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I was wondering if conjugate momenta and position can be the variables of the phase space or not.

I have $\frac{\mathrm{d}x_1}{\mathrm{d}t}$, $\frac{\mathrm{d}x_2}{\mathrm{d}t}$, $\frac{\mathrm{d}p_1}{\mathrm{d}t}$ and $\frac{\mathrm{d}p_2}{\mathrm{d}t}$ time evolutions and derivative of $x_1$ is equal to $x_2$.

Can one consider $\frac{\mathrm{d}x_1}{\mathrm{d}t}$ and $\frac{\mathrm{d}p_1}{\mathrm{d}t}$ in two dimensional phase space while also considering $\frac{\mathrm{d}x_2}{\mathrm{d}t}$ and $\frac{\mathrm{d}p_2}{\mathrm{d}t}$ in a two dimensional phase space and then try to draw the trajectories?

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There are two things to consider :

  1. Yes conjugate momenta and positions can be variables of the phase space. But their derivatives shouldn't : variables can describe the phase space only if they arise from a canonical transformation and thus preserve the hamiltonian structure of dynamic. In other words a hamiltonian must exists for these new variables. A way to find such transformation might be to use a generating function.
  2. You can draw a subset of the phase space knowing that it is of dimension 4 and so undrawable in reality. You can for instance make a projection on the plane $(x_1, p_1)$ and another on $(x_2,p_2)$ like you the good suggestion you did but without the derivatives. This is a classic way to deal with it.
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