Variational integrators for dynamic systems with holonomic constraints are proposed based on Hamilton’s principle. The variational principle is discretized by approximating the generalized coordinates and Lagrange multipliers by Lagrange polynomials, by approximating the integrals by quadrature rules. Meanwhile, constraint points are defined in order to discrete the holonomic constraints. The functional of the variational principle is divided into two parts, i.e., the action of the unconstrained term and the constrained term and the actions of the unconstrained term and the constrained term are integrated separately using different numerical quadrature rules. The influence of interpolation points, quadrature rule and constraint points on the accuracy of the algorithms is analyzed exhaustively. Properties of the proposed algorithms are investigated using examples. Numerical results show that the proposed algorithms have arbitrary high order, satisfy the holonomic constraints with high precision and provide good performance for long-time integration.

中文翻译：

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完整力学中的变分积分

基于汉密尔顿原理，提出了具有完整约束的动态系统的变分积分器。通过用拉格朗日多项式逼近广义坐标和拉格朗日乘数，通过用正交规则逼近积分来离散化变分原理。同时，定义约束点以离散完整约束。变分原理的功能分为两部分，即无约束项和受约束项的作用以及无约束项和受约束项的作用是使用不同的数字正交规则分别集成的。详尽分析了插值点，正交规则和约束点对算法精度的影响。通过实例研究了所提出算法的性质。数值结果表明，所提出的算法具有任意的高阶，满足了完整的完整约束，为长时间的集成提供了良好的性能。