A Global Instability Functional for The N-Body Gravitational Problem.

This paper introduces a novel scalar functional designed to quantify global dynamical instability in gravitational N-body systems. Instead of attempting to solve the equations of motion—known to lack general closed-form solutions for systems with three or more bodies—the work proposes an alternative approach based on the cumulative effect of pairwise gravitational interactions. The core construction begins with an interaction functional, denoted as B of t, defined as the sum of all pairwise gravitational terms, closely related to the system’s potential energy. This quantity is then integrated over time and embedded into an exponential form, yielding the instability functional, denoted as psi of t. This formulation encodes the history of interactions into a single scalar quantity. The paper establishes that psi of t is positive, bounded, and monotonically decreasing, and satisfies a simple first-order differential equation. These properties support its interpretation as a Lyapunov-type functional, providing a global measure of instability along system trajectories. A key result is that periods of strong interaction—such as close encounters between bodies—produce rapid decay in psi of t, allowing the identification of dynamically critical regimes. This makes the functional a useful qualitative diagnostic tool for detecting instability without requiring explicit trajectory solutions. Numerical experiments, including simulations of simplified systems such as the Sun–Earth–Moon configuration, demonstrate that the functional responds consistently to variations in interaction intensity. In particular, stronger gravitational interactions correspond to sharper decreases in psi of t, validating its effectiveness as an indicator of dynamical activity. Overall, the proposed functional provides a computationally simple and conceptually clear framework for analyzing instability in complex gravitational systems. While it does not replace full numerical integration, it offers a complementary method for identifying regions of strong interaction and potential instability. Future work may extend this approach to broader classes of dynamical systems and explore deeper connections with Lyapunov theory and chaos indicators.