A new look at linear (non-?) symplectic ion beam optics in magnets

作者：Baumgarten C^*

来源：Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment , 2014, 735: 546-551.

DOI：10.1016/j.nima.2013.09.066

• Beam optics
• Particle accelerators
• Cyclotrons

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 the 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 the entrance and the exit of solenoid magnets can be reinterpreted 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 in analogous 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 of a matrix method.

• 出版日期2014-1-21