The symplectiness of Maxwell’s equations

The connections between Maxwell´s equations and symplectic matrix are studied. First, we analyze the continuous- time Maxwell´s differential equations in free space and verify its time evolution matrix (TEMA) is symplectic-unitary matrix for complex space or symplectic-orthogonal matrix for real space. Second, the spatial differential operators are discretized by pseudo-spectral (PS) approach with collocated grid and by finite-difference (FD) method with staggered grid. For the PS approach, the TEMA conserves symplectic-unitary property. For the FD method, the TEMA conserves symplectic-orthogonal property. Finally, symplectic integration scheme is used in the time direction. In particular, we find the symplectiness of the TEMA also can be conserved. The mathematical proofs presented are helpful for deep researching the symplectic PSTD approach and the symplectic FDTD method.