Analytic Expressions for Exponentials of Specific Hamiltonian Matrices

Hamiltonian matrices appear in a variety or problems in physics and engineering, mostly related to the time evolution of linear dynamical systems as for instance in ion beam optics. The time evolution is given by symplectic transfer matrices which are the exponentials of the corresponding Hamiltonian matrices. We describe a method to compute analytic formulas for the matrix exponentials of Hamiltonian matrices of dimensions $4\times 4$ and $6\times 6$. The method is based on the Cayley-Hamilton theorem and the Faddeev-LeVerrier method to compute the coefficients of the characteristic polynomial. The presented method is extended to the solutions of $2\,n\times 2\,n$-matrices when the roots of the characteristic polynomials are computed numerically. The main advantage of this method is a speedup for cases in which the exponential has to be computed for a number of different points in time or positions along the beamline.