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

Reduction of invariant constrained systems using anholonomic frames

Abstract Related Papers Cited by
  • We analyze two reduction methods for nonholonomic systems that are invariant under the action of a Lie group on the configuration space. Our approach for obtaining the reduced equations is entirely based on the observation that the dynamics can be represented by a second-order differential equations vector field and that in both cases the reduced dynamics can be described by expressing that vector field in terms of an appropriately chosen anholonomic frame.
    Mathematics Subject Classification: 34A26, 37J60, 70G45, 70H03.

    Citation:

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

    A. M. Bloch with the collaboration of J. Baillieul, P. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control,'' Springer, 2003.

    [2]

    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.

    [3]

    A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasi-velocities and symmetries in nonholonomic systems, Dynamical Systems, 24 (2009), 187-222.doi: 10.1080/14689360802609344.

    [4]

    F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Camb. Phil. Soc., 132 (2002), 323-351.doi: 10.1017/S0305004101005679.

    [5]

    H. Cendra, J. E. Marsden and T. S. Ratiu, "Lagrangian Reduction by Stages,'' Memoirs of the American Mathematical Society 152, AMS 2001.

    [6]

    H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems, in "Mathematics Unlimited -- 2001 and Beyond'' (eds. B. Engquist and W. Schmid), Springer (2001), 221-273.

    [7]

    J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lecture Notes in Mathematics 1793, Springer, 2002.

    [8]

    M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901.doi: 10.1063/1.2885077.

    [9]

    M. Crampin and T. Mestdag, Reduction and reconstruction aspects of second-order dynamical systems with symmetry, Acta Appl. Math., 105 (2009), 241-266.doi: 10.1007/s10440-008-9274-7.

    [10]

    M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dynamical Systems, 25 (2010), 159-187.doi: 10.1080/14689360903360888.

    [11]

    M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' LMS Lecture Notes 59, Cambridge University Press, 1988.

    [12]

    R. H. Cushman, H. Duistermaat and J. Śniatycki, "Geometry of Nonholonomically Constrained Systems,'' Advanced Series in Nonlinear Dynamics 26, World Scientific, 2010.

    [13]

    M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308.doi: 10.1088/0305-4470/38/24/R01.

    [14]

    K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, "The Breadth of Symplectic Geometry'' (eds. J. E. Marsden and T. S. Ratiu), Birkhäuser, 2005, 75-116.

    [15]

    B. Jovanovic, Geometry and integrability of Euler-Poincaré-Suslov equations, Nonlinearity, 14 (2001), 1555-1567.doi: 10.1088/0951-7715/14/6/308.

    [16]

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

    [17]

    O. Krupková, Mechanical systems with non-holonomic constraints, J. Math. Phys., 38 (1997), 5098-5126.doi: 10.1063/1.532196.

    [18]

    J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Texts in Applied Mathematics 17, Springer, 1999.

    [19]

    J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41 (2000), 3379-3429.doi: 10.1063/1.533317.

    [20]

    T. Mestdag and M. Crampin, Invariant Lagrangians, mechanical connections and the Lagrange-Poincaré equations, J. Phys. A: Math. Theor., 41 (2008), 344015.doi: 10.1088/1751-8113/41/34/344015.

    [21]

    T. Mestdag and B. Langerock, A Lie algebroid framework for non-holonomic systems, J. Phys. A: Math. Gen., 38 (2005), 1097-1111.doi: 10.1088/0305-4470/38/5/011.

    [22]

    J. I. Neĭmark and N. A. Fufaev, "Dynamics of Nonholonomic Systems,'' Transl. of Math. Monographs 33, AMS, 1972.

    [23]

    W. Sarlet, F. Cantrijn and D. J. Saunders, A geometrical framework for the study of non-holonomic Lagrangian systems, J. Phys. A: Math. Gen., 28 (1995), 3253-3268.doi: 10.1088/0305-4470/28/11/022.

    [24]

    J. Vilms, Connections on tangent bundles, J. Diff. Geom., 1 (1967), 235-243.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(93) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return