the Euler-Lagrange equation
/ OY-ler lah-GRAHN-zh /
This is the workhorse of analytical mechanics. Once you have written down the Lagrangian, this single equation grinds out the equations of motion automatically, with no free-body diagrams and no resolving of forces into components. It is the mathematical condition that the action be stationary.
For each generalized coordinate q_i it reads d/dt (partial L / partial q_dot_i) - (partial L / partial q_i) = 0. The first term is the time rate of change of the generalized momentum; the second is the generalized force. It is a second-order differential equation in q_i. In Cartesian coordinates with L = (1/2) m v^2 - V it collapses to m x_ddot = -partial V / partial x, which is Newton's second law.
You meet the very same equation throughout the calculus of variations: it is the stationarity condition for any functional built as an integral of f(y, y', x) dx, and mechanics is just one instance. Its great practical advantage is that it holds in any generalized coordinates whatsoever. When constraints or velocity-dependent forces are present the bare form is augmented, with Lagrange-multiplier terms or a Rayleigh dissipation term.
For a pendulum, L = (1/2) m l^2 theta_dot^2 + m g l cos(theta), and the Euler-Lagrange equation gives theta_ddot = -(g/l) sin(theta).
One angular coordinate in, the exact pendulum equation out.
The bare form assumes independent coordinates; with holonomic constraints you either eliminate coordinates or add a Lagrange-multiplier term on the right-hand side.