Symplectic integration of nonlinear Hamiltonian systems

Symplectic numerical methods are commonly used in applications to Hamiltonian problems because they preserve the symplectic geometry inherent in these systems. While they do not conserve the Hamiltonian exactly, these numerical schemes do perform better than non-symplectic methods of comparable orders. This was formerly attributed to the preservation of a nearby Hamiltonian, but the existence of such a Hamiltonian has now been called into question.;Lessnick (1996) studied the limitations of symplectic methods using the simple harmonic oscillator as the main test case. In this linear example, she constructed a closed-form expression for the nearby Hamiltonian; this function was defined by a power series in h, the step-size. As the time-step was increased, this perturbed Hamiltonian progressed from a circle to an ellipse. Finally, it ceased to have physical meaning for ho ≥ 2, where o is the frequency of the oscillator.;It is natural to extend this analysis to the nonlinear domain. We used the pendulum as our test case since it serves as a model for more complicated non-linear systems. The introduction of a separatrix and homoclinic points greatly alter the dynamics. In this regime, the behavior of symplectic methods becomes increasingly complex and then chaotic as the step-size is increased. We saw the formation of "chaotic" island chains even far from the separatrix.;Furthermore, a closed-form expression for the perturbed Hamiltonian generally does not exist. This highlights the sharp difference in dynamics between linear and nonlinear systems. In our nonlinear example, the observed trajectories remain close to that of some Hamiltonian for a length of time that depends exponentially on the reciprocal of the time-step. Moreover, we observe that the Lyapunov exponent, which should be zero, also appears to scale exponentially with the reciprocal of the step-size.;While symplectic numerical methods do provide some advantages, they lack easily computable error bounds such as those available in time-adaptive methods. We derived practical error bounds by analyzing the Baker-Campbell-Hausdorff expansions of these methods. We discovered that the traditional method of halving the time-step and subtracting the difference still produces the most accurate error bound. With the low error variance in our problem, we found no substantial advantage in using a variable step-size.