- #1

- 69

- 2

I've attached the part from Landau & Lifschitz Mechanics where I got confused.

"The necessary condition for S(action) to have a minimum (extremum) is that these terms (called the first variation, or simply the variation, of the integral) should be zero. "

Why is this a necessary condition? If you could point me towards a definition AND explain intuitively that'd be great.

(Background: I've learned calculus and Taylor expansions-- I think this is related. But, visually, I don't see why a variation in the function at the minimum should be zero. Imagine a paraboloid. The bottom is the minimum. If you go a tiny bit in any direction (a variation) then the first-order change in the function is NOT zero. Maybe it's just that I don't understand what first-order change means.)

"The necessary condition for S(action) to have a minimum (extremum) is that these terms (called the first variation, or simply the variation, of the integral) should be zero. "

Why is this a necessary condition? If you could point me towards a definition AND explain intuitively that'd be great.

(Background: I've learned calculus and Taylor expansions-- I think this is related. But, visually, I don't see why a variation in the function at the minimum should be zero. Imagine a paraboloid. The bottom is the minimum. If you go a tiny bit in any direction (a variation) then the first-order change in the function is NOT zero. Maybe it's just that I don't understand what first-order change means.)