Numerically approximate schemes are frequently used to integrate classical equations of motions in the context of molecular dynamics simulations. The temporal integrations are sampling high-dimensional probability densities. Knowing the properties and accuracy of these numerical integration schemes is mandatory for the study of the averaging results. In this context the concept of the shadow Hamiltonian was introduced which relates the errors of the numerical integration scheme to an approximated Hamiltonian. The exact trajectory of the shadow Hamiltonian leads to the same trajectory like the slightly erroneous integration scheme. In the presentation a variational ansatz to compute the shadow Hamiltonian will be discussed.
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