I am having trouble understanding the following diagram from pg. 18 of Arnold's book:
I am unable to see why local maxima of the potential energy correspond to unstable equilibria (and, reciprocally, why local minima correspond to stable equilibria).
My reasoning leads me to the opposite conclusion - for a local maxima $x_M$, $f'(x) > 0$ for $x < x_M$, so that things should increase to $x_M$, whereas $f'(x) < 0$ for $x > x_M$, so that things should decrease to $x_M$. However, I am being imprecise when I say "things", since I am not sure whether what I refer to is the same "thing" which is considered in the phase portrait.