The Lagrangian Method

An interesting numerical method for dynamics foregoes the differential equations of motion working, instead, directly with the action. Since the object of a numerical evolution is to find a sequence of points that approximates the motion of the system, the question comes down to formulating the least action principle with repsect to a path defined piecewise by a given interpolation. Assuming the interpolation is fixed, this reduces to an optimization problem over the point sequence, itself. Solving this yields a series of iteration equations which can then be used to numerically evolve the system. Since the interpolation is fixed, this produces a suboptimal solution, but one optimal with respect to the constraint. The method may be thought of as the classical and computational version of the path integral formulation. The process is illustrated with application to the simple harmonic oscillator and the Kepler problem. A notable feature in the latter application is that the iteration involves logarithms, rather than polynomials!