\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Linear weakly Noetherian constants of motion are horizontal gauge momenta

Abstract Related Papers Cited by
  • noindent The notion of gauge momenta is a generalization of the momentum map which is relevant for nonholonomic systems with symmetry. Weakly Noetherian functions are functions which are constants of motion of all 'natural' nonholonomic systems with a given kinetic energy and any $G$-invariant potential energy. We show that, when the action of the symmetry group on the configuration manifold is free and proper, a function which is linear in the velocities is weakly-Noetherian if anf only if it is a gauge momenta which has a horizontal generator.
    Mathematics Subject Classification: 37J15, 37J05, 70F25.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    C. Agostinelli, Nuova forma sintetica delle equazioni del moto di un sistema anolonomo ed esistenza di un integrale lineare nelle velocità lagrangiane, Boll. Un. Mat. Ital. (3), 11 (1956), 1-9.

    [2]

    L. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308.doi: 10.1016/0034-4877(96)84069-9.

    [3]

    S. Benenti, A 'user-friendly' approach to the dynamical equations of non-holonomic systems, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 036, 33 pp.

    [4]

    A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.doi: 10.1007/BF02199365.

    [5]

    A. M. Bloch, "Nonholonomic Mechanics and Controls,'' Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003.doi: 10.1007/b97376.

    [6]

    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.

    [7]

    F. Cantrjn, M. de León, M. de Diego and J. Marrero, Reduction of nonholonomic mechanical systems with symmetries, Rep. Math. Phys., 42 (1998), 25-45.doi: 10.1016/S0034-4877(98)80003-7.

    [8]

    F. Cantrjn, J. Cortés, M. de León and M. de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Phil. Soc., 132 (2002), 323-351.

    [9]

    R. Cushman, D. Kemppainen, J. Śniatycki and L. Bates, Geometry of nonholonomic constraints, Proceedings of the XXVII Symposium on Mathematical Physics (Toruń, 1994), Rep. Math. Phys., 36 (1995), 275-286.doi: 10.1016/0034-4877(96)83625-1.

    [10]

    M. de León, J. C. Marrero and D. Martín de Diego, Mechanical systems with nonlinear constraints, Int. J. Th. Phys., 36 (1997), 979-995.doi: 10.1007/BF02435796.

    [11]

    F. Fassò, A. Ramos and N. Sansonetto, The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions, Reg. Ch. Dyn., 12 (2007), 579-588.doi: 10.1134/S1560354707060019.

    [12]

    F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics, Rep. Math. Phys., 62 (2008), 345-367.doi: 10.1016/S0034-4877(09)00005-6.

    [13]

    F. Fassò, A. Giacobbe and N. Sansonetto, On the number of weakly Noetherian constants of motion of nonholonomic systems, J. Geom. Mech., 1 (2009), 389-416.

    [14]

    Il. Iliev and Khr. Semerdzhiev, Relations between the first integrals of a nonholonomic mechanical system and of the corresponding system freed of constraints, J. Appl. Math. Mech., 36 (1972), 381-388.doi: 10.1016/0021-8928(72)90049-4.

    [15]

    Il. Iliev and P. Ilija, On first integrals of a nonholonomic mechanical system, J. Appl. Math. Mech., 39 (1975), 147-150.doi: 10.1016/0021-8928(75)90046-5.

    [16]

    J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rat. Mech. An., 118 (1992), 113-148.doi: 10.1007/BF00375092.

    [17]

    C.-M. Marle, On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints, in "Classical and Quantum Integrability'' (Warsaw, 2001), Banach Center Publ., 59, Polish Acad. Sci., Warsaw, (2003), 223-242

    [18]

    J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,'' Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004.

    [19]

    J. Śniatycki, Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys., 42 (1998), 5-23.doi: 10.1016/S0034-4877(98)80002-5.

    [20]

    D. V. Zenkov, Linear conservation laws of nonholonomic systems with symmetry, in "Dynamical systems and differential equations'' (Wilmington, NC, 2002), Discrete Contin. Dyn. Syst., 2003, suppl., 967-976.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(80) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return