Considering a 1D space, I'm given the graph

Potential energy as a function of position

where $U(\pmb{q})$ is the potential energy as a function of position $\pmb{q}$, and each $E_0, E_1,E_2$ is a different amount of total energy in the system. I'm first asked to draw every possible orbit in the phase space for a particle with energy $E_0$.

It is the first time a do this, and I'm certainly lost. The first thing I realize is that, given that $E_0 = K + U = \frac{p^2}{2m} + U$, I assume my graph should plot $p(q) = \sqrt{2m(E_0 - U(q))}$ for some given mass $m$. The resulting plot is

Orbit in the phase space for a particle with energy E_0

which is purely qualitative and, moreover, is nothing like an orbit. I get that the main reason for it not to be an orbit is that I'm thinking of $p$ as a function of $q$, which doesn't seem to be the correct way of thinking of an orbit in the phase space. I'm therefore pretty sure I'm doing this wrong. That's why I ask for your help.

What is wrong in my understanding, my reasoning and/or my plot?

  • $\begingroup$ You seem to be missing a square in your expression for $T$ and thus a square root in your expression for $p(q)$. $\endgroup$
    – jacob1729
    Oct 20, 2019 at 13:19
  • $\begingroup$ Is your system specifically $1$-D? (That is, one dimensinoal configuration space, two dimensional phase space)? $\endgroup$
    – jacob1729
    Oct 20, 2019 at 13:22
  • $\begingroup$ Oh, sure I am. Actually $p = \sqrt{2m(E_0 -U)}$. Now, would my graph be correct if I simply added a line symmetrical to the one I drew with respect to the axis $q$? $\endgroup$
    – Albert
    Oct 20, 2019 at 13:28
  • $\begingroup$ Yes, @jacob1729, the space is specifically 1-D. $\endgroup$
    – Albert
    Oct 20, 2019 at 13:36
  • $\begingroup$ An orbit in phase space is a curve $(p(t),q(t))$ in the $p-q$ plane (it doesn't need to be parametrised by time like this, but its probably best to think of it that way). You have identified (with a few issues) the curve corresponding to the case $E=E_0$ - the question simply wants you to draw such curves for all $E$. $\endgroup$
    – jacob1729
    Oct 20, 2019 at 15:02


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