A Variational Formulation of the Rotor Curvature Field

This paper develops a variational formulation for the Rotor Curvature Field ψ(x,t), the field variable describing circulating curvature within the vacuum manifold in the Rotor Dynamics Framework. A minimal Lagrangian density consistent with translational symmetry, rotational symmetry, and global phase invariance is constructed for the complex curvature field. Application of the Euler–Lagrange equations yields a nonlinear field equation governing the evolution of curvature amplitude and phase. The resulting dynamics describe spatial redistribution of curvature, vacuum response to curvature intensity, and nonlinear self-interaction that stabilizes localized rotor structures. Conservation laws associated with phase symmetry and spacetime translations are derived through Noether’s theorem, producing conserved curvature charge, energy, and momentum. The variational formulation provides a compact mathematical foundation for curvature dynamics and connects localized particle structures, propagating radiation modes, and collective coherent states within a single field-theoretic framework. Higher-order derivative terms are discussed as extensions describing additional curvature feedback processes.