(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

having a math oscillator, with

[tex]\varphi(t) = \varphi_0 cos(\frac{2 \pi}{T})t[/tex]

[tex]T=2 \pi \sqrt{\frac{l}{g}}[/tex]

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

2. Relevant equations

http://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian [Broken])

3. The attempt at a solution

[tex]dw=\int \rho dp d \varphi[/tex]

trying using liuville theorem:

[tex]\frac{\partial}{\partial \varphi}(\rho \dot{\varphi}}) + \frac{\partial}{\partial p}(\rho \dot{p}) = 0[/tex]

[tex]\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[/tex]

[tex]\frac{\partial \dot{\varphi}}{\partial \varphi} = 0; \frac{\partial \dot{p}}{\partial p} = m r(sin(\varphi)-cos(\varphi))[/tex]

[tex]\rho = \frac{2 \frac{\partial \rho}{\partial t}}{m r(sin(\varphi)-cos(\varphi))}[/tex]

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!

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# Homework Help: Probability of some state for oscillator

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