fundamental lemma of calculus of variations

In every source I read that lemma is presentet in such an abstract form that I cannot tell whether its the same as what I have called Jacob's lemma. There is a lemma that is referred to as 'the fundamental lemma of calculus of variations' With the above introduced I have arrived at the title of this question. In my answer on physics.stackexchange I proposed the name: 'Jacob's lemma', as Jacob Bernoulli was obviously the first to present it.īut then it occured to me: chances are this lemma already has a name. In that lecture is is not stated whether Feynman learned Jacob's lemma from a source, or whether he came up with that reasoning independently.) (Well, there is one physicist who does present it: Feynman, in the Feynman lectures. In none of them this reasoning is presented. I've checked out a lot of introductions to calculus-of-variations-in-physics. To my knowledge the above reasoning by Jacob Bernoulli is not known in the physics community. If the curve as a whole is an extremum then every subsection of that curve is an extremum, down to infinitisimally short subsections. The point would move along AGFDB in a shorter time than along ACEDB, If another segment of arc CFD were traversed in a shorter time, then Then the segment of arc CED isĪmong all segments of arc with C and D as end points the segment thatĪ heavy point falling from A traverses in the shortest time. Let ACEDB be the desired curve along which a heavy point falls from A to B in the shortest time, and let C and D be two points In the diagram below the line AB represents any curve that is an extremum of some property that depends of the form of the curve. In his treatment, before discussing the brachistochrone specifically, Jacob presented a more general observation. The way that Jacob Bernoulli approached that problem is historically important. I'm posting this question here as it is purely a mathematical question.Ībout a week ago, on Physics.stackexchange, I posted an answer to the following question: Euler-Lagrange Equation: From boundary value to initial value problemĪs is recounted everywhere, the first time a variational problem was brought to the attention of the entire mathematical community of the time was the Brachistochrone problem.

Fundamental lemma of calculus of variations