  Upon entering a star system, light sails are subject to both gravitational
forces and radiation pressure, and can use both in concert to modify their
trajectory. Moreover, stars possess significant magnetic fields, and if the
sail is in any way charged, it will feel the Lorentz force also.
  We investigate the dynamics of so-called "photogravimagnetic assists" of
sailcraft around $\alpha$ Centauri A, a potential first destination en route to
Proxima Centauri (the goal of the Breakthrough Starshot program). We find that
a 10m$^2$ sail with a charge-to-mass-ratio of around 10 $\mu$C/g or higher will
need to take account of magnetic field effects during orbital maneouvres. The
magnetic field can provide an extra source of deceleration and deflection, and
allow capture onto closer orbits around a target star.
  However, flipping the sign of the sailcraft's charge can radically change
resulting trajectories, resulting in complex loop-de-loops around magnetic
field lines and essentially random ejection from the star system. Even on
well-behaved trajectories, the field can generate off-axis deflections at
$\alpha$ Centauri that, while minor, can result in very poor targetting of the
final destination (Proxima) post-assist.
  Fortunately for Breakthrough Starshot, nanosails are less prone to charging
en route than their heavier counterparts, but can still accrue relatively high
charge at both the origin and destination, when travelling at low speeds.
Photogravimagnetic assists are highly non-trivial, and require careful course
correction to mitigate against unwanted changes in trajectory.
