Symmetric integrable-polynomial factorization for symplectic one-turn-map tracking

It was found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree by which Lie transformations can be evaluated exactly. By utilizing symplectic integrators, an integrable polynomial factorization is developed to convert a symplectic map in the form of Dragt-Finn factorization into a product of Lie transformations associated with integrable polynomials. A small number of factorization bases of integrable polynomials enables one to use high-order symplectic integrators so that the high-order spurious terms can be greatly suppressed. A symplectic map can thus be evaluated with desired accuracy