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

TL;DR

The paper develops a fourth-order, implicit, symplectic variational integrator by discretizing the action with Simpson quadrature and quadratic interpolation, designed for nonlinear systems with inseparable Hamiltonians. It applies the method to a chaotic nonlinear double pendulum and a three-degree-of-freedom Lagrange top, comparing against implicit midpoint and RK4, and demonstrates exact momentum preservation and strong long-time energy behavior. The results show fourth-order convergence in nutation and energy, with no artificial energy dissipation, highlighting the method's robustness for long-time simulations of nonlinear dynamics. The work provides a practical, structure-preserving tool for nonlinear mechanical systems and points to future extensions to higher-order Lobatto schemes and Lie-group formulations.

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.

Figures (14)

• Figure 1: Double pendulum affected by gravity. Two point masses $\{m_1, m_2\}$ are linked together by massless thin rigid rods of respective lengths $\{\ell_1,\ell_2\}$. Mass positions are given by the generalized coordinates $q=\left(q^1,q^2\right)$.
• Figure 2: Configuration parameters ($q$) evolution for the nonlinear double pendulum. Step size is $h=e-1$ for the symplectic integrators (Implicit midpoint and Simpson). RK4 gives the reference curve at $h=e-5$. Simpson's solutions follow the reference curve for longer simulations.
• Figure 3: Generalized momenta ($p$) evolution for the nonlinear double pendulum. Step size is $h=e-1$ for the symplectic integrators (Implicit midpoint and Simpson). RK4 gives the reference curve at $h=e-5$. Simpson's solutions follow the reference curve for longer simulations.
• Figure 4: Energy error convergence lines for the double pendulum motion. RK4 showcases a higher convergence rate than its characteristic one (of 4) because the error is high for the larger time steps. The convergence rates of the Implicit midpoint and Simpson methods are 2 and 4, respectively.
• Figure 5: Relative error on the conserved quantity $H(p,q)$ for the double pendulum. RK4's error grows with time. The symplectic methods (Implicit midpoint and Simpson) do not artificially dissipate energy.