Dual vectors are applied in Hamilton system of applied mechanics. Electric and magnetic field vectors are the dual vectors in electromagnetic field. The Hamilton system method is introduced into the analysis of electromagnetism waveguide with inhomogeneous materials. The transverse electric and magnetic fields are regarded as the dual. The basic equations are solved in Hamilton system and symplectic geometry. With the Hamilton variational principle, the symplectic semi_analytical equations are derived and preserve their symplectic structures. The given numerical example demonstrates the solution of LSE (Longitudinal Section Electric) mode in a dielectric waveguide. (
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