Implementing Hamilton's Law of Varying Action with Shifted Legendre Polynomials

The "Law of Varying Action," originally published by Hamilton in 1834, was recently employed by Bailey to generate power series solutions characterizing the motions of dynamical systems. Furthermore, this method enables the approximating series to be constructed in a simple and direct way, without using the associated differential equations of motion. The implementation of this new technique is presented here using, instead of ordinary power series, series with Shifted Legendre Polynomials as basis functions. The theory and applications are described in detail and numerical results are presented for two problems—(i) a lightly damped harmonic oscillator with one degree of freedom, (ii) two periodic orbits of the restricted three-body problem. The far superior performance of the Shifted Legendre Polynomial series is confirmed in both examples.