EXPLICITLY INTEGRABLE POLYNOMIAL HAMILTONIANS AND EVALUATION OF LIE TRANSFORMATIONS

We have found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree, with which each associat ed polynomial Hamiltonian is integrable, and the associated Lie transf ormation can be evaluated exactly. An integrable polynomial factorizat ion has thus been developed to convert a sympletic map in the form of a Dragt-Finn factorization into a product of exactly evaluable Lie tra nsformations associated with integrable polynomials. Having a small nu mber of factorization bases of integrable polynomials enables one to c onsider a factorization with the use of high-order symplectic integrat ors so that a symplectic map can always be evaluated with the desired accuracy. The results are significant for studying the long-term stabi lity of beams in accelerators.