# Drawing the orbits in the phase space of a particle with the following properties

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

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

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?

• You seem to be missing a square in your expression for $T$ and thus a square root in your expression for $p(q)$. Oct 20, 2019 at 13:19
• Is your system specifically $1$-D? (That is, one dimensinoal configuration space, two dimensional phase space)? Oct 20, 2019 at 13:22
• 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 $q$? Oct 20, 2019 at 13:28
• Yes, @jacob1729, the space is specifically 1-D. Oct 20, 2019 at 13:36
• 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 $E$. Oct 20, 2019 at 15:02