Non-differentiable variational principles |
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Authors: | Jacky Cresson |
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Affiliation: | Université de Franche-Comté, Equipe de Mathématiques de Besançon, CNRS-UMR 6623, Théorie des nombres et algèbre, 16 route de gray, 25030 Besançon cedex, France |
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Abstract: | We develop a calculus of variations for functionals which are defined on a set of non-differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non-differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler-Lagrange equation. We finally prove that solutions of the Schrödinger equation can be obtained as extremals of a non-differentiable variational principle, leading to an extended Hamilton's principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space-time. |
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Keywords: | Non-differentiable functions Variational principle Least-action principle Schrö dinger's equation |
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