基于对偶变量变分原理的完整约束系统保辛算法

Symplectic algorithm for holonomic constrained systems based on the dual variable variational principle

基于对偶变量变分原理,选择积分区间两端位移为独立变量,构造了求解完整约束哈密顿动力系统的高阶保辛算法。首先,利用拉格朗日多项式对作用量中的位移、动量及拉格朗日乘子进行近似;然后,对作用量中不包含约束的积分项采用Gauss积分近似,对作用量中包含约束的积分项采用Lobatto积分近似,从而得到近似作用量;最后,在此近似作用量的基础上,利用对偶变量变分原理,将求解完整约束哈密顿动力系统问题转化为一组非线性方程组的求解。算法具有保辛性和高阶收敛性,能够在位移的插值点处高精度地满足完整约束。算法的收敛阶数及数值性质通过数值算例验证。

Based on the dual-variable variational principle,symplectic algorithms for Hamiltonian systems with holonomic constraints are derived by taking the displacements at two ends of time intervals as the independent variables.The approximation of the action integral is obtained by approximating the displacements,momentums and Lagrange multipliers by Lagrange polynomial,by implementing Gauss quadrature rule on the integral corresponding to the Hamiltonian and by implementing Lobatto quadrature rule on the integral corresponding to constraints.Based on this approximation and using the dual-variable variational principle,the problem of solving holonomic constrained Hamiltonian systems is transformed into solving a set of nonlinear equations.The resulting algorithm is symplectic,has high convergence order and can satisfy the holonomic constraints with high-precision at interpolation points of the approximate displacements.The convergence order and numerical properties of the symplectic algorithms are shown by numerical examples.