Understanding mathematical concept behind phase space and phase portait I'd like to expose the problem through an example, which was what made me think about it. It's a rational mechanics problem.
Consider the one dimensional Cauchy problem $\begin{cases}m\ddot{x} = F(x,\dot{x},t) \\ x(0) = x_0 \\ \dot{x}(0) = v_0 \\\end{cases}$ where $F : \mathbb{R} \times \mathbb{R} \times \mathbb{R} \longrightarrow \mathbb{R}, x,x_0,v_0 \in \mathbb{R}$, and suppose exists a potential $V$, i.e a function such that $-V'(x) = F(x)$.
With the last sentence we made sure to consider only purely positional forces and as a consequence, "purely positional potentials". From this we can introduce thec concept of phase space and phase portrait. What I don't get about this last one, is the following :
We started from the differential equation above, which has solutions that are scalar function that depends on time, i.e $x = x(t)$, but when we use $V$ (which is a function of $x$) to deal with phase portrait, seems like $x$ has become now an idependant variable, forgetting about time.
So my question is what mathematical concept is behind this type of study, what allows to do so, and why.
Any help or reference would be appreciated, phase space seems a little bit misterious to me yet.
 A: Phase space is the space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
In your case, it is evident from the structure of the solutions of your ODE that solutions x(t) are two-parameter ($x_0,v_0$) families, so they can be represented on a 2d plane as lines (trajectories) ($x(t),\dot{x}(t)$) where t parameterizes the line completely characterized by the initial point ($x_0,v_0$). So these trajectories do not cross--they correspond to a unique IC ($x_0,v_0$) and the differential equation propagates ($x(t),\dot{x}(t)$) on them to a unique future location at a later time.
At any given time, a point of the system will be a point on this 2d plane, so it will belong to such a unique trajectory. Some of these trajectories may be periodic, as for the linear oscillator, F=-x; or open lines, for F=0; or spiral to a fixed point for the damped oscillator, $F=-x-\gamma \dot x$ (a further option you somehow chose to not consider).
All higher time derivatives of x(t) beyond the first are completely determined by the ODE and its time derivatives, and hence add no further information on the state. Think of plotting ($x,\dot{x}, \ddot{x}$) instead: You still get lines in your expanded 3d space, but with no further information in characterizing your states, their past, or their future,  so as to  contrast them to other families/types of states.  Periodic trajectories will still be periodic, open ones open, those tending to a fixed point very similar.

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*The takeaway is that the two independent constants of your solution set of the 2nd order ODE motivate a graphic-interface 2d space for plotting them to characterize their salient features.

