Mathematical and Computational Methods for the n-Body Problem

The gravitational n-body problem, considered one of the most classical and enduring challenges in celestial mechanics, seeks to describe the motion of n masses interacting through Newtonian gravity. Despite its deceptively simple formulation, the problem exhibits a profound interplay between order and chaos, giving rise to an extraordinary variety of dynamical behaviours. This Thesis develops an integrated framework that combines variational analysis, advanced numerical optimisation, and rigorous computer-assisted proofs (CAPs) to detect, classify, and validate periodic solutions of the n-body problem. The work begins by reformulating the search for periodic trajectories as a variational problem, where periodic orbits correspond to critical points of an action functional defined over the space of periodic loops. A discretised Fourier representation is then introduced, reducing the infinite-dimensional variational problem to a finite-dimensional optimisation task. Within this setting, novel hybrid algorithms are implemented, most notably the Multi-Population Adaptive Inflationary Differential Evolution Algorithm and the Lattice domain-partitioning scheme, which jointly enable efficient exploration and refinement of the action landscape under prescribed symmetry constraints. Once candidate periodic orbits are identified, their structure is analysed through a combination of the discrete Morse index, intra- and trans-level distance metrics, and topological invariants such as the winding numbers. These diagnostics reveal the local and global organisation of the action landscape, distinguishing between minimum points, saddle-type solutions, and families of related orbits. The Thesis further extends classical variational methods by implementing a constructive Mountain Pass algorithm, based on the Ambrosetti–Rabinowitz and Barutello–Terracini frameworks, to uncover non-minimising critical points corresponding to new periodic trajectories. The final part of the work establishes a rigorous computer-assisted validation framework. Using interval arithmetic, validated Fourier-series operations, and fixed-point theorems, numerically computed orbits are enclosed within certified bounds, thereby providing formal proofs of existence for true periodic solutions of Newton’s equations. This CAPs methodology transforms high-precision computations into mathematically guaranteed results, uniting analytical rigour with computational discovery. Overall, the Thesis advances the state of the art by integrating variational theory, numerical optimisation, and validated computation into a single, coherent methodology. The resulting framework not only expands the catalogue of known periodic orbits in the gravitational n-body problem but also establishes a reproducible and extensible foundation for rigorous exploration of nonlinear dynamics in celestial mechanics and beyond.