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
University of Western Australia
School of Mathematics and Statistics (M019)
35 Stirling Highway
Crawley 6009
Western Australia
Received 16 March 2005; revised 19 June 2006
Communicated by K. Wysocki


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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