Mathematics and Computers in Simulation 57 (2001) 139–145 Symplectification of truncated maps for Hamiltonian systems Serge Andrianov St. Petersburg State University, Bibliotechnaja pl.2, Petrodvoretz, St. Petersburg 198904, Russia Abstract The time–displacement operator for Hamiltonian dynamical systems M(t |t 0 ) is a symplectic transformation — Lie map. Any truncated Lie map (Lie approximant) loses the very important symplectic property. In this report a method of correcting the truncated map is given. The matrix representation for desired symplectic map is calculated in the symbolic form. The symplectification conditions have the form of simple linear algebraic equations. The symbolic solutions of these equations are computed in advance and are stored in a database. This approach allows us to simplify the calculation process for the evolution of dynamical systems, in particular, for long time evolution. © 2001 Published by Elsevier Science B.V. on behalf of IMACS. Keywords: Hamiltonian maps; Lie algebraic methods; Symplectic maps; Exact symplectic conditions; Computer algebra 1. Introduction The basic levels of map modeling, generated by Hamiltonian systems, are presented in Fig. 1. It is known that in the case of Hamiltonian dynamical systems, the vector fields generate the symplectic Lie algebra Sp(2n, R) and the associated transformations generate the one-parameter symplectic groups in Sp(2n, R) — the group of symplectic diffeomorphisms acting on the phase space. In this work we discuss the last level of this diagram — the problem of correction of approximates for Lie transformations in the form of a matrix series. This procedure is very important for problems in Hamiltonian dynamical systems theory. In particular, for long time evolution of dynamical systems, for example, for multiturn beam dynamics in storage rings, colliders and so on. In practice there are two following main problems. • The first problem is related to serving qualitative properties, for example, generation of the irreducible representations, the building of basis of corresponding algebras and so on. • The second problem is related to the truncation of generating series. This truncation leads to the loss of important properties, for example, symplecticity. E-mail address: serge.andrianov@pobox.spbu.ru (S. Andrianov). 0378-4754/01/$ – see front matter © 2001 Published by Elsevier Science B.V. on behalf of IMACS. PII:S0378-4754(01)00333-0