Euler-Lagrange equations

The Euler-Lagrange equation converts the global variational condition δS = 0 into a local differential equation for every generalised coordinate: d/dt(∂L/∂q̇ᵢ) = ∂L/∂qᵢ. It follows from integration by parts and the fundamental lemma of the calculus of variations.

For a Lagrangian L = T − V with T = ½mv² and V = V(q), the Euler-Lagrange equation reproduces exactly Newton's second law F = ma. The power of the formulation is that it works unchanged in any coordinate system: polar, spherical, rotating, constrained. Choose the coordinates that make the constraints vanish, crank through the equation, and the dynamics fall out.

Derived by Euler and Lagrange independently in the mid-18th century. Lagrange's 1788 Mécanique analytique brought the formulation to its modern state.