Non-differentiable variational principles

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:307

Article

Cresson, J

关键词: non-differentiable functions;    variational principle;    least-action principle;    Schrodinger's equation;   

DOI  :  10.1016/j.jmaa.2004.10.006

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

Free