# Probability of some state for oscillator

1. Dec 26, 2008

### LMZ

1. The problem statement, all variables and given/known data
having a math oscillator, with
$$\varphi(t) = \varphi_0 cos(\frac{2 \pi}{T})t$$
$$T=2 \pi \sqrt{\frac{l}{g}}$$

find the probability to find $$\varphi$$ in interval $$[\varphi, \varphi + d\varphi]$$

2. Relevant equations
http://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian)

3. The attempt at a solution
$$dw=\int \rho dp d \varphi$$

trying using liuville theorem:
$$\frac{\partial}{\partial \varphi}(\rho \dot{\varphi}}) + \frac{\partial}{\partial p}(\rho \dot{p}) = 0$$
$$\frac{\partial \rho}{\partial \varphi} \dot{\varphi}+\rho \frac{\partial \dot{\varphi}}{\partial \varphi} + \frac{\partial \rho}{\partial p} \dot{p}+\rho \frac{\partial \dot{p}}{\partial p} = 0$$
$$\frac{\partial \dot{\varphi}}{\partial \varphi} = 0; \frac{\partial \dot{p}}{\partial p} = m r(sin(\varphi)-cos(\varphi))$$
$$\rho = \frac{2 \frac{\partial \rho}{\partial t}}{m r(sin(\varphi)-cos(\varphi))}$$

don't know if I'm right till now. Next I think to introduce probability density (rho) in integral for probability (dw), and don't know how to integrate.

thanks for your time, I appreciate this!