Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory

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.

30 pages, LaTeX

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Cited by in corpus (3)

• A physically based connection between fractional calculus and fractal geometry
• Fractional differential equations solved by using Mellin transform
• Variational problems with fractional derivatives: Invariance conditions and Nöther's theorem