2D harmonic oscillator Problem

A particle of mass 1 moves in a 2D harmonic oscillator potential . If the position and velocity of the particle at time t = 0 are given by and ,

(a) Find the position and velocity of the particle at any time t > 0.

(b) Determine the period of the motion.

The period depends on r(t) or v(t) or both?

How to get this? Why?

(c) Find the total energy of the particle.

How to get this?

(d) Suppose that the potential is instead . Is there a period defined for the motion in this case? Explain why or why not.

Any advice and comments will be helpful, thanks a lot, guys