Phase space of a harmonic oscillator and a pendulum

[DannyJ108]

Homework Statement

Draw the phase space in new canonical coordinates and the phase space of a single varying mass pendulum

Relevant Equations

H=p^2/2m+(1/2)kx^2

Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this.

Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian as a function of them (which is done correctly, I think).

I have to obtain the phase space in (Q,P) of this oscillator, but I have no idea how to. My question is how do I proceed to do so?

Also, another exercise I have is to obtain the solution to the equations of motion for a single varying-mass pendulum assuming small oscillations (sinx = x). I've proceeded introducing the rocket equation into the eq. of motion I've got, but I don't know if this is the correct way to do so. I have to obtain the phase space of this pendulum too, but no idea how to.

I would appreciate any help I can get regarding this, it would be extremely helpful.

Thanks fellow physicists.

 

[anuttarasammyak]

Homework Statement:: Draw the phase space in new canonical coordinates and the phase space of a single varying mass pendulum
Relevant Equations:: H=p^2/2m+(1/2)kx^2

Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P).

Here I cannot find what Q and P are as function of normal coordinate q and momentum p.

 

[DannyJ108]

Here I cannot find what Q and P are as function of normal coordinate q and momentum p.

True. Sorry about that.

$q = \sqrt{2P}A^{-1/4}\ sinQ$ $p = \sqrt{2P}A^{1/4}\ cosQ$ Also, I have to apply this transformation to the Hamiltonian using $A = Km$ The resulting Hamiltonian I get is: $H = \omega P$ being $\omega = \sqrt \frac K m$ But as I said, I don't know how to represent the phase space in Q,P

 

[anuttarasammyak]

In mathematics xy plane can be divided by cells of area $dx\ dy$ or in other way by $r\ dr\ d\phi$.
Drawing r$\phi$ rectangle plane, $\phi$ is limited in [0,2$\pi$] and r is half limited in [0,+$\infty$]. Area of rectangle cells $dr\ d\phi$ are not just the product but proportional to the value of r.

In phase space here p, q corresponds to the former, P,Q corresponds to the latter where P to r and Q to $\phi$.

 

[DannyJ108]

In mathematics xy plane can be divided by cells of area $dx\ dy$ or in other way by $r\ dr\ d\phi$.
Drawing r$\phi$ rectangle plane, $\phi$ is limited in [0,2$\pi$] and r is half limited in [0,+$\infty$]. Area of rectangle cells $dr\ d\phi$ are not just the product but proportional to the value of r.

In phase space here p, q corresponds to the former, P,Q corresponds to the latter where P to r and Q to $\phi$.

Okay, but how do I proceed to get the equations of phase space? What variables do I need to group in order to know what to draw?
Also, would you know if the procedure I mentioned for the pendulum are correct?

 

[anuttarasammyak]

What do you mean by "the equations of phase space" ? Show them for original p and q to get the precise idea SVP.