  We analyze dynamics of the AdS$_{N+1}$ particle realized on the coset
SO$(2,N)/$SO$(1,N)$. Hamiltonian reduction provides the physical phase space in
terms of the coadjoint orbit obtained by boosting a timelike element of ${\frak
so}(2,N)$. We show equivalence of this approach to geometric quantization and
to the SO$(N)$ covariant oscillator description, for which the boost generators
entail a complicated operator ordering. As an alternative scheme, we introduce
dual oscillator variables and derive their algebra at the classical and the
quantum level. This simplifies the calculations of the commutators for the
boost generators and leads to unitary irreducible representations of ${\frak
so}(2,N)$ for all admissible values of the mass parameter. We furthermore
discuss an SO$(N)$ covariant supersymmetric extensions of the oscillator
quantization, with its realization for superparticles in AdS$_2$ and AdS$_3$
given by recent works.
