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Fractional generalized Hamiltonian mechanics

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Abstract

In this paper, we present a new fractional theory of dynamics, i.e., the dynamics of generalized Hamiltonian system with fractional derivatives (fractional generalized Hamiltonian mechanics). Based on the definition of Riemann–Liouville fractional derivatives, the fractional generalized Hamiltonian equations are obtained, the gradient representation and second-order gradient representation of the fractional generalized Hamiltonian system are studied, and then the conditions on which the system can be considered as a gradient system and a second-order gradient system are given, respectively. By using the method and results of this paper, the conditions under which a fractional generalized Hamiltonian equation can be reduced to a generalized Hamiltonian equation, a fractional Hamiltonian equation and a Hamiltonian equation are given, respectively, and then the existing conditions and their form of gradient equation and second-order gradient equation are investigated. Finally, an example of a fractional dynamical system is given to illustrate the method and results of the application.

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Authors and Affiliations

  1. Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou, 310018, People’s Republic of China

    Lin Li & Shao-Kai Luo

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  1. Lin Li
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  2. Shao-Kai Luo
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Correspondence to Shao-Kai Luo.

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Li, L., Luo, SK. Fractional generalized Hamiltonian mechanics. Acta Mech 224, 1757–1771 (2013). https://doi.org/10.1007/s00707-013-0826-1

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  • DOI: https://doi.org/10.1007/s00707-013-0826-1

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