### Abstract

In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the momentum constraint implicit throughout and we make use of a Routhian function defined on a certain submanifold of the Pontryagin bundle. Then, we show how the reduced implicit Lagrange-Routh equations can be described in the context of dynamical systems associated to Dirac structures, in which we fully utilize a symmetry reduction procedure for implicit Hamiltonian systems with symmetry.

Original language | English |
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Pages (from-to) | 291-304 |

Number of pages | 14 |

Journal | Journal of Geometry and Physics |

Volume | 104 |

DOIs | |

Publication status | Published - 2016 Jun 1 |

### Fingerprint

### Keywords

- Dirac structures
- Hamilton-Pontryagin principle
- Implicit Lagrange-Routh equations
- Routh reduction

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology

### Cite this

*Journal of Geometry and Physics*,

*104*, 291-304. https://doi.org/10.1016/j.geomphys.2016.02.010

**Implicit Lagrange-Routh equations and Dirac reduction.** / García-Toraño Andrés, Eduardo; Mestdag, Tom; Yoshimura, Hiroaki.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 104, pp. 291-304. https://doi.org/10.1016/j.geomphys.2016.02.010

}

TY - JOUR

T1 - Implicit Lagrange-Routh equations and Dirac reduction

AU - García-Toraño Andrés, Eduardo

AU - Mestdag, Tom

AU - Yoshimura, Hiroaki

PY - 2016/6/1

Y1 - 2016/6/1

N2 - In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the momentum constraint implicit throughout and we make use of a Routhian function defined on a certain submanifold of the Pontryagin bundle. Then, we show how the reduced implicit Lagrange-Routh equations can be described in the context of dynamical systems associated to Dirac structures, in which we fully utilize a symmetry reduction procedure for implicit Hamiltonian systems with symmetry.

AB - In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the momentum constraint implicit throughout and we make use of a Routhian function defined on a certain submanifold of the Pontryagin bundle. Then, we show how the reduced implicit Lagrange-Routh equations can be described in the context of dynamical systems associated to Dirac structures, in which we fully utilize a symmetry reduction procedure for implicit Hamiltonian systems with symmetry.

KW - Dirac structures

KW - Hamilton-Pontryagin principle

KW - Implicit Lagrange-Routh equations

KW - Routh reduction

UR - http://www.scopus.com/inward/record.url?scp=84962483081&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84962483081&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2016.02.010

DO - 10.1016/j.geomphys.2016.02.010

M3 - Article

VL - 104

SP - 291

EP - 304

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -