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 systems. We derive the governing equations 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. We then develop the principal analytic stability frameworks—Hill stability and the Gladman criterion, Lagrange stability, Lyapunov stability, angular-momentum deficit (AMD) stability, Laplace–Lagrange secular theory, the pendulum model of mean-motion resonance, and the Chirikov resonance-overlap criterion including Wisdom’s µ2/7 law—and connect them to chaos indicators (Lyapunov exponents, MEGNO, frequency-map analysis). A detailed treatment of numerical integration compares Runge–Kutta, Bulirsch–Stoer, symplectic (leapfrog, Wisdom–Holman, high-order compositions), and modern high-accuracy and hybrid schemes, with worked demonstrations of energy-error behavior and an illustrative test-particle stability map computed for this report. We review how transit photometry, radial velocities, transit-timing variations, direct imaging, and astrometry inform dynamical models, and we analyze five case studies in depth: TRAPPIST-1, Kepler-11, Kepler-36, HR 8799, and α Centauri, with shorter discussions of GJ 876 and resonant-chain systems. We close 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 numerical results generated for this report are explicitly identified as illustrative computations by the author and are distinguished throughout from published observational data.