How do I write something as a first order system, or vector field, on the phase plane

I have been asked to write d2s/dt^2 = −s as first order system, or vector field, on the phase plane.

and then compute its potential, I have no idea how to do either.

I have looked it up, I am currently on this website Pauls Online Notes : Differential Equations - Systems of Differential Equations and still don't understand. It's not in my course notes either.

Please help if you have any idea, I have an exam in a couple of week and NEED to know this.

Re: How do I write something as a first order system, or vector field, on the phase p

would there not be a constant of integration

Re: How do I write something as a first order system, or vector field, on the phase p

would it not be $\displaystyle \frac{1}{2} \dot{s}^2+\frac{s^2}{2}= constant=E$ for some E

to compute its potential

compare with $\displaystyle \ddot{s}=-\nabla(s)$

so

$\displaystyle \nabla(s)=s$

so

$\displaystyle V(s)=\frac{s^2}{2}+K$

then choose K to be equal to zero