Conservation laws and Hamilton’s equations for systems with long-range interaction and memory

2007/03/31 by Vasily E. Tarasov, George M. Zaslavsky
Mathematics · Physics and Astronomy · #Fractional Differential Equations Solutions #Mathematical and Theoretical Analysis #Model Reduction and Neural Networks #cond-mat.other #math-ph #math.DS #math.MP #nlin.CD #physics.class-ph

paper · pdf · doi:10.1016/j.cnsns.2007.05.017

30 pages, LaTeX

arxiv created 2007/04/24 · openalex publication_date 2007/05/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether's theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time-space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form.

Citations

• On the quantum theory of the electromagnetic field
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