# Homework Help: Hamiltons equations for a satellite

1. Nov 21, 2009

### anubis01

1. The problem statement, all variables and given/known data
Find Hamiltons canonical equations for a satellite of mass m moving about a star of mass M at the origin. Consider motion in a plane, using polar coordinates r,$$\theta$$.

Repeat the process using cartesian coordinates x,y from the start.

2. Relevant equations
H=T+V

3. The attempt at a solution
I did both questions but i'm not exactly sure if I defined the kinetic and potential energies correctly for each case. For polar coordinates I found T=m/2 * (r$$\dot{}$$2 +r2 $$\theta$$$$\dot{}$$2) and potential energy V=V(r)

For cartessian coordinates I found the Kinetic T=m/2(x$$\dot{}$$2)
and potential energy V=$$\frac{-GMm}{\sqrt{x^2+y^2}}$$.

So if anyone can confirm if I set up the kinetic and potential energy equations correctly that would be great.

Last edited: Nov 21, 2009
2. Nov 21, 2009

### Pengwuino

That is the correct kinetic energy in the Lagrangian formulation, what did you have for your kinetic energy and how did you form the Hamiltonian?

3. Nov 21, 2009

### anubis01

Taking care to rearrange the equation I found H=$$\frac{1}{2m}$$ [pr2+$$\frac{p\theta^2}{r^2}$$]. Using this I got the hamilton equations, which there 4 of in this case. I'm sure i did that part correctly i just wasn't 100% sure about setting up the kinetic and potential energy equations correctly.

Oh and when doing the same problem in cartessian coordinates was I correct in defining the Kinetic energy T=m/2(x$$\dot{}$$2)
and potential energy V=$$\frac{-GMm}{\sqrt{x^2+y^2}}$$.

4. Nov 21, 2009

### ideasrule

Not quite. H=T+V, and you only included the kinetic energy. The potential energy, V=-GMm/r, must also be included.

The potential energy is correct, but don't forget the y component of velocity counts towards the kinetic energy!

5. Nov 22, 2009

### anubis01

Ah, that makes more sense. Thanks for the help, its much appreciated.