- #1

fluidistic

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## Homework Statement

Write down the Hamiltonian and its corresponding Hamilton equations for a particle in a central potential. Find the solution to the Kepler problem in this description.

## Homework Equations

Hamiltonian.

The Hamilton equations or motion equations are [itex]\dot q _i = \frac{\partial H}{\partial p_i}[/itex] and [itex]\dot p_i =-\frac{\partial H}{\partial q_i}[/itex].

## The Attempt at a Solution

I've found as Hamiltonian of the central potential: [itex]H=\frac{1}{2m} \left ( p_r ^2 + \frac{p_ \theta ^2 }{r^2} \right ) +U(r)[/itex].

I know that for the Kepler problem, U(r) becomes k/r.

So that [itex]H=\frac{1}{2m} \left ( p_r ^2 +\frac{p_ \theta ^2 }{r^2} \right ) +\frac{k}{r}[/itex].

Now I'm stuck at finding and solving the equations of motion.

Here is my attempt:

[itex]\dot r = \frac{p _r}{m}[/itex].

And [itex]\dot p_r= mr \dot \theta ^2 -\frac{k}{r^2}[/itex].

I don't know if this is good so far. Nor how to continue.