Abstract

We address the design of time integrators for mechanical systems that are explicit in the forcing evaluations. Our starting point is the midpoint rule, either in the classical form for the vector space setting, or in the Lie form for the rotation group. By introducing discrete, concentrated impulses we can approximate the forcing impressed upon the system over the time step, and thus arrive at first-order integrators. These can then be composed to yield a second order integrator with very desirable properties: symplecticity and momentum conservation.

Highlights

  • It is well-known that the Verlet algorithm may be written as a composition of two first order algorithms, the symplectic Euler and its adjoint [2]

  • What is perhaps less known is that there is an interpretation of this composition in terms of an approximation of the midpoint rule, which is implicit in the evaluation of the forcing impulse

  • Our goal in this brief note is to point out that this mechanically inspired derivation yields the well-known second order explicit Verlet algorithm in the vector space setting, and an extremely accurate integrator for rigid body rotations when the midpoint rule is interpreted in the Lie sense

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Summary

Introduction

It is well-known that the Verlet algorithm (explicit Newmark for a certain choice of its parameters) may be written as a composition of two first order algorithms, the symplectic Euler and its adjoint [2]. What is perhaps less known is that there is an interpretation of this composition in terms of an approximation of the midpoint rule, which is implicit in the evaluation of the forcing impulse. Our goal in this brief note is to point out that this mechanically inspired derivation yields the well-known second order explicit Verlet algorithm in the vector space setting, and an extremely accurate integrator for rigid body rotations when the midpoint rule is interpreted in the Lie sense

Vector space midpoint rule approximation
Midpoint rule approximation on the rotation group
Example
Conclusions

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