Kinematic reduction and the Hamilton-Jacobi equation

María Barbero-Liñán; Manuel de León; David Martín de Diego; Juan C. Marrero; Miguel C. Muñoz-Lecanda

doi:10.3934/jgm.2012.4.207

Article Contents

2012, Volume 4, Issue 3: 207-237. Doi: 10.3934/jgm.2012.4.207

This issue Previous Article Foreword Next Article Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields

Kinematic reduction and the Hamilton-Jacobi equation

1.
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid
2.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049 Madrid
3.
Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid
4.
Unidad asociada ULL-CSIC, Geometría diferencial y mecánica geométrica, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, 38071 La Laguna, Tenerife, Canary Islands
5.
Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1, E-08034 Barcelona

Received: September 30, 2011

Revised: February 29, 2012

Published: August 2012

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Abstract

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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