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Zbl 1221.70024
Tarasov, Vasily E.; Zaslavsky, George M.
Conservation laws and Hamilton's equations for systems with long-range interaction and memory. (English)
[J] Commun. Nonlinear Sci. Numer. Simul. 13, No. 9, 1860-1878 (2008). ISSN 1007-5704

Summary: 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.

MSC 2000:: *70H05 Hamilton's equations
37K05 Hamiltonian structures, etc.
70H33 Symmetries
70S05 Lagrangian formalism and Hamiltonian formalism

Keywords: fractional differential equations; Noether's theorem; long-range interaction; long-term memory; conservation laws

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