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Tonelli Principle: Finite Reduction and Fixed Energy Molecular Dynamics Trajectories

We propose a novel theoretical and practical alternative to the Maupertuis functional in the field of molecular dynamics: the Tonelli functional. Our aim is to adapt this technique to the study of rare events where the initial and the final states of a system are known, and we look for transition paths. We reach it with a rigorous mathematical development of the functional and an efficient numerical algorithm. We couple the Tonelli functional with an exact finite dimensional reduction, and we prove error estimates for its implementation. This is not far from a multiscale approach; indeed the reduction will help the study of the high frequencies of the systems: it can be seen as a magnifying glass for atomic trajectories. We test our techniques first on simple models in order to show the details and the main features of these new tools: an harmonic oscillator, a one dimensional double-well potential, the well-tested Mueller potential, and a short oscillatory trajectory of a 4-atom cluster are studied under...