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Main Question or Discussion Point
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 firstorder change in the function is NOT zero. Maybe it's just that I don't understand what firstorder 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 firstorder change in the function is NOT zero. Maybe it's just that I don't understand what firstorder change means.)
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