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The Lagrangians and dissipation functions are proposed for use in theelectrodynamics of the double-negative and chiral metamaterials with finiteloss. The double-negative metamaterial considered here is the wires and splitrings periodic structure, while the chiral metamaterial is the single-resonancehelical resonators array. For either system, application of Legendretransformation leads to a Hamiltonian density identical to the energy densityobtained in our previous work based on the Poynting theorem and the mechanismof the power loss. This coincidence implies the correctness of the energydensity formulas obtained before. The Lagrangian description and Hamiltonianformulation can be further developed for exploring the properties of theelementary excitations or quasiparticles in dispersive metamaterials due tolight-matter interactions.

Lagrangian description and Hamiltonian density for the electrodynamics of dispersive metamaterials