arXiv:2607.10652v2 Announce Type: replace Abstract: This paper presents a symplectic Hamiltonian direct discontinuous Galerkin (DDG) method for approximating wave propagation problems, including the linear and semilinear wave equations. Within an auxiliary-variable-free DG framework, we prove that the symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure. It follows that methods such as the symmetric interior penalty, BR2, and symmetric DDG (SDDG) methods admit a discrete Hamiltonian structure, whereas the Baumann--Oden and DDG methods do not possess this property. Exploiting this structure, we construct fully discrete symplectic schemes by combining the SDDG spatial discretization with symplectic time integrators. We also develop a unified treatment of periodic, Dirichlet, Neumann, and Robin boundary conditions that preserves the discrete Hamiltonian structure of the SDDG scheme. We further derive error estimates for the SDDG method applied to semilinear wave equations under these boundary conditions, showing the optimal convergence rate for the displacement and the suboptimal convergence rate for the velocity. Numerical experiments validate the theoretical convergence rates and demonstrate that the symplectic Hamiltonian DDG method achieves superior long-time energy conservation and accuracy.