Calculus of Variations

🌀 Calculus of Variations

While standard calculus deals with finding the extremum of a function of numbers, the calculus of variations deals with finding the extremum of a functional—a mapping from a space of functions to the real numbers.

🟢 1. The Functional

A functional $J$ often takes the form of an integral: $J[y] = \int_{x_1}^{x_2} L(x, y(x), y'(x)) \, dx$ The goal is to find a function $y(x)$ that minimizes or maximizes $J$ .

Classical Examples

The Shortest Path: Finding the curve that minimizes distance between two points (a straight line).
Brachistochrone Problem: Finding the curve along which a particle slides under gravity in the shortest time.
Minimal Surface Area: Finding the shape of a soap film stretched between two loops.

🟡 2. The Euler-Lagrange Equation

The fundamental result of the calculus of variations is that if $y(x)$ is an extremum of $J$ , it must satisfy the Euler-Lagrange Equation: $\frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) = 0$

Derivation Sketch

We consider a small variation $\eta(x)$ around the optimal $y(x)$ and require that the first variation of $J$ be zero: $\delta J = \int_{x_1}^{x_2} \left( \frac{\partial L}{\partial y} \eta + \frac{\partial L}{\partial y'} \eta' \right) dx = 0$ Using integration by parts on the second term leads to the EL equation.

🔴 3. Applications in Physics and Control

Principle of Least Action

In classical mechanics, the path taken by a system is the one that minimizes the Action ( $S$ ), which is the integral of the Lagrangian ( $T - V$ ): $S = \int L(t, q, \dot{q}) \, dt$ This allows the derivation of Newton’s laws from a more fundamental principle.

Optimal Control Theory

In engineering, we want to find a control input $u(t)$ that minimizes a cost functional (e.g., fuel consumption) while moving a system (e.g., a rocket) from state A to state B.

Hamiltonian: $H = L + \lambda^T f$ , leading to Pontryagin’s Minimum Principle.