Concept
Optimization where the unknown is itself a function — the foundation of physical "principle of least action".
Understand it in one breath
Which path makes the first-order change of a quantity vanish under small variations? This stationarity condition can describe a minimum, maximum, or saddle point. It appears in minimal surfaces, geometrical optics, Lagrangian mechanics, and action principles in relativity, with different admissible paths and boundary conditions in each case.
At a glance
Problem | Quantity minimized | Solution | Contributor · year |
|---|---|---|---|
Brachistochrone | Descent time | Cycloid | Bernoulli, 1696 |
Shortest path between two points | Distance ∫ds | Straight line (plane) / geodesic (surface) | Euclid |
Largest area for a fixed perimeter | −area | Circle | Queen Dido (legend) |
Shape of a soap film | Area ∫dA | Minimal surface (mean curvature 0) | Plateau, 1849 |
Path of light through a varying refractive index | Stationary optical travel time | Fermat’s principle | Fermat, 1662 |
Planetary orbit | ∫(T − V)dt (action) | Newton’s equations of motion | Hamilton, 1834 |
General relativity | ∫R√−g d⁴x (constants omitted) | Einstein field equations | Einstein and Hilbert, 1915 |
The calculus of variations asks when an entire admissible path, rather than a single number, is stationary under small changes. Additional conditions decide whether a physical solution is actually a minimum.
Key formula
Key moments
Johann Bernoulli — the brachistochrone challenge
Which curve carries a sliding body between two points in the least time? Bernoulli challenged Europe’s mathematicians; the answer is a cycloid.
Newton answers overnight
Newton solved the brachistochrone problem rapidly and submitted his answer anonymously. Bernoulli reportedly recognized him “by the lion’s claw.”
The Euler–Lagrange equation
Work by Euler and later Lagrange produced a general method for variational problems, opening a path to expressing physical laws through stationary action.
Lagrange — Mécanique analytique
Lagrange rebuilt mechanics in an analytic form centered on generalized coordinates and variational principles, a language later extended throughout physics.
Modern applications
Lagrangian and Hamiltonian physics, geodesics in general relativity, loss minimization in machine learning, Fermat’s principle in optics, and economics.
Beyond MathVoyage
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