document: The dynamics of various physical phenomena, such as the movement of pendulums, planets, or water waves can be described in a variational framework. The development of variational principles for classical mechanics traces back to Euler, Lagrange, and Hamilton; an overview of this history can be found in [1,19]. This approach allows to express all the dynamics of a system in a single functional – the Lagrangian – which is an action integral. Hamiltonian mechanics is a reformulation of Lagrangian mechanics which provides a convenient framework to study the symmetry properties of a system. This is expressed by Noether's theorem which establishes the direct connection between the symmetry properties of Hamiltonian systems and conservation laws. When one approximates the system numerically, it is advantageous to preserve the Hamiltonian structure also at the discrete level. Given that Hamiltonian systems are abundant in nature, their numerical approximation is therefore a topic of significant relevance.