R. S. Harrington (1968)
Classic study of hierarchical triple-star evolution highlighting secular inclination–eccentricity coupling relevant to Lidov–Kozai cycles.
Harrington develops a Hamiltonian, averaged (von Zeipel) treatment of hierarchical triple stars showing the semimajor axes have no secular drift, yielding dynamical stability in the ideal point-mass problem. The long-period (Lidov–Kozai-type) evolution is solved in closed form using Weierstrass elliptic functions for the quadrupole Hamiltonian, predicting large periodic exchanges between inner eccentricity and mutual inclination while the outer eccentricity is nearly constant. For near-perpendicular configurations these cycles can push the inner periastron to very small values, creating a practical “quasi-instability” once finite stellar radii, tides, or mass transfer are considered. A phase-mixing argument then suggests secular evolution biases observed triples toward lower mutual inclinations and higher inner eccentricities.
Hamiltonian perturbation theory with elimination of short-period terms (von Zeipel) followed by an analytic quadrupole-level Lidov–Kozai solution using elliptic functions, plus simple statistical phase mixing.
Celestial mechanics of hierarchical triples (Hamiltonian/Delaunay variables) and the Lidov–Kozai mechanism in secular perturbation theory.
Nice application of von Zeipel method to triple stars.
— ES