An Exploratory Framework for Gravitation and Electrodynamics: A Lagrangian-Hamiltonian Perspective
Abstract
This paper presents an exploratory, force-based framework for gravitation and electrodynamics, motivated by a correspondence between Musakhail's aether dynamics and Einsteinian special relativity. These two perspectives are interpreted through the lens of Lagrangian-Hamiltonian duality, wherein force-based formulations (Lagrangian) and energy-based formulations (Hamiltonian) are treated as complementary descriptions of underlying physical dynamics. The aims of this work are threefold. First, to develop a force-based interpretation of gravitational interactions by examining Musakhail's force relation, F=c^2 (m-m_0 ) in parallel with the relativistic energy expression, E^2=(pc)^2+(m_0 c^2 )^2 highlighting their dual structure. Second, to introduce and analyze two exploratory electromagnetic four-vectors (J·E,E×B) and (ħω,v×B) employing an extremization principle as a heuristic tool for investigating structural analogies between dissipation, Poynting flux, and Lorentz-force dynamics. Third, to explore a minimal-scale electro-gravitational correspondence through helical flux-tube geometries and a constant-mass acceleration mechanism, suggesting possible shared features between fermionic transport and electromagnetic field configurations. The extremization procedure, in which the scalar component of a four-vector is set equal to the magnitude of its vector component, is applied heuristically to reveal formal parallels rather than to derive rigorous field equations. Within this phenomenological model, gravitational interactions are considered as collective nuclear-scale force processes, while electromagnetic energy transport is examined through Lorentz-force cancellation and Poynting-flow relations. The helical flux-tube structures provide a unifying geometric motif, with effective tension identified with Newtonian gravitational force. This work is intended as a conceptual and phenomenological exploration rather than a replacement for established relativistic field theories. Its physical relevance depends on further mathematical development and empirical validation. Several qualitative, testable consequences are outlined to motivate future theoretical refinement and experimental assessment.