  Kuramoto's original model describes the dynamics and synchronization behavior
of a set of interacting oscillators represented by their phases. The system can
also be pictured as a set of particles moving on a circle in two dimensions,
which allows a direct generalization to particles moving on the surface of
higher dimensional spheres. One of the key features of the 2D system is the
presence of a continuous phase transition to synchronization as the coupling
intensity increases. Ott and Antonsen proposed an ansatz for the distribution
of oscillators that allowed them to describe the dynamics of the transition's
order parameter with a single differential equation. A similar ansatz was later
proposed for the D-dimensional model by using the same functional form of the
2D ansatz and adjusting its parameters. In this paper we develop a constructive
method to find the ansatz, similarly to the procedure used in 2D. The method is
based on our previous work for the 3D Kuramoto model where the ansatz was
constructed using the spherical harmonics decomposition of the distribution
function. In the case of motion in a D-dimensional sphere the ansatz is based
on the hyperspherical harmonics decomposition. Our result differs from the
previously proposed ansatz and provides a simpler and more direct connection
between the order parameter and the ansatz.
