Lagrangian

The Lagrangian L of a mechanical system is a single scalar function of the generalised coordinates and velocities, equal (in the simplest case) to the kinetic minus the potential energy: L = T − V. From it, all the equations of motion of the system can be derived by imposing that the integral S = ∫L dt (the action) be stationary along the true physical path — the principle of least action.

The formalism was developed by Joseph-Louis Lagrange in his 1788 Mécanique analytique as an elegant restatement of Newtonian mechanics using generalised coordinates. It replaces vectors and forces with scalars and energies, and so simplifies the treatment of constrained systems (pendulums, rolling wheels, double pendulums) where Newton's laws become messy to apply. The resulting Euler–Lagrange equations, d/dt(∂L/∂q̇) − ∂L/∂q = 0, are the workhorses of modern classical mechanics.

The Lagrangian framework is also the natural setting for Noether's theorem. When the Lagrangian is invariant under a continuous transformation, the corresponding Noether charge is conserved. In quantum field theory the same object (extended to field variables) defines the theory entirely: the Standard Model is specified by writing down its Lagrangian. It is, in a precise sense, the most compact statement of what a physical theory is.