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

Colombo Leonardo J.; Irazú María Emma Eyrea; Andrés Eduardo García-Toraño

doi:10.3934/jgm.2020014

Article Contents

2020, Volume 12, Issue 2: 309-321. Doi: 10.3934/jgm.2020014

This issue Previous Article Nonholonomic and constrained variational mechanics Next Article

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

Early access: June 2020

Published: June 2020

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

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Abstract

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.

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

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Figure 1. A "billiard" with moving walls

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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

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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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References

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