Calculus of Variations: Brachistochrone & Geodesics
Optimizing functionals — and finding curves of fastest descent and shortest path.
The calculus of variations finds extremals of functionals
$$J[y] = \int_a^b F(x, y, y')\, dx, \quad y(a), y(b) \text{ prescribed}.$$Imposing $\delta J = 0$ for arbitrary $\delta y$ vanishing at endpoints yields the Euler–Lagrange equation:
$$\frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'} = 0.$$Classical applications:
- Geodesics: minimize arc length $\int \sqrt{1 + y'^2}\, dx$ → straight lines in $\mathbb R^2$.
- Brachistochrone (Bernoulli, 1696): minimize time of descent under gravity along a curve. Solution is an inverted cycloid — not the straight line!
- Catenary: hanging chain minimizing potential energy → $\cosh$ curve.
- Plateau problem: minimal surfaces (soap films) — Euler–Lagrange in 2D becomes minimal-surface PDE.
If $F$ doesn't depend on $x$ explicitly, the Beltrami identity
$$F - y' \frac{\partial F}{\partial y'} = \text{const}$$is a first integral. For the brachistochrone $F = \sqrt{1+y'^2}/\sqrt{y}$, this directly gives the cycloid in parametric form. Lagrangian mechanics, control theory, optimal transport are all variational.