SIMULATION OF ONE-DIMENSIONAL NOISY HAMILTONIAN-SYSTEMS AND THEIR APPLICATION TO PARTICLE STORAGE-RINGS

Stochastic differential equations are investigated which reduce in the deterministic limit to the canonical equations of motion of a Hamilto nian system with one degree of freedom. For example, stochastic differ ential equations of this type describe synchrotron oscillations of par ticles in storage rings under the influence of external fluctuating el ectromagnetic fields. In the first part of the article new numerical i ntegration algorithms are proposed which take into account the symplec tic structure of the deterministic Hamiltonian system. It is demonstra ted that in the case of small white noise the algorithm is more effici ent than conventional schemes for the integration of stochastic differ ential equations. In the second part the algorithms are applied to syn chrotron oscillations. Analytical approximations for the expectation v alue of the squared longitudinal phase difference between the particle and the reference particle on the design orbit are derived. These app roximations are tested by comparison with numerical results which are obtained by use of the symplectic integration algorithms.