I'm posting this question here as it is purely a mathematical question.

About a week ago, on Physics.stackexchange, I posted an answer to the following question: Euler-Lagrange Equation: From boundary value to initial value problem

As 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.

The way that Jacob Bernoulli approached that problem is historically important. In his treatment, before discussing the brachistochrone specifically, Jacob presented a more general observation.

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.


Lemma. 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 on it as close together as we like. Then the segment of arc CED is among all segments of arc with C and D as end points the segment that a heavy point falling from A traverses in the shortest time. Indeed, if another segment of arc CFD were traversed in a shorter time, then the point would move along AGFDB in a shorter time than along ACEDB, which is contrary to our supposition.

In my words:
If the curve as a whole is an extremum then every subsection of that curve is an extremum, down to infinitisimally short subsections.

The following context is important:
To my knowledge the above reasoning by Jacob Bernoulli is not known in the physics community. I've checked out a lot of introductions to calculus-of-variations-in-physics. In none of them this reasoning is presented.

(Well, there is one physicist who does present it: Feynman, in the Feynman lectures. 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.)

In my answer on physics.stackexchange I proposed the name: "Jacob's lemma", as Jacob Bernoulli was obviously the first to present it.

But then it occured to me: chances are this lemma already has a name.

With the above introduced I have arrived at the title of this question.
There is a lemma that is referred to as 'the fundamental lemma of calculus of variations'

Here is my problem: 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.

My background is physics; my mathematical experience is whatever is necessary to do mathematical physics (taking derivatives, integration, etc.), and in all of those situations my thinking is in terms of the physics that is represented by the mathematical expression.

In order for an answer to be helpful to me the description of the fundamental lemma needs to be translated to a verbal/visual form, to make it accessible to visual thinking.

Some more context:
There is the following derivation of the Euler-Lagrange equation by Preetum Nakkiran. Preetum Nakkiran points out: "This result is often proven using integration by parts – but the equation expresses a local condition, and should be derivable using local reasoning." In Preetum Nakkiran's derivation the steps in the reasoning have a physical interpretation.

In physics the most important application of the Euler-Lagrange equation is in the context of Hamilton's stationary action.
Obviously it is very important that the mathematics of Hamilton's stationary action is well understood.

A couple of months ago I posted an answer on physics.stackexchange, giving a visual demonstration of how Hamilton's stationary action is mathematically equivalent to the newtonian formalism. (There are of course many proofs, but the existing proofs are in a form such that the steps of the proof do not have a physical interpretation.)

The answer I posted on physics.stackexchange is an abbreviated version of an article that is on my website (educational website). The title of the article on my website is 'least action visualized

The source of the quoted material:
"A SOURCE BOOK IN MATHEMATICS, 1200-1800", edited by D. J. Struik.

  • $\begingroup$ According to Wikipedia, the “fundamental lemma of calculus of variations” doesn’t really look like it has much to do with Jacob Bernoulli’s reasoning. Wikipedia seems to suggest the keywords “dynamic programming” and “optimal substructure” (naturally much more recent). $\endgroup$
    – Aphelli
    Jul 12, 2020 at 22:52


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