# Is uniform circular motion an SHM?

I know the projection along a diameter is an SHM but is circular motion itself an SHM? If we consider the mean position to be the center of the circle then the centripetal acceleration is proportional to the distance and in opposite direction of the position of the particle. So shouldn't it be an SHM?

The multi-dimensional analog of simple harmonic motion is an object subject only to a harmonic potential, $U = \frac 1 2 k ||\vec r - \vec r_0||^2$, where $k$ is a positive constant (oftentimes called the spring constant), $\vec r$ is the object's position, and $\vec r_0$ is the position of the center of the potential. By choosing the origin to be the center of the potential, this simplifies to $U = \frac 1 2 k r^2$. Elliptical motion results from such a potential. One dimensional simple harmonic motion can be viewed as a degenerate ellipse.
Since a circle is a special case of an ellipse, a two dimensional harmonic oscillator can result in uniform circular motion. However, this does not mean that uniform circular motion is simple harmonic motion. For example, gravitation between two point masses can result in uniform circular motion. Gravitation is not harmonic; the harmonic potential is proportional to $r^2$ while gravitational potential is proportional to $1/r$. Any potential that is a function of radial distance only will result in uniform circular motion if the velocity vector has exactly the right magnitude for uniform circular motion and is normal to the radial vector.