Orbital Stability in Exoplanetary Systems

This report presents a self-contained, graduate-level review of the classical mechanics, analytic stability theory, numerical methods, and observational applications relevant to the long-term dynamical evolution of multi-planet exoplanetary systems. The governing equations are derived from first principles: the Newtonian N-body problem and its ten classical integrals, the two-body problem and Keplerian elements, the circular restricted three-body problem, the Jacobi integral, Lagrange equilibrium points, and the Hill sphere. The principal analytic stability frameworks are developed and critically compared — Hill stability and the Gladman criterion, Lagrange stability, Lyapunov stability, and angular-momentum deficit (AMD) stability — together with Laplace–Lagrange secular theory, the pendulum model of mean-motion resonance, and the Chirikov resonance-overlap criterion, including Wisdom's μ^(2/7) law, connecting resonance overlap to the onset of chaos. Numerical methods for long-term integration are reviewed with emphasis on the structural properties that make gigayear integrations trustworthy: Runge–Kutta methods, symplectic integrators and the shadow-Hamiltonian property, the Wisdom–Holman map, Bulirsch–Stoer, IAS15, and hybrid schemes. Illustrative calculations computed by the author (energy-error comparison of symplectic versus non-symplectic integrators; a test-particle survival map near a Jupiter-mass planet; a secular eigenmode solution; zero-velocity curves; a resonance phase portrait) demonstrate the analytic results and are explicitly labeled as pedagogical demonstrations rather than research-grade simulations. Observational methods (radial velocities, transits, transit-timing variations, direct imaging, astrometry) are reviewed in terms of the dynamical information they deliver, followed by detailed case studies of TRAPPIST-1, Kepler-11, Kepler-36, HR 8799, and Alpha Centauri, with shorter discussions of GJ 876 and other resonant chains. The report closes with a critical assessment of open problems, including the physical origin of the instability-time versus spacing relation, the reconciliation of resonant-chain survival with migration theory, and the role of machine-learned stability classifiers. All quoted system parameters originate in the cited peer-reviewed literature; no new observational data are presented. This work was prepared as an independent study.