It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.

翻译：众所周知，在使用变分积分器时，场论的拉格朗日描述与哈密顿描述在离散时间层面是等价的。除辛哈密顿结构外，许多物理系统在写成混合形式时也呈现出哈密顿结构。本文针对线性波传播现象，研究了拉格朗日、辛哈密顿与混合公式之间的离散等价性。在有限元满足相容性条件的前提下，拉格朗日公式与混合公式确实等价。时间离散方面，本文考虑了蛙跳格式与隐式中点法则。在处理波动问题的混合方法中，原始变量（例如力学中的位移或电磁学中的磁势）并非问题的未知量，而是通过其时间导数后验重构。当这一重构通过梯形法则实现时，上述时间离散方法将导向等价公式。