Abstract
We consider nonholonomic systems with linear, time-independent constraints subject to positional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator distribution ℜ°, the fibers of which are the annihilators of the set of all values taken by the reaction forces on the fibers of the constraint distribution. We show that this distribution, which can be effectively computed in specific cases, plays a central role in the study of first integrals linear in the velocities of this class of nonholonomic systems. In particular we prove that, if the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, then an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution ℜ°. Since the fibers of ℜ° contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved momenta than what was known so far. Some examples are given.
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References
Agostinelli, C., Nuova Forma Sintetica Delle Equazioni del Moto di un Sistema Anolonomo ed Esistenza di un Integrale Lineare Nelle Velocità, Boll. Un. Mat. Ital., 1956, vol. 11, pp. 1–9.
Kozlov, V.V. and Kolesnikov, N.N., On Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 28–33 (in Russian).
Arnol’d, V.I., Kozlov, V.V., and Neishtadt, A.I., Mathematical Aspects of Classical and Celestial Mechanics. Dynamical Systems, III. Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993.
Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., and Murray, R.M., Nonholonomic Mechanical Systems with Symmetry, Arch. Rational Mech. Anal., 1996, vol. 136, pp. 21–99.
Cantrjn, F., de Leon, M., de Diego, M., and Marrero, J., Reduction of Nonholonomic Mechanical Systems with Symmetries, Rep. Math. Phys., 1998, vol. 42, pp. 25–45.
de Leon, M., Cortes, J., de Diego, M. and Martínez, S., Introduction to Mechanics and Symmetry, in Bajo, I. and Sanmartin E. (eds.), Recent Advances in Lie Theory. Research and Exposition inMathematics Series, vol. 25, pp. 305–332, Heldermann Verlag, 2002.
Marle, C.-M., On Symmetries and Constants of Motion in Hamiltonian Systems with Nonholonomic Constraints, in Classical and Quantum Integrability (Warsaw, 2001), pp. 223–242, Banach Center Publ., vol. 59, Polish Acad. Sci., Warsaw, 2003.
Bloch, A.M., Nonholonomic Mechanics and Controls. Interdisciplinary Applied Mathematics, vol. 24, Systems and Control, New-York: Springer-Verlag, 2003.
Duistermaat, J.J., Chaplygin’s Sphere, arXiv:math/0409019, 2004.
Cushman, R., Kemppainen, D., Śniatycki, J., and Bates, L., Geometry of Nonholonomic Constraints, Proceedings of the XXVII Symposium on Mathematical Physics (Toruń, 1994), Rep. Math. Phys., 1995, vol. 36, pp. 275–286.
Śniatycki, J., Nonholonomic Noether Theorem and Reduction of Symmetries, Rep. Math. Phys., 1998, vol. 42, pp. 5–23.
Koiller, J., Reduction of Some Classical Nonholonomic Systems with Symmetry, Arch. Rat. Mech. An., 1992, vol. 118, pp. 113–138.
Cantrjn, F., Cortes, J., de Leon, M., and de Diego, M., On the Geometry of Generalized Chaplygin Systems, Math. Proc. Cambridge Phil. Soc., 2002, vol. 132, pp. 323–351.
Cortés Monforte, J., Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, vol. 1793, Berlin: Springer-Verlag, 2002.
Iliev, Il. and Semerdzhiev, Khr., Relations between the First Integrals of a Nonholonomic Mechanical System and of the Corresponding System Freed of Constraints., J. Appl. Math. Mech., 1972, vol. 36, pp. 381–388.
Iliev, Il., On First Integrals of a Nonholonomic Mechanical System, J. Appl. Math. Mech., 1975, vol. 39, pp. 147–150.
Neimark, Ju.I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, vol. 33, Providence: AMS, 1972.
Bullo F., and Lewis, A.D., Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, vol. 49, New-York: Springer-Verlag, 2005.
Arnold, V.I., Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, New-York: Springer-Verlag, 1989.
Abraham, R. and Marsden, J.E., Foundations of Mechanics, Benjamin, Reading, 1978.
Bates, L. and Śniatycki, J., Nonholonomic Reduction, Rep. Math. Phys., 1993, vol. 32, pp. 99–115.
Pars, L., A Treatise on Analytical Dynamics, New-York: Heinemann, 1965.
Bates, L., Graumann, H., and MacDonnell, C., Examples of Gauge Conservation Laws in Nonholonomic Systems, Rep. Math. Phys., 1996, vol. 37, pp. 295–308.
Zenkov, D.V., Linear Conservation Laws of Nonholonomic Systems with Symmetry, in Dynamical Systems and Differential Equations (Wilmington, NC, 2002), Discrete Contin. Dyn. Syst., 2003, suppl., pp. 967–976.
Fassò, F., Giacobbo, A., and Sansonetto, N., Gauge Integrals, the Nonholonomic Momentum Equation, and the Reaction-Annihilator Distribution (in preparation).
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Fassò, F., Ramos, A. & Sansonetto, N. The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions. Regul. Chaot. Dyn. 12, 579–588 (2007). https://doi.org/10.1134/S1560354707060019
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DOI: https://doi.org/10.1134/S1560354707060019