New way to construct high order Hamiltonian variational integrators |
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| Authors: | Minghui Fu Kelang Lu Weihua Li S. V. Sheshenin |
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| Affiliation: | 1. School of Engineering, Sun Yat-sen University, Guangzhou 510275, China;2. Guangdong Provincial Academy of Building Research Group Company Limited, Guangzhou 510500, China;3. College of Electromechanical Engineering, Guangdong Polytechnic Normal University, Guangzhou 510635, China;4. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow 119992, Russia |
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| Abstract: | This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton’s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and momentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive. |
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| Keywords: | unconventional Hamilton's variational principle nonlinear dynamics symplectic algorithm Hamiltonian system variational integrator |
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