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Lagrangian mechanics and variational integrators on two‐spheres
Author(s) -
Lee Taeyoung,
Leok Melvin,
McClamroch N. Harris
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2603
Subject(s) - variational integrator , variational principle , mathematics , lie group , lagrangian mechanics , equations of motion , spheres , hamilton's principle , integrator , calculus of variations , mathematical analysis , classical mechanics , analytical mechanics , geometry , physics , quantum mechanics , voltage , astronomy , quantum dynamics , quantum
Euler–Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two‐spheres. The geometric structure of a product of two‐spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler–Lagrange equations on two‐spheres, which are more compact than existing equations written in terms of angles. Since the variational integrators are derived from Hamilton's principle, they preserve the geometric features of the dynamics such as symplecticity, momentum maps, or total energy, as well as the structure of the configuration manifold. Computational properties of the variational integrators are illustrated for several mechanical systems. In addition, Lie group variational integrators can be used to integrate Lagrangian flows on more general homogeneous spaces. This is achieved by lifting the discrete Hamilton's principle on homogeneous spaces to a discrete variational principle on the Lie group that is constrained by a discrete connection. Copyright © 2009 John Wiley & Sons, Ltd.