A novel, computationally efficient, approach for Hamilton's law of varying action

A new technique providing improved computational efficiency in the implementation of Hamilton's Law of Varying Action is presented. Similar to previous methods, this efficient method allows one to construct an approximate solution for an initial value problem in general dynamics without reference to the theory of differential equations associated therewith. Additionally, in the present formulation, the approximating polynomials need not satisfy the initial conditions beforehand. Using orthogonal polynomials, a set of simultaneous equations is obtained that has fewer non-zero entries compared to formulations presented in existing literature. Such a sparse system of equations may result in substantial computational economy. A harmonic oscillator and a two-degrees-of-freedom system with and without damping are studied. The numerical results using present formulations compare well with alternative solutions.