Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study

Abstract We discuss some of the contributions, made by the authors and their research group, on the numerical methods for Hamiltonian systems. Our main concern will be the Hamiltonian algorithms, presenting the proper way for computing Hamiltonian dynamics. These algorithms are conceived developed and analysed systematically within the framework of symplectic geometry. This approach is natural since the dynamical evolution of Hamiltonian systems are exclusively symplectic transformations. The vast majority of conventional methods are non-symplectic; they inevitably imply artificial dissipation and other parasitic artifacts of non-hamiltonian distortions. The Hamiltonian algorithms are clean algorithms, free from all kinds of non-Hamiltonian pollutions. They actually give outstanding performance, far superior than the conventional non-symplectic methods, especially in the aspects of global, structural properties and long-term tracking capabilities. We give in detail some comparative numerical experimentation; in many cases the contrast is quite striking.

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