J. Aust. Math. Soc. 83 (2007), no. 1, pp. 105–125.

Elastica in \operatorname{SO}(3)

Tomasz Popiel Lyle Noakes
University of Western Australia
School of Mathematics and Statistics (M019)
35 Stirling Highway
Crawley 6009
Western Australia
Australia
popiet01@maths.uwa.edu.au
University of Western Australia
School of Mathematics and Statistics (M019)
35 Stirling Highway
Crawley 6009
Western Australia
Australia
lyle@maths.uwa.edu.au
Received 16 March 2005; revised 19 June 2006
Communicated by K. Wysocki

Abstract

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 \operatorname {SO}(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = \operatorname {SO}(3), we solve the Euler–Lagrange equation by quadratures.

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2000 Mathematics Subject Classification: primary 49K99, 49Q99
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2354??? Z'blatt-MATH: 1128.49019
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