Advanced Search
Article Contents
Article Contents

A note on Hybrid Routh reduction for time-dependent Lagrangian systems

Communicated by Manuel de León

L. Colombo was partially supported by I-Link Project (Ref: linkA20079) from CSIC, Ministerio de Economia, Industria y Competitividad (MINEICO, Spain) under grant MTM2016- 76702-P; "Severo Ochoa Programme for Centres of Excellence" in R & D (SEV-2015-0554). The project that gave rise to these results received the support of a fellowship from "La Caixa" Foundation (ID 100010434). M.E. Eyrea Irazú was partially supported by CONICET Argentina

Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • This note discusses Routh reduction for hybrid time-dependent mechanical systems. We give general conditions on whether it is possible to reduce by symmetries a hybrid time-dependent Lagrangian system extending and unifying previous results for continuous-time systems. We illustrate the applicability of the method using the example of a billiard with moving walls.

    Mathematics Subject Classification: Primary: 70S10, 37J15; Secondary: 70H03.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  A "billiard" with moving walls

    Figure 2.  Simulation for $ c = 0.25 $. The figure in the left corresponds with the reduced trajectory while the figure to the right corresponds with the reconstructed solution

    Figure 3.  Simulation for $ c = 0.10 $. he figure in the left corresponds with the reduced trajectory while the figure to the right corresponds with the reconstructed solution

  • [1] C. Albert, Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact, J. Geom. Phys., 6 (1989), 627-649.  doi: 10.1016/0393-0440(89)90029-6.
    [2] A. Ames and S. Sastry, Hybrid cotangent bundle reduction of simple hybrid mechanical systems with symmetry, American Control Conference, Minneapolis, MN, 2006. doi: 10.1109/ACC.2006.1656622.
    [3] A. Ames and S. Sastry, Hybrid Routhian reduction of Lagrangian hybrid systems, American Control Conference, Minneapolis, MN, 2006. doi: 10.1109/ACC.2006.1656621.
    [4] A. Ames, R. Gregg and M. Spong, A geometric approach to three-dimensional hipped bipedal robotic walking, 46th IEEE Conference on Decision and Control, New Orleans, LA, 2007, 5123–5130. doi: 10.1109/CDC.2007.4434880.
    [5] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Springer, New York, 2015. doi: 10.1007/978-1-4939-3017-3.
    [6] A. Bloch, W. Clark and L. Colombo, Quasivelocities and symmetries in simple hybrid systems, IEEE 56th Annual Conference on Decision and Control (CDC), Melbourne, VIC, Australia, 2017, 1529–1534. doi: 10.1109/CDC.2017.8263869.
    [7] L. J. Colombo and M. E. Eyrea Irazú, Symmetries and periodic orbits in simple hybrid Routhian systems, Nonlinear Anal. Hybrid Syst., 36 (2020), 14pp. doi: 10.1016/j.nahs.2020.100857.
    [8] L. Colombo, W. Clark and A. Bloch, Time reversal symmetries and zero dynamics for simple hybrid Hamiltonian control systems, Annual American Control Conference (ACC), Milwaukee, WI, 2018, 2218–2223. doi: 10.23919/ACC.2018.8431672.
    [9] J. CortésM. de LeónD. Martín de Diego and S. Martínez, Mechanical systems subjected to generalized non-holonomic constraints, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 651-670.  doi: 10.1098/rspa.2000.0686.
    [10] M. Crampin and T. Mestdag, Routh's procedure for non-abelian symmetry groups, J. Math. Phys., 49 (2008), 28pp. doi: 10.1063/1.2885077.
    [11] M. de León and M. Saralegi, Cosymplectic reduction for singular momentum maps, J. Phys. A, 26 (1993), 5033-5043.  doi: 10.1088/0305-4470/26/19/032.
    [12] A. Echeverría EnríquezM. C. Muñoz Lecanda and N. Román-Roy, Geometrical setting of time-dependent regular systems. Alternative models, Rev. Math. Phys., 3 (1991), 301-330.  doi: 10.1142/S0129055X91000114.
    [13] E. Eyrea Irazú, Geometric and Numerical Aspects of Mechanical Systems with Magnetic Terms, Ph.D. thesis, Universidad Nacional de La Plata, 2019.
    [14] K. Grabowska and P. Urbański, Geometry of Routh reduction, J. Geom. Mech., 11 (2019), 23-44.  doi: 10.3934/jgm.2019002.
    [15] R. Gregg and M. Spong, Reduction-based control with application to three-dimensional bipedal walking robots, American Control Conference, Seattle, WA, 2008,880–887. doi: 10.1109/ACC.2008.4586604.
    [16] A. IbortM. de LeónE. A. LacombaJ. C. MarreroD. M. de Diego and P. Pitanga, Geometric formulation of mechanical systems subjected to time-dependent one-sided constraints, J. Phys. A, 31 (1998), 2655-2674.  doi: 10.1088/0305-4470/31/11/014.
    [17] E. A. Lacomba and W. M. Tulczyjew, Geometric formulation of mechanical systems with one-sided constraints, J. Phys. A, 23 (1990), 2801-2813.  doi: 10.1088/0305-4470/23/13/019.
    [18] B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 20pp. doi: 10.1063/1.3277181.
    [19] B. Langerock, E. García-Toraño Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, J. Math. Phys., 53 (2012), 19pp. doi: 10.1063/1.4723841.
    [20] B. Langerock, T. Mestdag and J. Vankerschaver, Routh reduction by stages, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 31pp. doi: 10.3842/SIGMA.2011.109.
    [21] J. E. Marsden, G. Misioƚek, J.-P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages, Lecture Notes in Mathematics, 1913, Springer, Berlin, 2007. doi: 10.1007/978-3-540-72470-4.
    [22] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.
    [23] L. A. Pars, A Treatise on Analytical Dynamics, John Wiley & Sons, Inc., New York, 1965.
  • 加载中



Article Metrics

HTML views(398) PDF downloads(227) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint