Conventional numerical integration algorithms can not be used for long term stability studies of complicated nonlinear Hamiltonian systems since they do not preserve the symplectic structure of the system. Further, they can be very slow even if supercomputers are used. In this paper, we study the symplectic integration algorithm using solvable maps which is both fast and accurate and extend it to six dimensions. This extension enables single particle studies using all three degrees of freedom.
We discuss three methods to correct spherical aberration for a point to point imaging system. First, results obtained using Fermat's principle and the ray tracing method are described briefly. Next, we obtain solutions using Lie algebraic techniques. Even though one cannot always obtain analytical results using this method, it is often more powerful than the first method. The result obtained with this approach is compared and found to agree with the exact result of the first method.
In this paper, we construct an invariant metric in the space of homogeneous polynomials of a given degree ( 3). The homogeneous polynomials specify a nonlinear symplectic map which in turn represents a Hamiltonian system. By minimizing the norm constructed out of this metric as a function of system parameters, we demonstrate that the performance of a nonlinear Hamiltonian system is enhanced.
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