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Consider a particle of mass $m$ moving in the plane subject to a force given by $\textbf{f} (t, x, y) = (f(t, x, y), g(t, x, y))$. The coordinates $x(t)$ and $y(t)$ of its position satisfy $$ m \ddot{x} = f(t, x(t), y(t)) $$ and $$ m \ddot{y} = g(t, x(t), y(t)) $$ Recast this system as a system of four first-order differential equations

So the problem I have is just starting the question; I don't know which first-order differential equations I should be trying to find, and I can't figure out which ones it should be, or even how to begin.

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The standard way to do this is to relabel $\dot{x}$ and $\dot{y}$ as separate variables. Set $$ u = \dot{x}, \qquad v = \dot{y}, $$ then $$ m\dot{u} = f(t,x(t),y(t)) \\ m\dot{v} = g(t,x(t),y(t)). $$ The equations we used to define $u$ and $v$ are first-order, as are second two equations, and this is clearly equivalent to the original system.

Since the question doesn't ask for any properties, any first-order system you can find that is equivalent will do.

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