the Lagrangian
/ lah-GRAHN-zhee-un /
A single scalar function that encodes all the dynamics of a system. For ordinary mechanical systems it follows a strikingly simple recipe: kinetic energy minus potential energy. Feed this one function into a single universal equation and out come all the equations of motion.
It is written L = T - V, a function L(q, q_dot, t) of the generalized coordinates, the generalized velocities, and possibly time. Note carefully that it is not the total energy, which is T + V. The equations of motion come from demanding that the action, the time integral of L, be stationary. In Cartesian coordinates L = (1/2) m v^2 - V(x) yields F = -dV/dx = m a, exactly Newton's law.
Crucially, the Lagrangian is not unique. Adding the total time derivative of any function f(q, t), that is L -> L + df/dt, leaves the Euler-Lagrange equations completely unchanged; this is a kind of gauge freedom. And L = T - V is a common special case, not the definition: the general Lagrangian is whatever function makes the Euler-Lagrange equations reproduce the correct dynamics, and for velocity-dependent forces such as a charged particle in a magnetic field it is not simply T - V.
A mass on a spring has L = (1/2) m x_dot^2 - (1/2) k x^2, and the Euler-Lagrange equation gives m x_ddot = -k x, simple harmonic motion.
Kinetic minus potential, then one equation gives the motion.
Two common traps: the Lagrangian is not the energy (that is T + V), and it is defined only up to the total time derivative of a function of coordinates and time.