- #1
keniwas
- 59
- 1
Homework Statement
Show that in classical mechanics, for circular motion, in an arbitrary central potential
V (r), one has w = dE/dL where E is the energy, L is the angular momentum, and w is the
angular frequency of motion around the orbit.
This is the basis for a problem on understanding the correspondence between classical and quantum mechanis in the limit of large circular orbits (i.e. akin to Bohr's atom).
Homework Equations
Total Energy of the system in Circular Motion
[tex]E=\frac{1}{2}m(\dot{r}^2+\frac{L^2}{mr^2})+V(\vec{r})[/tex]
Where
[tex]L=r^2\omega m[/tex]
The Attempt at a Solution
Once I got the energy in terms of angular momentum, I simply took the derivative with respect to L, however this leaves me with the angular velocity [tex]\omega[/tex] and an additional term from the potential (which I can't find any argument as to why it should be zero in this case...)
[tex]dE/dL = 0+\frac{L}{mr^2}+\frac{dV}{dL}=\omega+\frac{dV}{dL}[/tex]
I am clearly missing somthing important in the problem... is my form for the total energy of the system wrong? or is there somthing special about the central potential I am missing?
Any input is greatly appreciated.
Last edited: