A New Look at Linear (Non-?) Symplectic Ion Beam Optics in Magnets

TL;DR

This paper revisits symplectic properties of ion beam optics in magnets, clarifying the role of canonical momenta and coordinate transformations, and providing new insights into transfer matrices in solenoid and bending magnets.

Contribution

It offers a new interpretation of transfer maps in magnets as transformations between canonical and mechanical momenta, enhancing understanding of symplectic motion in beam optics.

Findings

Reinterpreted non-symplectic entrance and exit maps as canonical-mechanical momentum transformations.

Derived the transfer matrix for bending magnets from the Lorentz force in Cartesian coordinates.

Showed that transfer matrices can be viewed as products of non-symplectic matrices related to coordinate transformations.

Abstract

We take a new look at the details of symplectic motion in solenoid and bending magnets and rederive known (but not always well-known) facts. We start with a comparison of the general Lagrangian and Hamiltonian formalism of the harmonic oscillator and analyze the relation between the canonical momenta and the velocities (i.e. the first derivatives of the canonical coordinates). We show that the seemingly non-symplectic transfer maps at entrance and exit of solenoid magnets can be re-interpreted as transformations between the canonical and the mechanical momentum, which differ by the vector potential. In a second step we rederive the transfer matrix for charged particle motion in bending magnets from the Lorentz force equation in cartesic coordinates. We rediscover the geometrical and physical meaning of the local curvilinear coordinate system. We show that analog to the case of solenoids - also the transfer matrix of bending magnets can be interpreted as a symplectic product of 3 non-symplectic matrices, where the entrance and exit matrices are transformations between local cartesic and curvilinear coordinate systems. We show that these matrices are required to compare the second moment matrices of distributions obtained by numerical tracking in cartesic coordinates with those that are derived by the transfer a matrix method.