A direct Newton method for calculus of variations

Consider m functions fi(x1,…,xn), the system of equations fi=0, i=1,…,m and the Newton iterations for this system that use the Moore–Penrose inverse of the Jacobian matrix. Under standard assumptions, the Newton iterations converge quadratically to a stationary point of the sum-of-squares ∑fi2. Approximating derivatives ẋ as differences Δx/Δt with Δt=h, we apply the Newton method to the system obtained by discretizing the integral ∫t0t1L(t,x,ẋ) dt. The approximate solutions yh of the discretized problem are shown to converge to a solution of the Euler–Lagrange boundary value problem (d/dt)∂L/∂ẋ=∂L/∂x with the degree of approximation linear in h, if the Lagrangian L(t,x,ẋ) is twice continuously differentiable. Higher continuous derivatives of L guarantee higher orders of approximation.