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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

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1965).Google Scholar
[2]Babelon, O., Bernard, D. and Talon, M., Introduction to Classical Integrable Systems (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[3]Carmo, M. P. do, Riemannian Geometry (Birkhäuser Boston Inc., Boston, MA, 1992).CrossRefGoogle Scholar
[4]Jurdjevic, V., ‘Non-Euclidean elastica’, Amer. J. Math. 117 (1995), 93–124.CrossRefGoogle Scholar
[5]Jurdjevic, V., Geometric Control Theory (Cambridge University Press, Cambridge, NY, 1997).Google Scholar
[6]Milnor, J., Morse Theory, Annals of Mathematics Studies No. 51 (Princeton University Press, Princeton, NJ, 1963).Google Scholar
[7]Noakes, L., ‘Null cubics and Lie quadratics’, J. Math. Phys. (3) 44 (2003), 1436–1448.CrossRefGoogle Scholar
[8]Noakes, L., ‘Non-null Lie quadratics in E3’, J. Math. Phys. (11) 45 (2004), 4334–4351.CrossRefGoogle Scholar
[9]Noakes, L., ‘Duality and Riemannian cubics’, Adv. Comput. Math. (11) 25 (2006), 195–209.CrossRefGoogle Scholar
[10]Noakes, L., ‘Lax constraints in semisimple Lie groups’, Q. J. Math. (4) 57 (2006), 527–538.CrossRefGoogle Scholar
[11]Noakes, L., Heinzinger, G. and Paden, B., ‘Cubic splines on curved spaces’, IMA J. Math. Control Inform. (4) 6 (1989), 465–473.CrossRefGoogle Scholar
[12]Noakes, L. and Popiel, T., ‘Quadratures and cubics in SO(3) and SO(1, 2)’, IMAJ. Math. Control Inform. (4) 23 (2006), 477–488.Google Scholar
[13]Varadarajan, V., Lie Groups, Lie Algebras, and their Representations (Springer-Verlag, New York, 1984).CrossRefGoogle Scholar