Hamiltonian Perturbation Theory and Symplectic Integrators

Abstract

Perturbation theory is in fact an outgrowth of the necessity to determine the orbits with ever greater accuracy. This problem can be solved today, but in what is for the theoretician a rather disappointing way. With mod-ern calculating machines, one is now able to compute directly results even more accurately than those provided by perturbation theory. (J. Moser 1978) . . . allows computer prediction of planetary positions far more accurate (by brute computation) than anything provided by classical perturbation theory. In a very real sense, one of the most exhalted of human endeavors, going back to the priests of Babylon and before, has been taken over by the machine. (S. Sternberg 1969) In this chapter we study the long-time behaviour of symplectic integrators, combin-ing backward error analysis and the perturbation theory of integrable Hamiltonian systems.