- #1
alliegator
- 3
- 0
Homework Statement
A particle of mass m moves on the surface of a paraboloidal bowl with position given by r=rcosθi+rsinθj+[itex]\frac{r^{2}}{a}[/itex]k
with a>0 constant. The particle is subject to a gravitational force F=-mgk but no other external forces.
Show that a suitable Lagrangian for the system is
L=[itex]\frac{1}{2}[/itex]m[itex](\dot{r^{2}}[/itex][itex]([/itex]1+4[itex]\frac{r^{2}}{a^{2}})[/itex][itex][/itex]+r[itex]^{2}[/itex][itex]\dot{θ^{2}}[/itex])-[itex]\frac{mgr^{2}}{a}[/itex]
Find two constants of the motion
If J[itex]^{2}[/itex]>2gp[itex]^{2}[/itex]/a where J and p are the initial values of the Jacobi function and the momentum conjugate to θ, show that in the subsequent motion the hight of the particle above to xy-plane varies between
h[itex]_{\pm}[/itex]=[itex]\frac{J\pm\sqrt{J^{2}-2gp^{2}/a}}{2mg}[/itex]
Homework Equations
Lagrangian= kinetic energy-potential energy
Kinetic energy=[itex]\frac{1}{2}[/itex]m||[itex]\dot{r}[/itex]||[itex]^{2}[/itex]
Jacobi=[itex]\sum[/itex]p[itex]_{j}[/itex]u[itex]_{j}[/itex]-L
The Attempt at a Solution
I found the Lagrangian and I found p[itex]_{θ}[/itex] to be a constant of the motion. I also found the Jacobi to be a constant of the motion because the Lagrangian has no explicit time dependence. Using the definition of the Jacobi (and also because in this case it is equal to the total energy) I found it to be [itex]\frac{1}{2}[/itex]m[itex](\dot{r^{2}}[/itex][itex]([/itex]1+4[itex]\frac{r^{2}}{a^{2}})[/itex][itex][/itex]+r[itex]^{2}[/itex][itex]\dot{θ^{2}}[/itex])+[itex]\frac{mgr^{2}}{a}[/itex]
I found p[itex]_{θ}[/itex] to be mr[itex]^{2}[/itex][itex]\dot{θ}[/itex]
I tried substituting [itex]\dot{θ}[/itex]=[itex]\frac{p_{θ}}{mr^{2}}[/itex] into the expression for the Jacobi and rearranging for [itex]\dot{r}[/itex] and then integrating to find r but I ended up with a complicated function which I couldn't integrate. I also noticed that the solution of h is in the form of the quadratic equation so I don't know if that's anything to do with it. Am I completely on the wrong track or have I done something stupid that complicates everything? Any help will be greatly appreciated.