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Erschienen in:
2012 | OriginalPaper | Buchkapitel
Variational Calculus on Lie Groups
verfasst von : Gregory S. Chirikjian
Erschienen in: Stochastic Models, Information Theory, and Lie Groups, Volume 2
Verlag: Birkhäuser Boston
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The calculus of variations is concerned with finding extremal paths of functionals in analogy with the way that classical calculus seeks to find critical points of functions. Variational calculus plays a central role in classical mechanics, connecting the “Principle of Least Action” and Lagrange’s equations of motion (also called the Euler– Lagrange equations). In that setting, generalized coordinates are introduced to describe the geometric configuration of a mechanical system. In this chapter, classical variational calculus is reviewed and extended to describe systems on Lie groups. Of course, the introduction of coordinates such as Euler angles to describe the orientation of a rigid body can be used to formulate classical variational problems at the expense of introducing singularities. However, it is possible to formulate variational problems on Lie groups
without
coordinates. This results in the Euler’Poincaré equations.