Long-term analysis of a variational integrator for charged-particle
dynamics in a strong magnetic field
Ernst Hairer and Christian Lubich,
Abstract. The differential equations of motion of a charged particle in
a strong non-uniform magnetic field have the magnetic moment as an adiabatic invariant.
This quantity is nearly conserved over long time scales covering arbitrary negative
powers of the small parameter, which is inversely proportional to the strength of
the magnetic field. The numerical discretisation is studied for a variational
integrator that is an analogue for charged-particle dynamics of the Störmer-Verlet
method. This numerical integrator is shown to yield near-conservation of a modified
magnetic moment and a modified energy over similarly long times.
The proofs for both the continuous and the discretised equations use modulated
Fourier expansions with state-dependent frequencies and eigenvectors.
Key Words. Charged particle, magnetic field,
adiabatic invariant,
modulated Fourier expansion.