Higher-Dimensional Constraint Structures in Fluid and Elastic Motion

This proposal investigates whether complex behaviours in fluid and elastic systems, such as droplet splashes, elastic rebounds, vortex transitions, and soliton interactions, may reflect hidden geometric or topological constraints embedded in higher-dimensional configuration spaces. Traditional models like the Navier-Stokes and elasticity equations accurately describe many local dynamics, yet often fall short in explaining why specific morphologies emerge repeatedly or why certain transitions occur suddenly across parameter regimes. The study integrates ultra-high-speed imaging with a modular AI discovery system designed to infer governing laws and structural features directly from motion data. Four canonical physical systems (droplet impacts, elastic rebounds, vortex rings, and shallow water solitons) are selected for their ability to exhibit repeatable, structured transitions under controlled conditions. The data is encoded into geometric and algebraic representations suitable for symbolic regression, manifold learning, and topological analysis. Mathematical tools, including fourth-order partial differential equations, Clifford algebra, and bifurcation models, are applied to interpret the inferred laws and classify their properties. Differential Galois Theory is used to assess solvability and symmetry of recovered equations. The aim is to determine whether consistent mathematical structure, if present, can offer a more compact or interpretable account of observed behaviours than traditional models. If successful, this approach may suggest that some forms of complexity in physical systems are shaped not by randomness alone, but by latent constraint structures that govern motion across both fluid and elastic regimes.