Phase Dynamics Framework: Variational Principle, Quantum Potential, Classical Limit, and Mapping to Hamilton–Jacobi and Madelung Formulations

This work presents a rigorous mathematical framework for phase dynamics in configuration space. All physical dynamics are derived from a smooth phase field Φ without assuming forces, particles, or fundamental time. The framework introduces the Euler–Phase equation, derives the unique quantum potential, and recovers classical mechanics in the limit κ → 0. It includes explicit mappings to Hamilton–Jacobi and Madelung formulations, gauge symmetry analysis, circulation quantization, and a comparison of Phase Dynamics with classical and quantum mechanics. This unified approach generalizes both CM and QM, allowing topologically nontrivial and multi-valued phase structures.