The Euler-Lagrange equation of classical mechanics was discovered during attempts to find a curve for which the time taken by a frictionless particle sliding down it under uniform gravity to its lowest point is independent of its starting point.
Did you know that Wikiversity can never have enough examples and solved problems? Wikiversity does not always have to re-create the wheel. Do you know a good website with mathematics example problems? Wikiversity needs branches of the Hunter-gatherers project for each academic subject area. Please make note of good online mathematics resources at Hunter-gatherers/Mathematics and link to these resources from the appropriate mathematics pages of Wikiversity.
Example. Did you know that there is a wiki for mathematics example problems? See www.exampleproblems.com. How do participants in the mathematics area of Wikipedia feel about linking to .com websites?