- #1
rolotomassi
- 53
- 0
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 [itex] L = R^2 \dot{\theta^2} + R^2 sin^2{\theta} \dot{\phi^2}[/itex]
Using the Euler Lagrange equations I obtain
[itex] sin^2{\theta} \dot{\phi} = A = const \ (1) \\ \ddot{\theta} - sin{\theta}cos{\theta} \dot{\phi^2} = 0 \ (2) [/itex]
and by substituting 1 into 2
[itex] \ddot{\theta} = A^2 cos{\theta}/sin^3{\theta} [/itex]
By integrating w.r.t time and using the fact [itex] dt = d{\theta}/\dot{\theta}[/itex] and that theta and its time derivative are treated as independent coordinates i get
[itex] \dot{\theta} + A^2/ 2 \dot{\theta} sin^2{\theta} \ = c_1 [/itex]
integrating w.r.t time again i get. [Using omega instead of theta dot now.]
[itex] \theta - A^2 cot{\theta}/\omega^2 \ = t c_1 + c_2[/itex]
I can't see anything wrong but I am supposed to get this into the form
[itex] \theta(t) = arccos ( \sqrt{1 - A^2/\omega^2} cos(\omega t + \theta_0)[/itex]
and I cant. If anyone can help I would appreciate it a lot. Thanks
The lagrangian is [itex] L = R^2 \dot{\theta^2} + R^2 sin^2{\theta} \dot{\phi^2}[/itex]
Using the Euler Lagrange equations I obtain
[itex] sin^2{\theta} \dot{\phi} = A = const \ (1) \\ \ddot{\theta} - sin{\theta}cos{\theta} \dot{\phi^2} = 0 \ (2) [/itex]
and by substituting 1 into 2
[itex] \ddot{\theta} = A^2 cos{\theta}/sin^3{\theta} [/itex]
By integrating w.r.t time and using the fact [itex] dt = d{\theta}/\dot{\theta}[/itex] and that theta and its time derivative are treated as independent coordinates i get
[itex] \dot{\theta} + A^2/ 2 \dot{\theta} sin^2{\theta} \ = c_1 [/itex]
integrating w.r.t time again i get. [Using omega instead of theta dot now.]
[itex] \theta - A^2 cot{\theta}/\omega^2 \ = t c_1 + c_2[/itex]
I can't see anything wrong but I am supposed to get this into the form
[itex] \theta(t) = arccos ( \sqrt{1 - A^2/\omega^2} cos(\omega t + \theta_0)[/itex]
and I cant. If anyone can help I would appreciate it a lot. Thanks