Dynamical Geometric Theory of Principal Bundle Constrained Systems: Strong Transversality Conditions and Variational Framework for Gauge Field Coupling

Abstract

This paper introduces a geometric mechanics framework for constrained systems on principal bundles. Our key innovation is the development and rigorous characterization of the strong transversality condition that establishes a fundamental connection between constraint distributions and principal bundle structures through a Lie algebra dual distribution function \(\lambda: P \to \mathfrak{g}^*\). We prove this condition is equivalent to a \(G\)-equivariant splitting of the Atiyah exact sequence. We establish the theoretical foundation by proving both existence theorems (for principal bundles satisfying \(\text{ad}^*\Omega\lambda = 0\) with parallelizable base manifolds) and uniqueness theorems (for semi-simple Lie algebras with trivial centers). From variational principles, we derive the dynamic connection equation \(\partial_t\omega = d^{\omega}\eta - \iota_{X_H}\Omega\), which reveals interaction mechanisms between gauge fields and matter fields in constrained systems. Our work introduces the Spencer cohomology mapping and proves its isomorphism with de Rham cohomology of the principal bundle, creating an exact correspondence between topological invariants and physical conservation laws. Through Zorn's lemma, we provide constructive proof of hierarchical fibrization, explaining topological mechanisms of constraint structure changes. The framework unifies gauge field theory and constrained mechanics, offering insights into non-ideal constraints, geometric phases, and topological invariants. Our analysis demonstrates that the strong transversality condition captures geometric effects from constraint-curvature coupling that standard approaches cannot detect. This theoretical framework opens new avenues for analyzing complex physical systems, applying to fluid dynamics and Yang-Mills theory.