# Phase Plot for Harmonic Oscillator

This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the $sin(\omega t)$:

However now that I am thinking about it, the phase is actually only the $\phi=2\pi f t$ part and therefore the real plot should look like this:

What is the problem with the second plot? Shouldn't the second one be the phase plot?

Note: Both of the plots are for the oscillator with $f=0.25$ and $t\in[0,10]$.

• I'd argue that neither of those look like a phase diagram for the harmonic oscillator. Could you give the actual formulae you are using in the above two plots? – Kyle Kanos Mar 12 '14 at 12:32
• Your $\phi$ should be equal to $\omega t$ or $2\pi ft$, but not $2\pi\omega t$. And there are different definitions of phase. – fibonatic Mar 12 '14 at 13:53

The problem you are having is that there are two different uses of the word "phase". One is, as you point out, the argument of the $sin$ function. The other use of the word is the ordered pair $(x, p)$, that is, coordinate and momentum. $(x,p)$ specifies a point in a two-dimensional "phase space" where one axis is $x$ and the other $p$. The state of a one-dimensional mechanical system is completely specified by its point in phase space. The location of the point in phase space changes with time as the system evolves.
One can also specify the state differently, using the coordinate and it time derivative: $(x,\dot{x})$. This two-dimensional space is called "state space".
So a phase plot is a plot of $p(t)$ vs. $x(t)$. That is, the points $(x,p)$ for every instant of time.
• @Cupitor: So is saying that matter has 4 distinct phases also absurd? The fact is physics reuses words and letters/symbols to represent different things. Here, $\phi$ reflects the offset from a purely sinusoidal function while phase space describes a set of all possible states a system could possess. – Kyle Kanos Mar 12 '14 at 14:06
• Well, this link from mathworld (via @Cupitor) says phase space is $(x,\dot{x})$, but in all my experience in physics, phase space is $(x,p)$ where $p=\frac{\partial\mathcal{L}}{\partial x}$ is the momentum conjugate to $x$. For a harmonic oscillator consisting of a mass and a spring, $p=m\dot{x}$, so in this case $p$ and $\dot{x}$ differ only by a factor of $m$, the mass. – garyp Mar 13 '14 at 16:44