Journal of Lie Theory 25 (2015), No. 1, 215--231
Copyright Heldermann Verlag 2015
Elastic Helices in Simple Lie Groups
Oscar J. Garay
Department of Mathematics, University of the Basque Country, UPV/EHU, 48080 Bilbao, Spain
oscarj.garay@ehu.es
Lyle Noakes
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia
Lyle.Noakes@uwa.edu.au
Elastic curves (elastica) are classical variational objects with many applications in physics and engineering. Elastica in real space forms are well understood, but in other ambient spaces there are few known explicit examples, except geodesics. The main purpose of the present paper is to construct new families of elastica when the ambient space is a simple Lie group G with a bi-invariant Riemannian metric. Then, using Lie reduction, we give a criterion for a pointwise product of one-parameter subgroups to be an elastic curve. This characterisation is applied first when G is the real space form SU(2), and comparisons can be made with classical results. We then focus on G = SU(3), for which very little is known. Analysis of our criterion leads to large families of new elastica in SU(3) which are helices, namely our new examples have constant Frenet curvatures. Elastic helices are also constant-speed tension-cubics, solving a different variational problem.
Keywords: Elastica, Riemannian cubics, Lie group, Riemannian manifold, Hopf map.
MSC: 49K99, 53A99, 53C42; 58J32, 49Q10, 58E99, 49S05
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