Order-by-order symplectification of truncated Lie maps

It is known that the symplecticity property for Hamiltonian systems is lost for truncated Lie maps. In the case of long time evolution this fact can lead to spurious effects appearance and/or to real effects vanishing. In this report an order-by-order symplectification method for truncated Lie maps is described. This method is based on the matrix formalism for Lie algebraic tools. According to this formalism truncated Lie map presented as a set of two-dimensional matrices corresponded to nonlinear aberrations up to N order. Matrix elements can be evaluated using computer algebra codes and Kronecker sum and production tools. These block-matrices for accelerator lattices are elements of a corresponding database. The additional conservative conditions (in our case it is symplectic conditions) lead to linear homogeneous algebraic equations for matrix elements. Choosing basis elements (calculated using the matrix formalism algorithms) one can calculate the others. Resulting block-matrices guarantee the symplecticity of the truncated Lie map up to the N-th order. These linear relations can be calculated in advance and stored in a symbolic database. Finally, this method is applied to some practical problems of particle physics