# How can one construct a phase space for the time evolution trajectories in Hamiltonian?

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?

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.