Second derivative=0; stable/unstable equilibrium?

[saraaaahhhhhh]

Homework Statement

Describe how to determine whether an equilibrium is stable or unstable when [d2U/dx2]_0 = 0

From Classical Dynamics - Ch 2 #45 - Marion Thornton

2. Homework Equations AND 3. The Attempt at a Solution

When second derivative positive, equilibrium is stable. When second derivative negative, equilibrium unstable.

When second derivative is 0, you have to do another test. I was under the impression that you could find a third derivative, fourth derivative, until you get a positive/negative value for x=0 and use the results to get stability, with the same rule as above? And if second and subsequent derivatives are 0, it's a neutral equilibrium.

The main reason I'm confused is the book had a weird solution that didn't make sense to me. They expanded the potential about the equilibrium point (how? why?), found the first derivative, and said that it's stable when n is even and unstable when n is odd. But that doesn't make sense to me. How could they make a conclusion like that if they don't know what U is?

Maybe I'm misunderstanding the question's notation - I understood it to mean "second derivative at x=0 is 0."

 

[kuruman]

What is n? Is your potential of the form U = kxⁿ (n > 0)? If so, sketch the potential for even and odd values of n and see for yourself what happens at x = 0.

 

[gabbagabbahey]

I was under the impression that you could find a third derivative, fourth derivative, until you get a positive/negative value for x=0 and use the results to get stability, with the same rule as above? And if second and subsequent derivatives are 0, it's a neutral equilibrium.

Ermmm...what the heck is a "neutral equilibrium"?

The main reason I'm confused is the book had a weird solution that didn't make sense to me. They expanded the potential about the equilibrium point (how? why?), found the first derivative, and said that it's stable when n is even and unstable when n is odd. But that doesn't make sense to me. How could they make a conclusion like that if they don't know what U is?

I'm guessing that the expansion is a Taylor expansion...you know how to do those right? What is the derivative of potential equal to physically? What should that quantity be near a stable/unstable equilibrium point?