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Hi,
I'm stuck on what I suspect is a really easy proof. I'd appreciate some rapid help!
For a trajectory with given initial position and velocity, we can always rotate the system of coordinates such that initially d\theta / dt = 0 and \theta= \pi/2 say, t = 0.
Using the Euler-Lagrange equations, show that \theta remains at \pi/2 for all times t.
Having obtained the Lagrangian (L=mv^2/2 + U, rewriting v as \dot{r}^{2}+r^{2}(\dot{\phi}^{2}sin^{2}\theta + \dot{\theta}^{2}) in spherical coordinates, and taken partial derivatives with respect to \dot{\theta} and \theta, and \phi and \dot{\phi}, I have found the following relations (I think they're correct?)
d/dt [mr^2\dot{\theta}] = mr^2\dot{\phi^2}cos(\theta)sin(\theta)
d/dt [mr^{2} \dot{\phi} sin^{2} (\theta)] = 0
(There is a third expression, from the partial derivatives wrt. r and \dot{r}, but I doubt it's relevant here.)
I'd be tempted to integrate both sides of the first to get an expression for \dot{\theta}, but that's where I become unstuck just now... I'm not sure how to integrate the RHS with respect to time, given what I know about the components. But presumably integration is going to be involved here, or the initial conditions would never come into play (?). I can deduce from the second that mr^{2} \dot{\phi} sin^{2} (\theta) = const = c, and so at t = 0 mr^{2} \dot{\phi} = c, but nothing more of interest seems to be following at the moment...
Incidentally, I take the geometrical meaning to be that the motion is confined to a plane orthogonal to L, ang. momentum is conserved.
Advice gratefully received (as soon as possible!) Many thanks :-)
I'm stuck on what I suspect is a really easy proof. I'd appreciate some rapid help!
Homework Statement
For a trajectory with given initial position and velocity, we can always rotate the system of coordinates such that initially d\theta / dt = 0 and \theta= \pi/2 say, t = 0.
Using the Euler-Lagrange equations, show that \theta remains at \pi/2 for all times t.
The Attempt at a Solution
Having obtained the Lagrangian (L=mv^2/2 + U, rewriting v as \dot{r}^{2}+r^{2}(\dot{\phi}^{2}sin^{2}\theta + \dot{\theta}^{2}) in spherical coordinates, and taken partial derivatives with respect to \dot{\theta} and \theta, and \phi and \dot{\phi}, I have found the following relations (I think they're correct?)
d/dt [mr^2\dot{\theta}] = mr^2\dot{\phi^2}cos(\theta)sin(\theta)
d/dt [mr^{2} \dot{\phi} sin^{2} (\theta)] = 0
(There is a third expression, from the partial derivatives wrt. r and \dot{r}, but I doubt it's relevant here.)
I'd be tempted to integrate both sides of the first to get an expression for \dot{\theta}, but that's where I become unstuck just now... I'm not sure how to integrate the RHS with respect to time, given what I know about the components. But presumably integration is going to be involved here, or the initial conditions would never come into play (?). I can deduce from the second that mr^{2} \dot{\phi} sin^{2} (\theta) = const = c, and so at t = 0 mr^{2} \dot{\phi} = c, but nothing more of interest seems to be following at the moment...
Incidentally, I take the geometrical meaning to be that the motion is confined to a plane orthogonal to L, ang. momentum is conserved.
Advice gratefully received (as soon as possible!) Many thanks :-)
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