Abstract
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.
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Notes
Note that \(X_f\) is not a Hamiltonian vector field in the usual sense since we work with an almost Poisson bracket. It would be more accurate to talk about an almost Hamiltonian vector field. We eliminate the “almost” in our terminology for brevity.
An everyday life example of such type of rigid body is a shoe (without heel).
References
Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin/Cummings, Reading (1978)
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)
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)
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)
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)
Brylinski, J.L., Zuckerman, G.: The outer derivation of a complex Poisson manifold. J. Reine Angew. Math. 506, 181–189 (1999)
Cantrijn, F., de León, M., de Martín Diego, D.: On almost-Poisson structures in nonholonomic mechanics. Nonlinearity 12, 721–737 (1999)
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)
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)
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)
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)
Dufour, J.P., Haraki, A.: Rotationnels et structures de Poisson quadratiques. C. R. Acad. Sci. Paris Sér. I Math. 312, 137–140 (1991)
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)
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)
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)
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)
García-Naranjo, L.C., Maciejewski, A.J., Marrero, J.C., Przybylska, M.: The inhomogeneous Suslov problem. Phys. Lett. A 378, 2389–2394 (2014)
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)
Grabowski, J., Marmo, G., Perelomov, A.M.: Poisson structures: towards a classification. Modern Phys. Lett. A 8, 1719–1733 (1993)
Grabowski, J.: Modular classes of skew symmetric relations. Transform. Gr. 17, 989–1010 (2012)
Ibort, A., de León, M.: Marrero J C and Martín de Diego D Dirac brackets in constrained dynamics. Fortschr. Phys. 47, 459–492 (1999)
Jovanović, B.: Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on SO(4). J. Phys. A 31, 1415–1422 (1998)
Koiller, J.: Reduction of some classical nonholonomic systems with symmetry. Arch. Ration. Mech. Anal. 118, 113–148 (1992)
Koiller, J., Ehlers, K.: Rubber rolling over a sphere. Regul. Chaot. Dyn. 12, 127–152 (2007)
Koon, W.S., Marsden, J.E.: Poisson reduction of nonholonomic mechanical systems with symmetry. Rep. Math. Phys. 42(1/2), 101–134 (1998)
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)
Liu, Z.J., Xu, P.: On quadratic Poisson structures. Lett. Math. Phys. 26, 33–42 (1992)
Marle, Ch.: M Various approaches to conservative and nonconservative nonholonomic systems Rep. Math. Phys. 42, 211–229 (1998)
Marrero, J.C.: Hamiltonian dynamics on Lie algebroids, unimodularity and preservation of volumes. J. Geom. Mech. 2(3), 243–263 (2010)
Marsden, J.E., Ratiu, T.S.: Introduction to mechanics with symmetry. Texts in Applied Mathematics, vol. 17. Springer (1994)
Ortega, J.P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction. Progress in Mathematics, vol. 222. Birkhuser Boston Inc., Boston (2004)
Routh, E.D.: Dynamics of a System of Rigid Bodies. 7th edn., revised and enlarged. Dover Publications Inc, New York (1960)
Schneider, D.: Non-holonomic Euler-Poincaré equations and stability in Chaplygin’s sphere. Dyn. Syst. 17, 87–130 (2002)
Stübler, E.: Zeinschriftür Math. und Phys. B 57, 260–271 (1909)
Vaisman, I.: Lectures on the geometry of Poisson manifolds. Progress in Mathematics, vol. 118. Birkhäuser Verlag, Basel (1994)
Van der Schaft, A.J., Maschke, B.M.: On the Hamiltonian formulation of non-holonomic mechanical systems. Rep. Math. Phys. 34, 225–233 (1994)
Veselov, A.P., Veselova, L.E.: Integrable nonholonomic systems on lie groups. Mat. Notes 44(5–6), 810–819 (1988)
Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23, 379–394 (1997)
Woronetz, P.: Uber die Bewegung eines starren Korpers, der ohne Gleitung auf einer beliebigen Flache rollt. (German). Math. Ann. 70(3), 410–453 (1911)
Woronetz, P.: Uber die Bewegungsgleichungen eines starren Korpers. (German). Math. Ann. 71(3), 392–403 (1911)
Xu, P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200, 545–560 (1999)
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)
Zenkov, D.V., Bloch, A.M.: Invariant measures of nonholonomic flows with internal degrees of freedom. Nonlinearity 16, 1793–1807 (2003)
Acknowledgments
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.
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Communicated by Anthony Bloch.
Appendix: Volume Forms on Vector Bundles
Appendix: Volume Forms on Vector Bundles
Let \(\tau : E \rightarrow Q\) be an orientable vector bundle, over an orientable manifold \(Q\). If \(\alpha \) is a section of \(\tau : E \rightarrow Q\), we can define its vertical lift \(\alpha ^\mathbf{v}\), which is the vector field on \(E\) given by
If \(\{e^{\beta }\}\) is a local basis of sections of \(E\) and \(\alpha = \alpha _{\beta }e^{\beta }\) then
where \(p_{\beta }\) are the coordinates on the fibers of \(E\) obtained using the basis \(\{e^{\beta }\}\).
Lemma 7.1
Marrero (2010) Let \(\nu \) be a volume form on \(Q\) and \(\Omega \) be a volume form on the fibers of \(E^*\). Then, there exists a unique volume form \(\nu \wedge \Omega \) on \(E\) such that
for \(\alpha _{1}, \dots , \alpha _{n} \in \Gamma (\tau )\) and \(\tilde{Z}_{1}, \dots , \tilde{Z}_{m}\) vector fields on \(E\) which are \(\tau \)-projectable on the vector fields \(Z_{1}, \dots , Z_{m}\) on \(Q\).
Locally, if \((q^i)\) are local coordinates on an open subset \(U \subseteq Q\) and \(\{e_{\alpha }\}\) is a basis of sections of \(E^*\) such that
then
A volume form \(\Phi \) on \(E\) is said to be of basic type if
Using (7.3), it is easy to prove that the volume form \(\nu \wedge \Omega \) is of basic type. In fact, we have the following result:
Proposition 7.2
A volume form \(\Phi \) on \(E\) is of basic type if and only if there exists a volume form \(\nu \) on \(Q\) and a volume form \(\Omega \) on the fibers of \(E^*\) such that
Proof
Suppose that \(\Phi \) is a volume form on \(E\) of basic type.
Let \(\nu _0\) be an arbitrary volume form on \(Q\) and \(\Omega _0\) a volume form on the fibers of \(E^*\). Then we can assume, without the loss of generality, that
Now, using (7.4), it follows that
which implies that \(\tilde{\sigma }\) is a basic function with respect to the vector bundle projection \(\tau : E \rightarrow Q\). In other words, there exists \(\sigma \in C^{\infty }(Q)\) such that \(\tilde{\sigma } = \sigma \circ \tau \).
Thus, if we take
we have that \(\Phi = \nu \wedge \Omega \). \(\square \)
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Fedorov, Y.N., García-Naranjo, L.C. & Marrero, J.C. Unimodularity and Preservation of Volumes in Nonholonomic Mechanics. J Nonlinear Sci 25, 203–246 (2015). https://doi.org/10.1007/s00332-014-9227-4
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DOI: https://doi.org/10.1007/s00332-014-9227-4
Keywords
- Nonholonomic mechanical systems
- Linear almost Poisson structure
- Modular vector field
- Unimodularity
- Invariant volume forms
- Symmetries
- Reduction