# Elliptical Motion

1. Nov 24, 2014

### camorin

1. The problem statement, all variables and given/known data
A particle of mass m is subjected to an isotropic, two dimensional, harmonic central force, F=-kr.
(r=(x,y)). At t=0 the particle is at r=A and has velocity V yhat.

Find r' r'' θ' θ'' in 2D spherical polar coordinates.

2. Relevant equations
Fx=-kx
Fy=-ky
3. The attempt at a solution
I have found the equations of motion, as well as the particular solutions using the initial conditions.
So far I have:
x(t)=-Acos(wt-π)
y(t)=V/wcos(wt-π/2)

From here I found r' to be Awsin(wt-π)-Vsin(wt-π/2)
and r'': Aw2cos(wt-π)-Vwcos(wt-π/2)
Im fairly certain everything up to this point is correct, but I have no idea what to do about theta.

I know in spherical polar coordinates θ=arctan(y/x) but I don't think I can just take the derivative in this form. I have tried setting x(t)=-Acos(wt) and y(t)=V/wsin(wt) by using the relation between sines and cosines offset by pi/2.This has brought me to
θ=arctan((V/wa)tan(wt)). Again, I hit a wall with finding the derivatives.

2. Nov 24, 2014

### Simon Bridge

Is the primed notation a space or a time derivative?

Write down an expression for theta and take the appropriate derivative.
Why not? Did you try?
... what happens when you take the appropriate derivative of that then?

3. Nov 24, 2014

### camorin

Im not sure of how to take the derivative of that arctan expression. Is there a way to simplify it? I've tried finding methods but I'm having trouble.

The primed notation is a time derivative, sorry.

4. Nov 25, 2014

### Simon Bridge

... that's what everyone else does.