  We study the canonical variables based numerical schemes of a hybrid model
with kinetic ions and mass-less electrons. Two equivalent formulations of the
hybrid model are presented with the vector potentials in different gauges and
the distribution functions depending on canonical momentum (not velocity),
which constitutes a pair of canonical variables with the position variable.
Particle-in-cell methods are used for the distribution functions, and the
vector potentials are discretized by the finite element methods in the
framework of finite element exterior calculus. Splitting methods are used for
the time discretizations. It is illustrated that the second formulation is
numerically superior and the schemes constructed based on the anti-symmetric
bracket proposed have better conservation properties and lower noise, although
the filters can be used to improve the schemes of the first formulation.
