Write the following equation as a first order system, then find the potential energy and sketch it

Question

I'm asked to write the following as a first order system:

$$\ddot s=s-s^3$$

In order to do this I have set $v=\dot s.$ This then gives me $$\dot v=s-s^3.$$

Is this correct?

Next I'm asked to compute the potential energy and sketch it. I'm given $m=1.$ To do this I have done the following:

$$F=ma=\ddot s=s-s^3=\frac{-dV}{ds}=\frac{-d}{ds}\bigg(\frac{s^4}{4}-\frac{s^2}{2}\bigg).$$

So $V=\frac{s^4}{4}-\frac{s^2}{2}.$

So the total energy is $$E=\frac{1}{2}\dot s^2+\frac{s^4}{4}-\frac{s^2}{2}$$

How do I sketch this?

Answer 1

As a system you are almost there.

$$\begin{pmatrix} \dot{s} \\ \dot{v} \end{pmatrix} = \begin{pmatrix} v \\ s - s^3 \end{pmatrix}$$

Answer 2

Yes, the two new equations you wrote together comprise the first-order system.