- #1
rolotomassi
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I have a free particle moving on the surface of a sphere of fixed radius R. Gravity is ignored and m/2 is left out since its constant.
The lagrangian is L = R^2 \dot{\theta^2} + R^2 sin^2{\theta} \dot{\phi^2}
Using the Euler Lagrange equations I obtain
sin^2{\theta} \dot{\phi} = A = const \ (1) \\ \ddot{\theta} - sin{\theta}cos{\theta} \dot{\phi^2} = 0 \ (2)
and by substituting 1 into 2
\ddot{\theta} = A^2 cos{\theta}/sin^3{\theta}
By integrating w.r.t time and using the fact dt = d{\theta}/\dot{\theta} and that theta and its time derivative are treated as independent coordinates i get
\dot{\theta} + A^2/ 2 \dot{\theta} sin^2{\theta} \ = c_1
integrating w.r.t time again i get. [Using omega instead of theta dot now.]
\theta - A^2 cot{\theta}/\omega^2 \ = t c_1 + c_2
I can't see anything wrong but I am supposed to get this into the form
\theta(t) = arccos ( \sqrt{1 - A^2/\omega^2} cos(\omega t + \theta_0)
and I cant. If anyone can help I would appreciate it a lot. Thanks
The lagrangian is L = R^2 \dot{\theta^2} + R^2 sin^2{\theta} \dot{\phi^2}
Using the Euler Lagrange equations I obtain
sin^2{\theta} \dot{\phi} = A = const \ (1) \\ \ddot{\theta} - sin{\theta}cos{\theta} \dot{\phi^2} = 0 \ (2)
and by substituting 1 into 2
\ddot{\theta} = A^2 cos{\theta}/sin^3{\theta}
By integrating w.r.t time and using the fact dt = d{\theta}/\dot{\theta} and that theta and its time derivative are treated as independent coordinates i get
\dot{\theta} + A^2/ 2 \dot{\theta} sin^2{\theta} \ = c_1
integrating w.r.t time again i get. [Using omega instead of theta dot now.]
\theta - A^2 cot{\theta}/\omega^2 \ = t c_1 + c_2
I can't see anything wrong but I am supposed to get this into the form
\theta(t) = arccos ( \sqrt{1 - A^2/\omega^2} cos(\omega t + \theta_0)
and I cant. If anyone can help I would appreciate it a lot. Thanks
