Modification of logarithmic Hamiltonians and application of explicit symplectic-like integrators

We modify the logarithmic Hamiltonian of Mikkola and Tanikawa by adding a constant (or function) to both the kinetic energy and the force function. Explicit symplectic algorithms are available when the logarithmic Hamiltonian has two separable parts of coordinates and momenta. However, they are not if the logarithmic Hamiltonian is inseparable. Fortunately, they are still efficient by manipulating the logarithmic Hamiltonian as a new separable Hamiltonian in an extended phase space. In fact, they belong to symplectic-like integrators. The choice of mixing maps affects the performance of the considered symplectic-like integrators. It is shown that two maps about sequent permutations of coordinates and momenta are inferior to a map with mid-point permutations in some cases. The choice of the constant (or function) added also exerts some influence on the performance of the algorithms. As a result, with the help of the mid-point permutations and a suitable choice for the constant (or function) included, the logarithmic Hamiltonian methods bring an increase in accuracy compared to the non-logarithmic ones, particularly for highly eccentric orbits.