The phase plane and potential energy

Question

I think I have spotted a mistake in my notes, however I need help verifying my assumption:

I am given the equation:

$d^2s/dt^2=-s$

m=1 for simplicity

I have recast it at a first order system: $dv/dt=-s$

Now I must compute the potential energy. The solution gives $V(s)=s^2/2$, however I got $(1/8)*s^4$. Which is correct?

Answer 1

You multiply $\frac{d^2s}{dt^2} = -s$ through by $\frac{ds}{dt}$ and integrate (by observing the chain rule formula). You will have $\frac{1}{2}\left(\frac{ds}{dt}\right)^2 + \frac{s^2}{2} = $constant. The left most term is kinetic energy, the right term is potential energy, the sum is the constant energy of the particle.