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.‎