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Elastica in SO(3)

Part of: Manifolds Optimality conditions

Published online by Cambridge University Press:  09 April 2009

Tomasz Popiel  and
Lyle Noakes
Tomasz Popiel
Affiliation:
University of Western AustraliaSchool of Mathematics and Statistics (M019)35 Stirling Highway Crawley 6009 Western [email protected], [email protected]
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Abstract

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In a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.

MSC classification

Secondary: 49K99: None of the above, but in this section 49Q99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 105 - 124
Copyright
Copyright © Australian Mathematical Society 2007

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