June  2020, 12(2): 309-321. doi: 10.3934/jgm.2020014

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

1. 

Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Calle Nicolás Cabrera 13-15, Cantoblanco, 28049, Madrid, Spain

2. 

Department of Mathematics, Universidad Nacional de La Plata, Calle 1 y 115, La Plata 1900, Buenos Aires, Argentina

3. 

Departamento de Matemática, Universidad Nacional del Sur, Av. Alem, 1253, 8000 Bahía Blanca, Argentina

Communicated by Manuel de León

Received  December 2019 Revised  March 2020 Published  June 2020 Early access  June 2020

Fund Project: 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

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.

Citation: Leonardo J. Colombo, María Emma Eyrea Irazú, Eduardo García-Toraño Andrés. A note on Hybrid Routh reduction for time-dependent Lagrangian systems. Journal of Geometric Mechanics, 2020, 12 (2) : 309-321. doi: 10.3934/jgm.2020014
References:
[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.

show all references

References:
[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.

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
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