Conservation laws and Hamilton’s equations for systems with long-range interaction and memory |
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| Institution: | 1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA;2. Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia;3. Department of Physics, New York University, 2–4 Washington Place, New York, NY 10003, USA;1. Institute of Continuous Media Mechanics, Academician Korolev Street, 1, 614013, Perm, Russia;2. Perm State University, Bukireva Street, 15, 614990, Perm, Russia;1. Institute of Mathematics, USC, RAS, Russia;2. Institute of Mathematics, University of Potsdam, Germany;1. Nanjing University of Science and Technology, Nanjing 210094, China;2. Beijing Institute of Technology, Beijing 100081, China;1. Lancaster University Management School, UK;2. Athens University of Economics and Business, Greece;3. London School of Economics, Systemic Risk Centre, UK;4. National Technical University of Athens, School of Applied Mathematics and Physics, Greece |
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| 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. |
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