Fractional Variational Framework for Systems With Higher‐Order Elements. Part I: Lagrangian Construction and Equations of Motion

This work presents a physics‐consistent methodology for constructing Lagrangians of electrical and electrochemical systems composed of higher‐order elements arranged according to Chua's periodic table. By extending Riewe's fractional‐order treatment of dissipation beyond classical resistive elements, the proposed framework enables the incorporation of both nonlinear dynamics and memory effects into a single variational formulation. Conditions are identified under which a Lagrangian can be constructed even in the presence of nonlinear and dissipative components. The approach is illustrated using a broad class of systems that are highly relevant to modern electronic and functional materials, including memristive devices, fractional elements, electrochemical interfaces, supercapacitors, lithium‐ion batteries, and biological tissues. These systems are characterized by nonlocal charge transport, fractional impedance behavior, and energy storage in distributed structures, which cannot be adequately described using conventional integer‐order models. By embedding fractional‐order elements into a variational framework, the proposed method provides a physically interpretable foundation for distinguishing energy storage from dissipation in complex material systems exhibiting memory and fractional dynamics. The results constitute a first step toward a Hamiltonian‐based energy analysis of advanced electronic materials with distributed parameters, opening new perspectives for predictive modeling of electrochemical energy storage devices and bio‐electronic systems.