Hamiltonian of a Point particle on a frictionless plane

[S1000]

Homework Statement

Find the Hamiltonian of a Point particle on a frictionless plane with a given potential (kr^2)/2

Relevant Equations

L = T - V

dot rep time derivative

L = (m/2) * ( rdot^2 + r ^2 θdot^2) - V (r,θ)
Lagrange eq
d/dt (∂L/∂xdot) = ∂L/∂x
H = Σ pi qdoti - L

I am stuck on Question e and then how to proceed to f. I cannot seem to show this using the steps in the prior questions. My answers are:

a)

b)

c)

c) continued - and d) at the bottom of the page
d)

I am not sure where I have gone wrong, as I am not sure how to apply the relevant Hamilton's eq to the Hamiltonian in e)
I can sub in p_r into the Hamiltonian in e) however, I cannot use the p_theta 'dot' expression.
and then also

for f) how to show the energy is equal to kr_0.

 

[strangerep]

Most people here won't have the patience to wade through that ugly mess.
You'll need to learn quickly how to use Latex on this forum.
(Do a search for "latex" to find instructions.)

 

[PhDeezNutz]

I agree with your Hamiltonian Expression, to complete the problem (the last few parts) realize that $\dot{r} = 0$ if we're talking about circular orbit $r = r_0$. Which from the EL equation involving $r$ should tell you $p_r = 0$ (Which in turn tells you $\dot{p_r} = 0$) (THIS IS THE KEY PART)

This cancels out a term in your Hamiltonian leaving.

$H = \frac{p_\theta^2}{2mr^2} + \frac{kr^2}{2}$

Take the expression for $H$ and find $\dot{p_r} = - \frac{\partial H}{\partial R}$ set it equal to $0$ and solve for $\frac{p_\theta^2}{2mr^2}$ and plug it back into $H$ your answer should pop out.

Your assignment is likely turned in by now but you may find this helpful anyway.

But yes, use Latex next time, It is awesome.