- #1

The book says in one dimension F=-dU/dr(p.185). From this, the system is stable at distance a when U'(a)=0 and U''(a)>0 where U is differentiated with respect to r.(p.217)

My question arises from the instance of a pendulum where a massless rigid rod of length l and a mass m are considered(p.217).

The book says the system is stable if the pendulum hangs downward and unstable if it hangs upward, both vertically.

I also agree the result, but I can't understand the reasoning in the book since the derivatives of potential energy in the reasoning are with respect to the angle theta, not r.

I have tried to solve it with the chain rule, indeed, we have dU/dr=dU/d(theta) × d(theta)/dr and d²U/dr²=dU²/d(theta)² × (d(theta)/dr)² + dU/d(theta)×d²(theta)/dr². But stuck due to the trouble with identifying d(theta)/dr.

To try again, at this time, I made a guess that generalises the stability of the system to higher dimension. The system is also stable at the point of minimum potential energy, by investigating small displacement along every axis. Is this OK?