- #1
Matt atkinson
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Homework Statement
By finding the Lagrangian and using the metric:
[tex]\left(\begin{array}{cc}R^2&0\\0&R^2sin^2(\theta)\end{array}\right)[/tex]
show that:
[tex]\theta (t)=arccos(\sqrt{1-\frac{A^2}{\omega^2}}cos(\omega t +\theta_o))[/tex]
Homework Equations
The Attempt at a Solution
So I got the lagrangian to be [itex] L=R^2 \dot{\theta^2} +R^2sin^2(\theta)\dot{\phi^2}[/itex] and then used the E-L equation to find the equations of motion and the fact that [itex] 2R^2sin^2(\theta) \dot{\phi}=const=p [/itex].
Using this and substituting into the equation i get for [itex]\theta[/itex] I get:
[tex]\frac{d}{dt}(2R^2\dot{\theta})=\frac{p^2}{2R^2}cot(\theta)csc^2(\theta)[/tex]
which I then integrate using the substitution [itex]dt=d\theta / \dot{\theta}[/itex] to get:
[tex]\dot{\theta}=\frac{p}{2R^2}\sqrt{c-\frac{1}{2}sin^{-2}(\theta)}[/tex]
Where c is the integration constant. Now if I separate variables to attempt to get a solution for [itex]\theta[/itex] i get:
[tex]\int _{\theta_o}^{\theta} \frac{d\theta}{\sqrt{c-\frac{1}{2}sin^{-2}}}=\frac{tp}{2R^2}[/tex]
But i have absolutely no idea how to solve that integral. Please any pointers would be appreciated.
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