Abstract
In this article we describe how to obtain symplectic ``slice'' maps for combined-function magnets, by using a method of generating functions. A feature of this method is that one can use an unexpanded and unsplit Hamiltonian. From such a slice map we obtain a first-order map which is symplectic at the closed orbit. We also obtain a symplectic kick map. Both results were implemented into the widely used program MAD-X to regain, in particular, the twiss parameters for the sliced model of the Proton Synchrotron at CERN. In addition, we obtain recursion equations for symplectic maps of general time-dependent Hamiltonians, which might be useful even beyond the scope of accelerator physics.
Highlights
Combined-function magnets (CFMs) are magnetic structures consisting of the superposition of a bending magnet with multipoles of higher order
IV we discuss in more detail how we implemented the formulas into MAD-X; we provide the explicit formula of the vector potential we have used and perform some numeric tests: We verify that the chromaticities in the Proton Synchrotron (PS) after the slicing process are converging to the same values obtained by Lie-transformation techniques using the Polymorphic Tracking Code (PTC), which is a symplectic integrator for tracking purposes developed by Forest [1]
Hereby the map used for computing the twiss parameters is first-order in the longitudinal step length Δs and symplectic at the origin, while the kick map used for tracking is symplectic everywhere
Summary
Combined-function magnets (CFMs) are magnetic structures consisting of the superposition of a bending magnet with multipoles of higher order. We will see how the coefficients are generated by the partial derivatives of the Hamiltonian As usual by such GF techniques, we are led to a system of two families of equations, where one of them is of implicit nature, and by construction, the map is symplectic in any order. A PYTHON demonstration code has been written to apply our result to specific Hamiltonians, for example, a Hamiltonian describing a driven 1D KVenvelope equation, and make some basic tests [5] After these general considerations we patch all things together and construct symplectic slice maps in Sec. III: We derive a map which emerges from the full unexpanded Hamiltonian by using a first-order expansion of the GF in Subsec.
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