Phase space is a mathematical space that represents all possible states of a physical system. It is a multidimensional space where each point corresponds to a unique combination of position and momentum coordinates of the system. For a particle moving in a circle of fixed radius with constant speed, the phase space has four dimensions.


In order to define the phase space for a particle moving in a circle, we need to consider both the position and momentum of the particle. The position of the particle can be described by its angular position on the circle, which can be represented by a single coordinate. The momentum of the particle, on the other hand, has two components - angular momentum and radial momentum.


The angular position of the particle can be represented by a single coordinate, such as the angle θ measured from a reference point on the circle. This coordinate describes the position of the particle along the circumference of the circle.


The angular momentum of the particle is a vector quantity that describes the rotational motion of the particle. It is given by the product of the particle's mass, its velocity, and the radius of the circle. The angular momentum has two components - one along the direction of motion (tangential to the circle) and the other perpendicular to the direction of motion (radial to the circle).


The radial momentum of the particle is the momentum component along the direction perpendicular to the circle. It is determined by the mass of the particle and its velocity in the radial direction.


Based on the above considerations, the phase space for a particle moving in a circle with constant speed has four dimensions:
1. Angular position (1 dimension)
2. Angular momentum (2 dimensions)
3. Radial momentum (1 dimension)

These four coordinates together describe the complete state of the particle in the phase space. Each point in the phase space corresponds to a unique combination of these four coordinates, representing a specific state of the particle. Therefore, the dimension of the phase space for a particle moving in a circle with constant speed is four.