Discontinuous Galerkin methods for the wave equation are typically defined by using the fact that the equation can be expressed as a symmetric hyperbolic system. Although these methods can be devised to be high-order accurate, they are naturally dissipative and cannot be used for long-time simulations. We show that it is possible, by taking advantage of the Hamiltonian structure of the wave equation, to overcome this drawback and obtain Discontinuous Galerkin (and other finite element) methods which maintain their original high-order accuracy while conserving the discrete space-discretization energy. We sketch the extension of this approach to other systems of equations with Hamiltonian structure including equations modeling linear and nonlinear elastic, electromagnetic and water wave equations.