Journal of Nonlinear Science

, Volume 25, Issue 1, pp 203–246 | Cite as

Unimodularity and Preservation of Volumes in Nonholonomic Mechanics

  • Yuri N. Fedorov
  • Luis C.  García-Naranjo
  • Juan C. MarreroEmail author


The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, which unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.


Nonholonomic mechanical systems Linear almost Poisson structure Modular vector field Unimodularity Invariant volume forms Symmetries Reduction 

Mathematics Subject Classification

37C40 37J60 70F25 70G45 70G65 



This work has been partially supported by MEC (Spain) Grants MTM2009-13383, MTM2011-15725-E, MTM2012-34478, MTM2012-31714 and the project of the Canary Government ProdID20100210. All the authors are grateful to their institutions for funding our research visits, which allowed the completion of the present article.


  1. Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin/Cummings, Reading (1978)zbMATHGoogle Scholar
  2. Arnold, V.I., Kozlov, V.V., Nishtadt, A.I.: Mathematical aspects of classical and celestial mechanics. Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fundamentalnye Napravleniya, vol. 3. VINITI, Moscow (1985). English transl.: Encyclopadia of Math. Sciences, vol. 3. Springer, Berlin (1989)Google Scholar
  3. Bloch, A.M., Krishnapasad, P.S., Marsden, J.E., Murray, R.M.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136, 21–99 (1996)CrossRefzbMATHGoogle Scholar
  4. Borisov, A.V., Mamaev, I.S.: The rolling motion of a rigid body on a plane and a sphere Hierarchy of dynamics. Regul. Chaotic Dyn. 7(2), 177–200 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 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(2), 201–218 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. Brylinski, J.L., Zuckerman, G.: The outer derivation of a complex Poisson manifold. J. Reine Angew. Math. 506, 181–189 (1999)zbMATHMathSciNetGoogle Scholar
  7. Cantrijn, F., de León, M., de Martín Diego, D.: On almost-Poisson structures in nonholonomic mechanics. Nonlinearity 12, 721–737 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. Cantrijn, F., Cortés, J.: de León M and Martín de Diego D On the geometry of generalized Chaplygin systems Math. Proc. Cambridge Philos. Soc. 132(2), 323–351 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. Chaplygin, S.A.: On a ball’s rolling on a horizontal plane. Regul. Chaot. Dyn. 7, 131–148 (2002); original paper in Math Sb., 24, 139–168 (1903)Google Scholar
  10. Chaplygin, S.A. On a motion of a heavy body of revolution on a horizontal plane. Collected works, vol. I. Theoretical mechanics. Mathematics (Russian), 51–57, Gos. Izd. Tekhn.-Teoret. Lit., Moscow, 1948 [see MR0052352 (14,609i)]. English translation. In: Regul. Chaotic Dyn. 7(2), 119–130 (2002)Google Scholar
  11. de León, M., Marrero, J.C., de Martín Diego, D.: Linear almost Poisson structures and Hamilton-Jacobi theory. J. Geom. Mech. 2(2), 159–198 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. Dufour, J.P., Haraki, A.: Rotationnels et structures de Poisson quadratiques. C. R. Acad. Sci. Paris Sér. I Math. 312, 137–140 (1991)zbMATHMathSciNetGoogle Scholar
  13. Evens, S., Lu, J.H., Weinstein, A.: Transverse measures, the modular class and a cohomology pairing for Lie algebroids. Q. J. Math. Oxford 50, 417–436 (1999)Google Scholar
  14. Fedorov, Y.N., García-Naranjo, L., Marrero, J.C.: Invariant measures in nonholonomic mechanics. Applied Dynamics and Geometric Mechanics (organized by A.M. Bloch, T. Ratiu, J. Scheurle). Oberwolfach Rep. 8(3), 2256–2260 (2011)Google Scholar
  15. Fedorov, Y.N., Jovanović, B.: Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces. J. Nonlin. Sci. 14(4), 341–381 (2004)CrossRefzbMATHGoogle Scholar
  16. García-Naranjo, L., Marrero, J.C.: Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane. Regul. Chaot. Dyn. 18, 372–379 (2013)CrossRefzbMATHGoogle Scholar
  17. García-Naranjo, L.C., Maciejewski, A.J., Marrero, J.C., Przybylska, M.: The inhomogeneous Suslov problem. Phys. Lett. A 378, 2389–2394 (2014)CrossRefGoogle Scholar
  18. Grabowski, J., de León, M., Marrero, J.C., Martín de Diego, D.: Nonholonomic constraints: a new viewpoint J. Math. Phys. 50(1), 013520 (2009) (17 pp)Google Scholar
  19. Grabowski, J., Marmo, G., Perelomov, A.M.: Poisson structures: towards a classification. Modern Phys. Lett. A 8, 1719–1733 (1993)Google Scholar
  20. Grabowski, J.: Modular classes of skew symmetric relations. Transform. Gr. 17, 989–1010 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. Ibort, A., de León, M.: Marrero J C and Martín de Diego D Dirac brackets in constrained dynamics. Fortschr. Phys. 47, 459–492 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  22. Jovanović, B.: Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on SO(4). J. Phys. A 31, 1415–1422 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  23. Koiller, J.: Reduction of some classical nonholonomic systems with symmetry. Arch. Ration. Mech. Anal. 118, 113–148 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  24. Koiller, J., Ehlers, K.: Rubber rolling over a sphere. Regul. Chaot. Dyn. 12, 127–152 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  25. Koon, W.S., Marsden, J.E.: Poisson reduction of nonholonomic mechanical systems with symmetry. Rep. Math. Phys. 42(1/2), 101–134 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  26. Kozlov, V.V.: Invariant measures of the Euler-Poincaré equations on Lie algebras. Funkt. Anal. Prilozh. 22, 69–70 (Russian); English trans. Funct. Anal. Appl. 22, 58–59 (1988)Google Scholar
  27. Liu, Z.J., Xu, P.: On quadratic Poisson structures. Lett. Math. Phys. 26, 33–42 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  28. Marle, Ch.: M Various approaches to conservative and nonconservative nonholonomic systems Rep. Math. Phys. 42, 211–229 (1998)zbMATHMathSciNetGoogle Scholar
  29. Marrero, J.C.: Hamiltonian dynamics on Lie algebroids, unimodularity and preservation of volumes. J. Geom. Mech. 2(3), 243–263 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  30. Marsden, J.E., Ratiu, T.S.: Introduction to mechanics with symmetry. Texts in Applied Mathematics, vol. 17. Springer (1994)Google Scholar
  31. Ortega, J.P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction. Progress in Mathematics, vol. 222. Birkhuser Boston Inc., Boston (2004)Google Scholar
  32. Routh, E.D.: Dynamics of a System of Rigid Bodies. 7th edn., revised and enlarged. Dover Publications Inc, New York (1960)Google Scholar
  33. Schneider, D.: Non-holonomic Euler-Poincaré equations and stability in Chaplygin’s sphere. Dyn. Syst. 17, 87–130 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  34. Stübler, E.: Zeinschriftür Math. und Phys. B 57, 260–271 (1909)zbMATHGoogle Scholar
  35. Vaisman, I.: Lectures on the geometry of Poisson manifolds. Progress in Mathematics, vol. 118. Birkhäuser Verlag, Basel (1994)Google Scholar
  36. Van der Schaft, A.J., Maschke, B.M.: On the Hamiltonian formulation of non-holonomic mechanical systems. Rep. Math. Phys. 34, 225–233 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  37. Veselov, A.P., Veselova, L.E.: Integrable nonholonomic systems on lie groups. Mat. Notes 44(5–6), 810–819 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  38. Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23, 379–394 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  39. Woronetz, P.: Uber die Bewegung eines starren Korpers, der ohne Gleitung auf einer beliebigen Flache rollt. (German). Math. Ann. 70(3), 410–453 (1911)CrossRefzbMATHMathSciNetGoogle Scholar
  40. Woronetz, P.: Uber die Bewegungsgleichungen eines starren Korpers. (German). Math. Ann. 71(3), 392–403 (1911)CrossRefMathSciNetGoogle Scholar
  41. Xu, P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200, 545–560 (1999)CrossRefzbMATHGoogle Scholar
  42. Yaroshchuk, V.A. New cases of the existence of an integral invariant in a problem on the rolling of a rigid body, without slippage, on a fixed surface. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. (6), 26–30 (1992)Google Scholar
  43. Zenkov, D.V., Bloch, A.M.: Invariant measures of nonholonomic flows with internal degrees of freedom. Nonlinearity 16, 1793–1807 (2003)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yuri N. Fedorov
    • 1
  • Luis C.  García-Naranjo
    • 2
  • Juan C. Marrero
    • 3
    Email author
  1. 1.Department de Matematica Aplicada IUniversitat Politecnica de CatalunyaBarcelonaSpain
  2. 2.Departamento de Matemáticas y MecánicaIIMAS-UNAMMexico CityMexico
  3. 3.ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento de Matemática Fundamental, Facultad de MatemáticasUniversidad de La LagunaCanary IslandsSpain

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