Positional Probability of Periodic Object Motion

[SSGD]

If you know the velocity of an object as a function of position Can you use a uniform distribution over one period and the object velocity to perform a change of variables for the positional probability.

Example.
X(t)=Asin(wt)
V(t)=Awcos(wt)
V(X)=+-Aw(1-(X/A)^2)^(1/2)
P(t)=1/T
T=Period

Change of Variables
P(X)=P(t)|dt/dX|=P(t)V^-1=
(w/(2pi))/(Aw(1-(X/A)^2)^(1/2)

It looks like you could do this over any time interval t=(a,b) if you know the velocity as a function of position.

 

[RUber]

You will probably want to be careful of your divide by zero errors when x = +/- a.