I'm having a bit of trouble understanding why looking at the idea of a potential function, V(x), can help one understand a differential equation $x'$ = $f(x)$.
Given $x'$ = $f(x)$, the author (Strogatz -- Non Linear Dynamics and Chaos, Ch. 2.7) defines the following:
$x'$ = $f(x)$
$f(x)$ = -$dV/dx$
Next the author shows that V(x(t)) (or V(t)) decreases along trajectories. This comes straight from the definition that $x'$ = $-dv/dx$ and the chain rule.
Then he explains that local minima of V(x) are stable fixed points while local maxima are unstable fixed points. Which is also clear.
My main questions are:
1.) Why do we bother trying to find V(x)? It's clear that we can analyze the slope of V(x) after we find it and quickly realize how the derivate $x'$ changes at a certain equilibrium point $x^*$. But why go through the trouble of finding V(x) when we can solve for where $x'$ = 0 and then do a phase diagram analysis?
2.) The author does say that potentials are a useful way to visualize the dynamics of a DE. However, the only (small) thing I see gained by this setup is the ability to look at the graph of V(x), put an imaginary ball somewhere and then let it fall to the lowest point. This would show that wherever the ball ends up is a stable equilibrium (if such a equilibrium exists).
Is there something I'm missing on the benefits of looking at a potential function V(x)?
Thanks!