Chapter 19: The Principle of Least Action

The nineteenth chapter of Volume II from The Feynman Lectures on Physics introduces the profound and fundamental concept known as the Principle of Least Action. This principle offers a sophisticated alternative to Newtonian mechanics by asserting that the actual physical path a particle takes between two points in time is the one that minimizes, or makes stationary, a quantity called the Action (S). Action is mathematically defined as the integral over time of the difference between the kinetic energy (KE) and the potential energy (PE); this difference is specifically termed the Lagrangian. Determining this path requires the application of the calculus of variations, a mathematical discipline used to find the specific function or curve that provides an extremum value to an integral. The criterion for the true path is met when a small variation in the path results in a change in action (delta S) that is zero in the first order, meaning any change in action is proportional to the square of the path deviation. Through this variational method, the derived equation of motion is shown to be equivalent to Newton's Second Law (F equals ma) for systems involving conservative forces. The principle is highly generalizable, applicable not only to classical particle mechanics in three dimensions but also extended to relativistic motion, requiring a modified action integral that incorporates the rest mass and electromagnetic potentials. Furthermore, the concept forms a crucial link to quantum mechanics, where the probability amplitude for a given path is proportional to the exponential of (i times S divided by Planck's constant), indicating that the path that minimizes the classical action is the one where neighboring paths interfere constructively. The chapter concludes by applying this minimum principle to field theory, demonstrating that finding the potential function (phi) that minimizes the total electrostatic energy (U star) leads directly to the core field equation, Poisson’s equation (nabla squared phi equals negative charge density divided by epsilon naught), illustrating this technique practically by accurately approximating the capacitance of a cylindrical capacitor.