Spin-orbit gravity equations extended from two bodies to N bodies

Gravitational wave models for compact-object mergers rely on post-Newtonian approximations that have mostly been worked out for two-body systems. Extending two independent effective-field-theory derivations to N spinning bodies, the authors find that all new physics beyond the binary case reduces to three-body interaction diagrams — no more complicated structures appear at next-to-leading order. Both routes yield Hamiltonians that match the established ADM result of Hartung and Steinhoff up to a canonical transformation, providing a consistency check across methods. The framework is relevant for modeling hierarchical triple systems and globular-cluster environments where multi-body spin effects may matter.