Conservative systems (Strogatz - Non-linear dynamics and chaos) I'm reading a book on Non-linear dynamics by Steven Strogatz and I came across an example in the Conservative Systems' chapter.
The question given is: 
Consider a particle of mass m=1 moving in a double-well potential V(x) = -$\frac{1}{2}$$\ x^2$ + $\frac{1}{4}$$\ x^4$. Find and classify all the equilibrium points for the system.
Since V(x) = -$\frac{1}{2}$$\ x^2$ + $\frac{1}{4}$$\ x^4$, we differentiate the potential to get the force and obtain F(x) = $\ x$ - $\ x^3$.
We also know that: 

$\ m$ $\ddot x$ = F(x)

Therefore, as $\ m$ = 1;

$\ddot x$ = $\ x$ - $\ x^3$.


Now, that's the part I don't understand. In the book, it says that the last equation can be rewritten as the vector field
$\dot x$ = $\ y$
$\dot y$ = $\ x$ - $\ x^3$.
Where does the 

$\ y$ and $\dot x$ = $\ y$

come from ?

It is a common trick. A new variable $y$ is introduced to transform the second order ODE into two first order ODEs, because it is easier to deal with.
