
Journal of Lie Theory 25 (2015), No. 1, 215231 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 biinvariant Riemannian metric. Then, using Lie reduction, we give a criterion for a pointwise product of oneparameter 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 constantspeed tensioncubics, solving a different variational problem. Keywords: Elastica, Riemannian cubics, Lie group, Riemannian manifold, Hopf map. MSC: 49K99, 53A99, 53C42; 58J32, 49Q10, 58E99, 49S05 [ Fulltextpdf (327 KB)] for subscribers only. 