Simpson variational integrator for nonlinear systems: a tutorial on the Lagrange top

TL;DR

This paper introduces a Simpson quadrature-based variational integrator for nonlinear mechanical systems, demonstrating its accuracy, symplecticity, and applicability to complex systems like the Lagrange top and double pendulum.

Contribution

The paper develops a novel implicit, symplectic, fourth-order integrator using Simpson's rule for variational problems in nonlinear mechanics, with comprehensive comparisons to existing methods.

Findings

The integrator is implicit, symplectic, and fourth-order accurate.

It performs well on nonlinear systems like the Lagrange top and double pendulum.

Convergence and accuracy are validated against analytical solutions.

Abstract

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.