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Conservation laws and Hamilton’s equations for systems with long-range interaction and memory. (English) Zbl 1221.70024
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:
 70H05 Hamilton’s equations 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc. 70S05 Lagrangian formalism and Hamiltonian formalism