Force free Möbius motions of the circle. (English) Zbl 1264.53016

Summary: Let \(\mathcal M\) be the Lie group of Möbius transformations of the circle. Suppose that the circle has initially a homogeneous distribution of mass and that the particles are allowed to move only in such a way that two configurations differ in an element of \(\mathcal M\). We describe all force free Möbius motions, that is, those curves in \(\mathcal M\) which are critical points of the kinetic energy. The main tool is a Riemannian metric on \(\mathcal M\) which turns out to be not complete (in particular not invariant, as happens with non-rigid motions) given by the kinetic energy.


53A30 Conformal differential geometry (MSC2010)
22E70 Applications of Lie groups to the sciences; explicit representations
53Z05 Applications of differential geometry to physics