# Phase space of a harmonic oscillator and a pendulum

• DannyJ108
In summary: I would appreciate any help I can get regarding this, it would be extremely helpful.Thanks fellow physicists.
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.

DannyJ108 said:
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.

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

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##.

Last edited:
anuttarasammyak said:
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?

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

## 1. What is the phase space of a harmonic oscillator?

The phase space of a harmonic oscillator is a mathematical representation of all possible states of the oscillator. It is a two-dimensional space where the position and velocity of the oscillator are plotted against each other.

## 2. How is the phase space of a harmonic oscillator different from a pendulum?

While both the harmonic oscillator and the pendulum have a phase space that represents their states, the main difference is in their equations of motion. The harmonic oscillator follows a linear equation, while the pendulum follows a non-linear equation due to the presence of gravity.

## 3. How does the energy of a system affect its phase space?

The energy of a system directly affects the shape and size of its phase space. A higher energy level will result in a larger phase space, while a lower energy level will result in a smaller phase space. This is because the energy determines the amplitude and frequency of the oscillations, which in turn affect the position and velocity of the oscillator in the phase space.

## 4. Can the phase space of a harmonic oscillator or a pendulum be visualized?

Yes, the phase space of a harmonic oscillator or a pendulum can be visualized using a phase portrait. This is a plot of the position and velocity of the oscillator over time, which gives a visual representation of the oscillator's motion in the phase space.

## 5. How does the phase space of a damped harmonic oscillator differ from an undamped one?

The phase space of a damped harmonic oscillator is different from an undamped one because the presence of damping results in a decrease in the amplitude of the oscillations over time. This leads to a spiral-like trajectory in the phase space, instead of a closed loop as seen in an undamped oscillator.

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