A direct Newton method for calculus of variations
The paper deals with a direct Newton method for calculus of variations. Here the system of equations: \(f_i(x_1,x_2,\dots, x_n)= 0\), \(i= 1,2,\dots, m\) and Newton iterations for this system that use the Moore-Penrose inverse of the Jacobian matrix, is considered. Under standard assumptions, the Newton interactions converge quadratically to a stationary point of the sum-of-squares \(\sum_i f^2_i\). Approximating the derivatives \(\dot x\) as differences \((\Delta x/\Delta t)\) with \(\Delta t= h\), the Newton method to the system obtained by discretizing the integral \(\int^{t_1}_{t_0} (t,x,\dot x) dt\), is applied. Main result: The approximate solutions \(y_{h_n}\) of the discretized problem are shown to converge to a solution of the Euler-Lagrange boundary value problem \((d/dt)\partial L/\partial\dot x=\partial L/\partial x\) with the degree of approximation linear in \(h\), if the Lagrangian \(L(t,x,\dot x)\) is twice continuously differentiable. Higher continuous derivatives of \(L\) guarantee higher-order of approximation.
scientific article; zbMATH DE number 1743069