The Lagrangian is a function L(q, q̇) of a system's generalized coordinates and their time derivatives, defined as kinetic energy minus potential energy, whose time integral over a trajectory is called the action. The principle of least action states that a physical trajectory is the one for which the action is stationary (its first variation vanishes), and applying the calculus of variations to this condition yields the Euler-Lagrange equation, which reproduces the classical equations of motion. In quantum field theory this classical mechanics formalism is extended and is foundational because it is manifestly Lorentz-covariant, unifies classical and quantum descriptions, underlies canonical quantization, connects symmetries to conserved quantities via Noether's theorem, and supports the path-integral formulation and renormalization.
The Lagrangian is a function L(q, q̇) of a system's generalized coordinates and their time derivatives, defined as kinetic energy minus potential energy, whose time integral over a trajectory is calle…