ABSTRACT
A mathematical model is given for the discrete motion of a particle in the presence of a conservative, polynomial potential energy function; that is, a discrete nonlinear oscillator. The model is in the form of a pair of first order difference equations. These difference equations are not derived by discretizing a differential equation. The model is derived directly form physics principles. The reason for doing this is that it is well known that numerical approximations to differential equations do not necessarily simultaneously preserve conservation laws and symplectic structures [3]. The forward orbit of the mathematical model given here conserves energy. In other words, there exists a Hamiltonian function for this system. The system is explicit for polynomial potential functions of degree 4 and is implicit for degree 5 and greater. The explicit system is a symplectic mapping.
