Nature is parsimonious: every physical system evolves along the unique path that makes its action stationary.
The Action ($S$) is a single scalar number — the time-integral of the Lagrangian $L = T - V$ along a path between two fixed states. Hamilton's Principle asserts that the path actually taken by a physical system is the one for which the action is stationary ($\delta S = 0$): not necessarily a minimum, but an extremum.
Applying calculus of variations to this condition yields the Euler-Lagrange equations — the equations of motion for any generalized coordinate $q_i$, free of constraint forces.
Just as a ray of light always travels the path that minimizes travel time (Fermat's Principle), a particle travels the path that makes mechanical action stationary. Both are instances of the same deep variational logic.
By demanding $\delta S = 0$ for all small path variations $\delta q_i(t)$ that vanish at the endpoints, integration by parts produces the Euler-Lagrange equations:
These second-order ODEs replace Newton's $\mathbf{F} = m\mathbf{a}$ in any coordinate system. For a free particle ($V = 0$) they give uniform motion; for a pendulum they reproduce the nonlinear swing equation — all from one scalar $L$.
Because the Lagrangian is a scalar, the Euler-Lagrange equations hold in any generalized coordinates — polar, spherical, or curvilinear — without rewriting force components. Constraints are handled automatically.
The principle extends far beyond classical mechanics. In optics, Fermat's principle minimizes optical path length. In quantum mechanics, Feynman's path integral formulation sums $e^{iS/\hbar}$ over all paths — the classical stationary path dominates as $\hbar \to 0$. In General Relativity, the Einstein–Hilbert action $S = \int R\sqrt{-g}\, d^4x$ yields Einstein's field equations.
The simulation below shows how Feynman's sum-over-paths concept works: many random paths, each weighted by a phase $e^{iS/\hbar}$, with constructive interference selecting the classical trajectory.
Every continuous symmetry of the action produces a conserved quantity. Time-translation symmetry → energy conservation. Spatial symmetry → momentum. Rotational symmetry → angular momentum. The action principle is the bridge between symmetry and conservation laws.