  We analyze particle dynamics on $N$ dimensional one-sheet hyperboloid
embedded in $N+1$ dimensional Minkowski space. The dynamical integrals
constructed by $SO_\uparrow (1,N)$ symmetry of spacetime are used for the
gauge-invariant Hamiltonian reduction. The physical phase-space parametrizes
the set of all classical trajectories on the hyperboloid. In quantum case the
operator ordering problem for the symmetry generators is solved by
transformation to asymptotic variables. Canonical quantization leads to unitary
irreducible representation of $SO_\uparrow (1,N)$ group on Hilbert space
$L^2(S^{N-1})$.
