Adjustment of force-gradient operator in symplectic methods
  Many force-gradient explicit symplectic integration algorithms have been
designed for the Hamiltonian $H=T (\mathbf{p})+V(\mathbf{q})$ with kinetic
energy $T(\mathbf{p})=\mathbf{p}^2/2$ in the existing references. When the
force-gradient operator is appropriately adjusted as a new operator, they are
still suitable for a class of Hamiltonian problems
$H=K(\mathbf{p},\mathbf{q})+V(\mathbf{q})$ with \emph{integrable} part
$K(\mathbf{p},\mathbf{q}) = \sum_{i=1}^{n}
\sum_{j=1}^{n}a_{ij}p_ip_j+\sum_{i=1}^{n} b_ip_i$, where
$a_{ij}=a_{ij}(\textbf{q})$ and $b_i=b_i(\textbf{q})$ are functions of
coordinates $\textbf{q}$. The newly adjusted operator is not a force-gradient
operator but is similar to the momentum-version operator associated to the
potential $V$. The newly extended (or adjusted) algorithms are no longer
solvers of the original Hamiltonian, but are solvers of slightly modified
Hamiltonians. They are explicit symplectic integrators with symmetry or
time-reversibility. Numerical tests show that the standard symplectic
integrators without the new operator are generally poorer than the
corresponding extended methods with the new operator in computational
accuracies and efficiencies. The optimized methods have better accuracies than
the corresponding non-optimized counterparts. Among the tested symplectic
methods, the two extended optimized seven-stage fourth-order methods of
Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result,
one of the two optimized algorithms is used to study the orbital dynamical
features of a modified H\'{e}non-Heiles system and a spring pendulum. These
extended integrators allow for integrations in Hamiltonian problems, such as
the spiral structure in self-consistent models of rotating galaxies and the
spiral arms in galaxies.
