Map and Lie Methods for Accelerator Physics: A Tale of Two Symmetries

Charged particle motion in E and B fields is characterized by two symmetries. First is Lorentz invariance, which is well understood. Second is the symplectic symmetry inherent in the Hamiltonian nature of the Lorentz-invariant equations that generate this motion. (The relation between initial and final conditions obtained by integration of Hamilton's equations is what is called a symplectic map. Thus the effect of any beam-line element in an accelerator is a symplectic map.) Its importance was first studied by Hamilton and Jacobi in the context of generating functions, but its full implications are still not fully appreciated or understood. Since discovery Lie algebras have been an important branch of Mathematics. Now they are also important for Physics both in Relativity and in Particle Theory. This talk aims to describe how Lie methods can also be used to exploit symplectic symmetry. A major goal of map and Lie methods is to treat linear and nonlinear behavior with equal facility. Lie tools have been developed for representing, computing, manipulating, multiplying, and applying symplectic maps. These maps can also be analyzed by normal-form methods (a nonlinear generalization of matrix diagonalization) both to determine and optimize expected accelerator performance and to motivate new designs. Additional applications include fast long-term tracking and the concept and calculation of eigen emittances.