The simplest system that I can think of, from classical point of view is a single particle moving in one dimension. Even for this system one needs two coordinates to describe the state, its position and its momentum. Even if the particle is at rest, we still need to specify its momentum (who knows at what time it may decide to move).

I was wondering are there systems whose state can be specific by a phase line rather than a plane. Can anyone give some examples?

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    $\begingroup$ Yes. You may have one degree of freedom if you put some constrain. For example, an harmonic oscillator in 1D has just one degree of freedom because $ m \overset{..}{x} = - k x$ gives a relationship between position and momentum. $\endgroup$ – MBolin Jan 27 '18 at 12:15
  • $\begingroup$ But for a harmonic oscillator as well, its phase space is 2-D. In fact it's a 2-D sphere with radius $\sqrt{2mE}$. Also your statement creates another doubt, do we represent momentum and position separately only when they are unconstrained ? $\endgroup$ – Ankur Singh Jan 27 '18 at 12:28
  • $\begingroup$ I understand a circle has dimension 1. If you express it in polar coordinates you have fixed radius and a movement along the angle phase $\endgroup$ – MBolin Jan 27 '18 at 12:36

If you confine a particle on a line, you confine it in a 2D subset of phase space (a plane).

If you want to confine it to a 1D subset phase space, one possibility is to confine its position or momentum to a point, i.e. to fix $p$ or $q$.

Another possibility, as suggested by Miguel Bolín in the comments, is to introduce a relation between position and momentum, $f(p,q)=0$, which will result in a curve in phase space. One example is the harmonic oscillator:

$$\frac{p^2}{2m}+\frac 1 2 {m \omega^2 q^2} - E = 0$$

whose trajectory in phase space is an ellipse.


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