Abstract
When the Poisson matrix of Poisson system is non-constant, classical symplectic methods, such as symplectic Runge-Kutta method, generating function method, cannot preserve the Poisson structure. The non-constant Poisson structure was transformed into the symplectic structure by the nonlinear transform. Arbitrary order symplectic method was applied to the transformed Poisson system. The Euler equation of the free rigid body problem was transformed into the symplectic structure and computed by the mid-point scheme. Numerical results show the effectiveness of the nonlinear transform.
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References
Hong Jialin. A novel numerical approach to simulate nonlinear Schrödinger equations with varing coefficients [J]. Applied Mathematical Letter, 2003,16:759–765.
Feng K. Difference schemes for Hamiltonian formulation and symplectic geometry [J]. J Comp Math, 1986,4(3):279–289.
Feng K, Wu H M, Qin M Z, Wang D L. Construction of canonical difference schemes for Hamiltonian formalism via generating functions [J]. J Comp Math, 1989,7(1):71–96.
Qin Mengzhao, Li S T. A note for Lie-Poisson Hamiltonian-Jacobi equation and Lie-Poisson integrator [J]. Computers Mathematical Application, 1995,30(7):67–74.
McLachlan R I. Explicit Lie-Poisson integration and the Euler equations [J]. Phys Rev Lett, 1993,71:3043–3046.
Li S T, Qin Mengzhao. Lie-Poisson integration for rigid body dynamics [J]. Computers Mathematical Application, 1995,30 (9): 105–118.
Zhu W, Qin M. Poisson schemes for Hamiltonian system on Poisson manifolds [J]. Computers Mathematical Application, 1994,27(12):7–16.
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Communicated by !GU Yuan-xian
Project supported by the National Natural Science Foundation of China (Nos. 10401033, 90103003 and 10471145)
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Sun, Jq., Ma, Zq., Tian, Ym. et al. Symplectic structure of poisson system. Appl. Math. Mech.-Engl. Ed. 26, 1484–1490 (2005). https://doi.org/10.1007/BF03246255
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DOI: https://doi.org/10.1007/BF03246255