## 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.

### MSC:

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

### Keywords:

distribution of mass; particles; kinetic energy