Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 307, Issue 1, Pages 48-64Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2004.10.006
Keywords
non-differentiable functions; variational principle; least-action principle; Schrodinger's equation
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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 Schrodinger 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. © 2004 Elsevier Inc. All rights reserved.
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