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We present a method for studying the secular gravitational dynamics ofhierarchical multiple systems consisting of nested binaries, which is valid foran arbitrary number of bodies and arbitrary hierarchical structure. We derivethe Hamiltonian of the system and expand it in terms of the -- assumed to be --small ratios $x_i$ of binary separations. At the lowest nontrivial expansionorder (quadrupole order, second order in $x_i$), the Hamiltonian consists ofterms which, individually, depend on binary pairs. At higher orders, inaddition to terms depending on binary pairs, we also find terms which,individually, depend on more than two binaries. In general, at order $n$ in$x_i$, individual terms depend on at most $n-1$ binaries. We explicitly derivethe Hamiltonian including all terms up and including third order in $x_i$(octupole order), and including the binary pairwise terms up and includingfifth order in $x_i$. These terms are orbit averaged, and we present a newalgorithm for efficiently solving the equations of motion. This algorithm ishighly suitable for studying the secular evolution of hierarchical systems withcomplex hierarchies, making long-term integrations of such systems feasible. Weshow that accurate results are obtained for multiplanet systems with semimajoraxis ratios as large as $\approx 0.4$, provided that high-order terms areincluded. In addition to multiplanet systems with a single star, we apply ourresults to multistar systems with multiple planets.

monthly notices of the royal astronomical society

Secular dynamics of hierarchical multiple systems composed of nested  binaries, with an arbitrary number of bodies and arbitrary hierarchical  structure. First applications to multiplanet and multistar systems