Abstract
We present a novel variational view at Lagrangian mechanics based on the minimization of weighted inertia-energy functionals on trajectories. In particular, we introduce a family of parameter-dependent global-in-time minimization problems whose respective minimizers converge to solutions of the system of Lagrange’s equations. The interest in this approach is that of reformulating Lagrangian dynamics as a (class of) minimization problem(s) plus a limiting procedure. The theory may be extended in order to include dissipative effects thus providing a unified framework for both dissipative and nondissipative situations. In particular, it allows for a rigorous connection between these two regimes by means of Γ-convergence. Moreover, the variational principle may serve as a selection criterion in case of nonuniqueness of solutions. Finally, this variational approach can be localized on a finite time-horizon resulting in some sharper convergence statements and can be combined with time-discretization.
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Acknowledgements
U.S. and M.L. are partly supported by FP7-IDEAS-ERC-StG Grant # 200947 BioSMA. U.S. acknowledges the partial support of CNR-AVČR Grant SmartMath, and the Alexander von Humboldt Foundation. Furthermore, M.L. thanks the IMATI-CNR Pavia, where part of the work was conducted, for its kind hospitality. Finally, we gratefully acknowledge some interesting discussion with Giovanni Bellettini and Alexander Mielke which eventually motivated us to consider some minimal regularity assumptions on the potential U. The authors are also indebted to the anonymous referees for their careful reading of the manuscript.
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Communicated by R.V. Kohn.
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Liero, M., Stefanelli, U. A New Minimum Principle for Lagrangian Mechanics. J Nonlinear Sci 23, 179–204 (2013). https://doi.org/10.1007/s00332-012-9148-z
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DOI: https://doi.org/10.1007/s00332-012-9148-z