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Hamiltonian systems are related to numerous areas of mathematics and have a lot of application branches, such as classical and quantum mechanics, statistics, optical, astronomy, molecular dynamic, plasma physics, etc. In general, the integration of these systems requires the use of geometric integrators. In this paper, we introduce a new variational approach for models which are formulated naturally as conservative systems of ODEs, most importantly Hamiltonian systems. Our variational method for Hamiltonian systems, which is proposed here, is in some sense symplectic and energy preserving. In addition to introducing the technique, we briefly indicate its most basic properties, and test its numerical performance in some simple examples

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