Dynamical invariants of rigid motions on the hyperbolic plane
Let \({\mathcal H}^ 2\) be a hyperbolic plane. A particle \(\{\) P,m\(\}\) is a pair given by a point \(P\in {\mathcal H}^ 2\) and a positive real number m; they are called the center and mass of the particle. A rigid particle system \(\{P_ k,m_ k\}^ n_{k=1}\) (n\(\geq 2)\) is a finite set of particles with given distances \(d(P_ i,P_ j)\), \(i,j=1,2,...,n\), of centers. It is well known that the only dynamical invariants of a rigid particle system on a Euclidean plane are the mass center and the ratio of mass and inertia. This means that these data and the initial position and velocity determine uniquely the force-free motion. The situation is much more complicated on the hyperbolic plane. This paper is devoted to the investigation of the geometry of left invariant metrics on the isometry group of the hyperbolic plane determined by the kinetic energy of a rigid particle system.
scientific article; zbMATH DE number 4192544