Simpson变分积分器用于非线性系统：关于拉格朗日陀螺的教程
Simpson variational integrator for nonlinear systems: a tutorial on the Lagrange top
AI总结：
本文介绍了一种基于辛普森求积的四阶隐式辛普尔积分器，用于求解非线性系统，通过双摆和拉格朗日陀螺的例子验证了其精度和收敛性。
摘要：
本文提出了一种基于辛普森求积的积分方法。该积分器设计用于由变分原理导出的有限维非线性机械系统。通过二次有限元插值状态和辛普森求积对作用量进行离散化，得到离散运动方程。该方案是隐式、辛普尔和四阶准确的。所提出的积分器在两个具有不可分离哈密顿量的系统例子上与隐式中点变分积分器进行了比较。首先，非线性双摆的例子展示了该方法如何应用于多体系统。然后，利用拉格朗日陀螺的解析解作为参考，分析数值方法的精度、收敛性和精度。还提出并求解了一个简化的拉格朗日陀螺系统，并用经典四阶方法求解。其解与完整系统的辛普森解进行比较，两者之间的收敛阶与经典方法的阶数一致。
英文摘要：
This contribution presents an integration method based on the Simpson quadrature. The integrator is designed for finite-dimensional nonlinear mechanical systems that derive from variational principles. The action is discretized using quadratic finite elements interpolation of the state and Simpson's quadrature, leading to discrete motion equations. The scheme is implicit, symplectic, and fourth-order accurate. The proposed integrator is compared with the implicit midpoint variational integrator on two examples of systems with inseparable Hamiltonians. First, the example of the nonlinear double pendulum illustrates how the method can be applied to multibody systems. The analytical solution of the Lagrange top is then used as a reference to analyze accuracy, convergence, and precision of the numerical method. A reduced Lagrange top system is also proposed and solved with a classical fourth-order method. Its solution is compared with the Simpson solution of the complete system, and the convergence order of the difference between both is consistent with the order of the classical method.
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