Hamiltons equations for a satellite

[anubis01]

Homework Statement

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.

Homework Equations

H=T+V

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.

 

[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?

 

[anubis01]

Taking care to rearrange the equation I found H=\frac{1}{2m} [p_r²+\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}}.

 

[ideasrule]

Taking care to rearrange the equation I found H=\frac{1}{2m} [p_r²+\frac{p\theta^2}{r^2}]. Using this I got the hamilton equations, which there 4 of in this case.

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

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}}.

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

 

[anubis01]

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