  We investigate the massless scalar particle dynamics on $AdS_{N+1} ~ (N>1)$
by the method of Hamiltonian reduction. Using the dynamical integrals of the
conformal symmetry we construct the physical phase space of the system as a
$SO(2,N+1)$ orbit in the space of symmetry generators. The symmetry generators
themselves are represented in terms of $(N+1)$-dimensional oscillator
variables. The physical phase space establishes a correspondence between the
$AdS_{N+1}$ null-geodesics and the dynamics at the boundary of $AdS_{N+2}$. The
quantum theory is described by a UIR of $SO(2,N+1)$ obtained at the unitarity
bound. This representation contains a pair of UIR's of the isometry subgroup
SO(2,N) with the Casimir number corresponding to the Weyl invariant mass value.
The whole discussion includes the globally well-defined realization of the
conformal group via the conformal embedding of $AdS_{N+1}$ in the ESU
$\rr\times S^N$.
