets in constrained dynamics, Fortschr. Phys. 47 (1999), 459-492.
  26. Jotz M., Ratiu T.S., Dirac structures, nonholonomic systems and reduction, Rep. Math. Phys. 69 (2012), 5-56, arXiv:0806.1261.
  27. Marle C.-M., Reduction of constrained mechanical systems and stability of relative equilibria, Comm. Math. Phys. 174 (1995), 295-318.
  28. Marle C.-M., Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys. 42 (1998), 211-229.
  29. Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts in Applied Mathematics, Vol. 17, 2nd ed., Springer-Verlag, New York, 1999.
  30. Neimark Ju.I., Fufaev N.A., Dynamics of nonholonomic systems, Translations of Mathematical Monographs, Vol. 33, Amer. Math. Soc., Providence, RI, 1972.
  31. Ortega J.-P., Ratiu T.S., Momentum maps and Hamiltonian reduction, Progress in Mathematics, Vol. 222, Birkhäuser Boston, Inc., Boston, MA, 2004.
  32. Pars L.A., A treatise on analytical dynamics, Heinemann Educational Books Ltd., London, 1968.
  33. Śniatycki J., Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys. 42 (1998), 5-23.
  34. van der Schaft A.J., Maschke B.M., On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys. 34 (1994), 225-233.
  35. Zenkov D.V., Linear conservation laws of nonholonomic systems with symmetry, Discrete Contin. Dyn. Syst. (2003), suppl., 967-976.

Previous article  Next article   Contents of Volume 12 (2016)