Hi folks,(adsbygoogle = window.adsbygoogle || []).push({});

I am a bit confused with the extreme condition used in the calculus of variations:

δ = 0

I don't understand this rule to find extreme solutions (maximum or minimum)

If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the minimum or maximum points the tangent line is horizontal. Since the angle of the tangent line is related to the derivative, it is straightforward to obtain the extreme points, we equal y' = 0. And then we check if it is a minimum, maximum or saddle point using the second derivative.

But with variations it is completely different, we have a functional, and we don't know which function minimizes or maximizes the calculation (normally a definite integral) . I have seen several approaches to find the extreme solution:

-You imagine that you know the solution, and then you express an unknown function in terms of this solution + ε *η. Where ε is a finite quantity, as small as required and η is another function. Then you use the normal differential calculus rules to find the function that minimizes-maximizes the functional, I mean you make derivative equal to zero for ε=0.

I understand what they mean but, How do I know a solution exists?

- Another reasoning talks about expanding the solution in terms of first order, second order... variations and imposing δ=0 because there are no first order changes (a similar idea to normal differential calculus but applied for the variations) for the right solution. Instead, they are second order in the right one. Or in other words "if a function is the solution of our problem, this function will have nearby many other solutions that give almost the same result".

Does anybody know any book that shows this geometrically?

The only one I found with some geometry was on Feynman Lectures on Physics but his reasoning looks really strange to me. I just copy and paste to avoid any kind of confusion on his own words:

(there is an image attached at the end of the post)

26-5 "A more precise statement of Fermat's principle"

Actually, we must make the statement of the principle of least time a little more accurately. It was not stated correctly above. It is incorrectly called the principle of least time and we have gone along with the incorrect description for convenience, but we must now see what the correct statement is. Suppose we had a mirror as in fig 1. What makes the light think it has to go to the mirror? The path of least time is clearly AB. So some people might say, "Sometimes it is a maximum time". It is not a maximum time, because certainly a curved path would take a still longer time! The correct statement is the following: a ray going in a certain particular path has the property that if we make a small change (say a one percent shift) in the ray in any manner whatever, say in the location at which it comes to the mirror, or the shape of the curve, or anything, there will be no first order change in the time; there will be only a second-order change in the time. In other words, the principle is that light takes a path such that there are many other paths nearby which take almost exactly the same time".

So he is discarding the straight line AB and saying that instead light travels from A to B passing through C... a result I can't just believe.

Honestly I think that if I have a light at point A and I turn on it pointing to the point B, the light is not going to go through C, it will take the straight line AB.

Assuming that mysteriously the light decides suddenly going to the mirror, reflecting at point C and going finally to point B.

Aren't the paths nearby the straight line AB taking almost exactly the same time to travel that AB?

If you think only about this "nearby paths" reasoning I can't even see any difference between any paths. Since all of them are straight lines with constant speed, small changes on the paths will make small changes in the times of travel, and that will be the same for AB, or for ACB, or for AEB, or for ADB, so why is that δ=0 gives us the right (according to Feynman) result ACB instead of AB or even any other one?

THanks for your time!

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# I Extremal condition in calculus of variations, geometric

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