Solving the n-body Problem

General Physics, a California-based corporation committed to advancing fundamental physics and computational methods, is pioneering a transformative mathematical approach to a problem that has plagued classical dynamics for centuries: the general n-body problem. Traditional analytical solutions for n greater than 3 have proven elusive, necessitating computationally intensive simulations. Our research proposes a shift from continuous differential equations in position space to a discrete, topological framework utilizing connected graph optimization coupled with advanced Fourier analysis. By representing particle interactions not as vectors in R3 but as a dynamic, weighted connected diagram, we seek to fundamentally re-engineer the problem's solvability, thereby unlocking robust, analytic solutions for complex systems, including the challenging three-body scenario.

A critical theoretical endeavor within this research is establishing the precise relation of the connected diagram, position, and the continuum. We hypothesize that the topological structure of particle interactions is isomorphic to the position space itself. This means the graph's nodes correspond to particles, and its edges represent the dynamic influence or field between them, existing on a topology that is a discrete mapping of the continuous physical space. The goal of connected graph optimization is thus to find the most stable, minimal-energy configuration of this discrete topology, which must correspond to a stable solution in the continuous position space. This requires mapping the optimization landscape of the discrete graph directly back into the manifold of classical mechanics, a sophisticated bridge between graph theory and general relativity principles.

To fully capture the dynamics within this connected framework, a rigorous analysis of exchange within the connected diagrams is necessary. This exchange represents the fundamental interaction between particles—be it gravitational, electromagnetic, or otherwise. We model these interactions using concepts analogous to group theory, specifically exploring the analytical properties of phase and conservation laws, reminiscent of an abstract SU(1) symmetry acting on the graph's edges. By focusing on the analytics of symmetric connected diagrams, General Physics aims to reduce the intractable general n-body problem to a manageable set of well-defined, closed-form solutions governed by the inherent symmetries of the interaction topology. This involves defining a new algebra on the graph space that preserves key physical invariants.

The key to unlocking the algebraic complexity of these symmetric connected diagrams lies in the application of Fourier analysis. By taking the Fourier transform of the discrete connected structure, we aim to convert complex spatial relationships into a spectrum of characteristic "frequencies" or eigenvalues, offering a simpler domain for analysis. It is here that we posit our most ambitious mathematical hypothesis: the Fourier transform of a connected diagram is a Riemann Zeta function, or a direct analytical extension thereof. Furthermore, we hypothesize that the stable energy levels or maximal nodes in this spectrum are directly analogous to the prime distribution. If validated, this suggests that the stable configurations of complex physical systems are fundamentally dictated by the distribution of prime numbers, establishing an unexpected and powerful connection between dynamic stability and fundamental number theory.

The work underway at General Physics represents a convergence of theoretical physics, graph theory, and number theory, promising to yield a unified, analytical approach to the n-body problem. By transforming the continuous problem into a discrete, symmetric, and Fourier-analyzable topological structure, we are actively solving for the analytical forms of these symmetric diagrams. Success in this endeavor will not only provide robust, long-term predictive models for celestial mechanics and particle dynamics but will also introduce entirely new classes of mathematical functions and optimization techniques. General Physics is dedicated to pursuing this frontier, confident that the secrets to cosmic stability and particle interaction are hidden within the topology of number and structure.