What does it mean to say that "an exactly equivalent way to express Hooke's Law is by the potential energy"?

Question

From Taylor, Classical Mechanics

What does that last line mean? Force and energy do not have equivalent units, so I don't understand how the author relates them. Intuitively, I can see how both the force and the potential energy oscillate, but other than that I'm not sure what mathematical connection there is.

They are equivalent. $U(x)=kx^2/2$ implies $F(x)=-kx$, and vice versa. — Cave Johnson, Oct 15 '16 at 02:40
@CaveJohnson "equivalent" and "implies" are not the same thing. — whatwhatwhat, Oct 15 '16 at 19:09

score 2 · Accepted Answer · answered Oct 15 '16 at 02:55

2

Surely you are familiar with: $-\nabla U(x)=F(x)$.

answered Oct 15 '16 at 02:55

Alex R.

32,771

$-U'(x)=-2kx \neq -kx$. – whatwhatwhat Oct 15 '16 at 21:19
@whatwhatwhat: Where did the $1/2$ go? – Alex R. Oct 17 '16 at 18:53

What does it mean to say that "an exactly equivalent way to express Hooke's Law is by the potential energy"?

1 Answers1