On the Numerical Solution of the Euler–Lagrange Equations

The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the constraint manifold. The algorithm guarantees that the constraints are automatically satisfied and requires a minimal number of evaluations of second-order derivative terms. In fact, second-order derivatives are involved only through the second fundamental tensor of the constraint manifold. This tensor may be computed either explicitly when second derivatives are available or via an approximation procedure with excellent accuracy. Examples are given along with comparisons with state-of-the-art software.