Non Linear Dynamics Study of the CLIC Damping Rings Using Sympletic Integrators

Symplectic integrators are extensively used for single particle tracking in accelerators. A symplectic integrator scheme, which involves only positive steps, for perturbed Hamiltonians of the form H = A + B, where both A and B are integrable, was proposed [1], and all order of such integrators were derived [2]. A particular member of the family, namely the SABA2 integrator has already proved to be very efficient for the numerical study of astronomical [2], as well as accelerator models [3, 4]. An orbit of a Hamiltonian system of N degrees of freedom H( p, q), with p = (p1, . . . , pN), q = (q1, . . . , qN ) and qi, pi, i = 1, . . . , N the generalized coordinates and momenta respectively, is defined by a vector x(t) = (x1(t), . . . , x2N (t)), with xi = pi, xi+N = qi, i = 1, . . . , N . Defining the Poisson bracket of functions f( p, q), g( p, q) by: