ISEGORIA / MATH ENCYCLOPEDIA
Calculus of variations: paths and principles
Extremal paths, action, and geodesics.
Before you begin: Calculus and mechanics
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Euler–Lagrange
A ball is thrown up at t = 0 and caught at the same height at t = 1, with Lagrangian L = ½q̇² − gq. Bend the trial path q* + εη away from the stationary path, keeping both ends fixed, and choose the shape of the variation η. The action S(ε) has zero slope at ε = 0 for every shape, and the lower strip shows why: the Euler–Lagrange residual q̈ + g vanishes only on the stationary path.
Worked example. For L = ½q̇² − gq the equation is q̈ = −g, so with q(0) = q(1) = 0 the stationary path is q* = ½g t(1 − t), here 4t(1 − t) with g = 8.
Watch out. Stationary does not always mean a global minimum.
What is being varied?
The whole path q(t), while endpoints are held fixed.
2. Geodesics on a sphere
Two cities share a latitude. On a longitude–latitude map the parallel joining them is a straight line, but it is not the shortest route. Drag the handle to deform the path through a family that runs from the parallel through the great circle and beyond: the length is stationary, and least, at the great circle, which bulges toward the nearer pole on the map.
Worked example. At latitude 40° with a longitude gap of 100°, the parallel has length 1.337 on the unit sphere, about 8,520 km on Earth, while the great circle has length 1.254, about 7,990 km.
Watch out. A coordinate straight line need not be geometrically straight.
Why does the shortest route look curved on the map?
The map stretches east–west distances by 1/cos φ, so a path saves length by moving toward the pole, where degrees of longitude are shorter.
3. Least action
A harmonic oscillator, L = ½q̇² − ½q², travels from q(0) = 0 to q(T) = 1. Add a variation ε·sin(kπt/T) to the classical path and compare the change of action for three different k. For T < π every variation raises the action. Once T passes π, the first conjugate point, the slowest variation lowers it: the classical path is still stationary, but it is a saddle, not a minimum.
Worked example. For T = 3π/2 and k = 1 the coefficient is (3π/8)(4/9 − 1) ≈ −0.65, so the action decreases along that variation even though δS = 0.
Watch out. The phrase least action is shorthand; the action is stationary, not always least.
What stays fixed in the variation?
The endpoints and the time interval.