Newton's 1685 resistance problem birthed variational calculus
Calculus of variations finds functions that maximize or minimize functionals—maps from functions to numbers, often integrals of a curve and its derivatives. Straight lines win shortest paths without constraints; soap films and Dirichlet principles need richer answers. Newton posed a minimal-resistance problem in 1685 and printed it in the 1687 Principia.
Newton's resistance problem was the first in the field to be both posed and correctly solved, and it stayed among the hardest attacked by these methods before 1900. Johann Bernoulli followed in 1696 with the brachistochrone, the curve of fastest descent, echoing a question Galileo had raised in 1638 without solving it by calculus. Bernoulli cracked it with the principle of least time, while Newton solved it variationally in 1697, and Jacob Bernoulli and the Marquis de l'Hôpital soon joined in.
Across the 19th century Gauss, Poisson, Ostrogradsky, Jacobi and Cauchy all refined the tests, but Karl Weierstrass's course is credited with first putting the subject on a firm foundation. Hilbert's list of problems in 1900 spurred further work, and Emmy Noether, Tonelli, Lebesgue and Hadamard contributed. Marston Morse turned the methods into what is now Morse theory, Pontryagin and others adapted them to optimal control, and Richard Bellman's dynamic programming offers an alternative route.
The core tool mirrors ordinary calculus. Where a function's peaks sit where its derivative is zero, a functional's extremes sit where its functional derivative vanishes, which leads to the Euler–Lagrange equation, usually a second-order differential equation. To derive it, one nudges a candidate curve by a small function that is zero at the endpoints, integrates by parts, and applies the fundamental lemma of the subject. The equation is necessary but not sufficient for an extremum. Strong extrema also require continuous first derivatives, so every strong extremum is weak but not the reverse.
On a curved surface the shortest paths are geodesics, possibly many. Light obeys Fermat's principle of shortest optical path, and mechanics has least action. Soap films dipped on wire frames solve Plateau's problem physically, though the mathematics allows several local minima with complicated topology.
Source: Calculus of variations
