  We take into account the dynamics of three types of models of rotating
galaxies in polar coordinates in a rotating frame. Due to non-axisymmetric
potential perturbations, the angular momentum varies with time, and the kinetic
energy depends on the momenta and spatial coordinate. The existing explicit
force-gradient symplectic integrators are not applicable to such Hamiltonian
problems, but the recently extended force-gradient symplectic methods proposed
in a previous work are. Numerical comparisons show that the extended
force-gradient fourth-order symplectic method with symmetry is superior to the
standard fourth-order symplectic method but inferior to the optimized extended
force-gradient fourth-order symplectic method in accuracy. The optimized
extended algorithm with symmetry is used to explore the dynamical features of
regular and chaotic orbits in these rotating galaxy models. The gravity effects
and the degree of chaos increase with an increase of the number of the radial
terms in the series expansions of the potential. There are similar dynamical
structures of regular and chaotical orbits in the three types of models for the
same number of the radial terms in the series expansions, energy and initial
conditions.
