Calculus of Variations and Fermat's Principle

References:

Wilkipedia on {\em Calculus of Variations}

Fermat's Principle

We are accustomed in Physics classes to seeing the laws of Physics stated in terms of forces, fields, and differential equations. As we have seen, such a formulation of a problem can provide a powerful means of obtaining the solution.

However, all of the familiar equations of elementary physics-Newton's laws, Hamilton's equations, Maxwell's equations, and Schrödinger's, to name a few--can equivalently be expressed in variational form--that is, as a statement that the state of the system (or the trajectory of a particle) is the one that minimizes some global property, usually expressed as an integral. For example,

• Hamilton's principle states that the motion $x(t)$ of a particle from time $t_1$ to time $t_2$ is the one that minimizes the action $\int_{t_1}^{t_2}\,L\,dt$, where $L = T - V$ is the Lagrangian of the system. It is easily shown that this principle is equivalent to the Lagrangian formulation of classical mechanics.
• The quantum-mechanical wave function $\psi$ of a system with Hamiltonian $H$ is the one that minimizes the quantity $\langle\psi\vert H\psi\rangle = \int\,dx\,\psi^\ast H\psi$ subject to the constraint that $\langle\psi\vert\psi\rangle$ is constant. This is equivalent to Schrödinger's equation.

There are many more examples. Often, a variational formulation is the most convenient or direct way of expressing the problem. The calculus of variations was developed precisely to handle such problems. Here we will adopt a Monte-Carlo approach to their solution. We will focus on two examples: Fermat's principle (described below) and energy minimization (next). In each case, we are seeking the configuration of a system that minimizes some integral property of the system.