  Symplectic integrators are the tool of choice for many researchers studying
dynamical systems because of their good long-term energy conservation
properties. For systems with a dominant central mass, symplectic integrators
are also highly efficient. In this chapter, I describe the theory of symplectic
integrators in terms of Lie series. I show how conventional symplectic
algorithms have been adapted for use in binary-star systems to study problems
such as the dynamical stability of multi-planet systems and the accretion of
planets from planetesimals. This is achieved by devising new coordinate systems
for the wide-binary and close-binary cases separately. I show how the
performance of these algorithms can be improved at little extra cost using
symplectic correctors. Finally, I discuss drawbacks of these algorithms, in
particular in dealing with close encounters with one or both members of the
binary, and the prospects for overcoming these problems.
