Hamilton's principle
/ HAM-il-tun /
Newton tells you how a particle moves by pushing on it instant by instant: force equals mass times acceleration, applied at every moment. Hamilton's principle tells you the very same motion in a completely different language — globally, all at once. Of all the conceivable paths a system could take between its starting configuration at one time and its ending configuration at a later time, the path nature actually follows is the one that makes a single number, the action, stationary. The system 'chooses' its whole trajectory by an extremal principle rather than by moment-to-moment pushing.
The action S is a functional: S = integral over time of the Lagrangian L dt, where for ordinary mechanics L = (kinetic energy) minus (potential energy), T - V. Hamilton's principle says the true motion makes the first variation of S vanish, delta S = 0, among all paths with the same fixed endpoints in time. Apply the Euler-Lagrange machinery to this action and out come the equations of motion. For a single particle with L = (1/2) m v^2 - V(x), the Euler-Lagrange equation is m x'' = -dV/dx, which is exactly Newton's second law. The same recipe, with the right Lagrangian, generates the dynamics of pendulums, planets, fields, and more.
The power of Hamilton's principle is that it is coordinate-free and unifying. You write one scalar, the Lagrangian, in whatever coordinates are convenient — angles, distances, rotating frames — and the Euler-Lagrange equations automatically produce the correct equations of motion, with constraint forces and fictitious forces handled gracefully. This is why analytical mechanics, field theory, and even general relativity and the Standard Model of particle physics are all framed as 'the action is stationary'. One honest clarification: the popular name 'least action' overstates it — the action is made stationary, and is genuinely a minimum only for sufficiently short time intervals; over longer times the true path can be a saddle of the action, not a minimum. Stationary, not necessarily least, is the precise statement.
For a mass on a spring, L = (1/2) m x'^2 - (1/2) k x^2. The action is integral of L dt, and delta S = 0 gives the Euler-Lagrange equation m x'' + k x = 0 — simple harmonic motion, recovered without ever drawing a free-body diagram.
Make the action stationary and the equations of motion appear — here, simple harmonic motion.
'Least action' is a misnomer: the principle requires the action to be stationary, delta S = 0, not minimal. The true path is a minimum only over short enough times; over longer intervals it is often a saddle.