THE NECESSARY CONDITION FOR A RUNGE-KUTTA SCHEME TO BE SYMPLECTIC FORHAMILTONIAN-SYSTEMS

In this paper, beginning with the expansion of a Runge-Kutta scheme (L emma 1), the author introduces the so-called Y-T-type array for each i rreducible one (Lemmas 2-3). As a study of the properties of Y-T-type arrays, the author establishes Lemmas 4-5, and then the key Theorem 1. The interesting Theorem 2 is concluded from Lemma 1. The last result (Theorem 3) is obtained from Theorem 2 and Theorem 1; thus the author solves the hard problem on the necessary condition for a Runge-Kutta s cheme to be symplectic for Hamiltonian systems. Hence, a sufficient an d necessary condition is given. Finally, Remark 1 is given as a necess ary supplement of Theorem 3.