this is my first post here, so please excuse me if I do anything wrong
I'm a high school student, and my question is about phase planes, which I got interested in after learning about differential equations. In Arnold's ODE book, he introduces the notion of a phase plane here:
This phase plane seems to be different from other vector fields I've seen, which has normally axes $x$ against $t$, or $y$ against $x$. It seems to be different because it is '2 dimensional' and that this case arises for second order differential equations. Gilbert Strang's course on YouTube also introduces this, but doesn't explicitly explain why the axes of the phase plane are $y'$ and $y$. A comment in the video said that it is because 'any second order ODE can be transformed into a system of 2 first order ODEs'.
My question is that I do not understand the process of defining the coordinates $x_1$ and $x_2$ of the phase space to be the position and velocity, instead of maybe the velocity and time, or using a three dimensional vector field, with position velocity and time. I know that time is specific to this case, which is time-dependent, but is time somehow built into the solution to the differential equation? I am not sure where time fits into this and how we can interpret the phase space without time.
It would be helpful enough to point me in any further reading I would have to do to understand this, instead of a full answer if possible. I have tried to look for an explanation which relates creating phase plots to transforming second order equations to a system of first order ones, but to no avail. Thanks very much to anyone who can help!
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