A particle of mass 1 moves in a 2D harmonic oscillator potential $\displaystyle V(x, y)=8(x^2+4y^2)$. If the position and velocity of the particle at time t = 0 are given by $\displaystyle r_{0}=2i-j$and $\displaystyle v_{0}=4i+8j$,

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

$\displaystyle r(t)=(2cos4t+sin4t)i+(-cos8t+sin8t)j$

$\displaystyle v(t)=(-8sin4t+4cos4t)i+(8sin8t+8cos8t)j$

(b) Determine the period of the motion.

$\displaystyle T=\frac{\pi}{2}$ The period depends on r(t) or v(t) or both?

How to get this? Why?

(c) Find the total energy of the particle.

$\displaystyle E=104$ How to get this?

(d) Suppose that the potential is instead $\displaystyle V(x,y)=8(x^2+2y^2)$. 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