- #1
bobred
- 173
- 0
Homework Statement
Derive the equations of motion and show that the equation of motion for [itex]\phi[/itex] implies that [itex]r^2\dot{\phi}=K[/itex] where [itex]K[/itex] is a constant
Homework Equations
Using cylindrical coordinates and [itex]z=\alpha r[/itex]
The kinetic and potential energies are
[tex]T=\dfrac{m}{2}\left[\left(1+\alpha^{2}\right)\dot{r}^{2}+r^{2}\dot{\phi}^{2}\right][/tex] and
[tex]V=mg\alpha r[/tex]
The Lagrangian is
[tex]L\left(r,\dot{r},\dot{\phi}\right)=\dfrac{m}{2}\left[\left(1+\alpha^{2}\right)\dot{r}^{2}+r^{2}\dot{\phi}^{2}\right]-mg\alpha r[/tex]
The equation of motion is
[tex]\dfrac{d}{d t}\left(\dfrac{\partial L}{\partial\dot{\mathbf{q}_{k}}}\right)-\dfrac{\partial L}{\partial\mathbf{q}_{k}}=0[/tex]
The Attempt at a Solution
The equation of motion for [itex]r[/itex]
[tex]\ddot{r}+\alpha^{2}\ddot{r}-r\dot{\phi}^{2}+g\alpha =0[/tex]
The equation of motion for [itex]\phi[/itex]
[tex]r^{2}\dot{\phi}-r^{2}\ddot{\phi} =0[/tex]
Is this correct? If so how does it imply that [itex]r^2\dot{\phi}=K[/itex] is a constant?
Thanks