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Lie group foliations: Dynamical systems and integrators

Robert I. McLachlan, Matthew Perlmutter, G. Reinout W. Quispel

TL;DR

This work addresses preserving foliations of phase space under numerical integration, focusing on foliations generated by Lie group actions. It shows that foliate vector fields often decompose as a tangential part along group orbits and an invariant part, enabling the design of foliation-preserving integrators such as projection methods, discrete-gradient schemes, and Lie group–based Runge-Kutta methods. Key contributions include conditions guaranteeing the $X_{\mathrm{tan}} + X_{\mathrm{inv}}$ decomposition (e.g., isometric actions, invariant complements, free and proper actions, and slices) and concrete integrators for both simple foliations and Lie-group foliations (including Lie-Euler and RKMK schemes) with explicit matrix-group examples. The framework advances geometric integration for systems with symmetry or first integrals, providing robust, structure-preserving tools for long-time simulations.

Abstract

Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action of a Lie group has a particularly nice structure, which we study in detail, giving conditions under which all foliate vector fields can be written as the sum of a vector field tangent to the orbits and a vector field invariant under the group action. This allows the application of many techniques of geometric integration, including splitting methods and Lie group integrators.

Lie group foliations: Dynamical systems and integrators

TL;DR

This work addresses preserving foliations of phase space under numerical integration, focusing on foliations generated by Lie group actions. It shows that foliate vector fields often decompose as a tangential part along group orbits and an invariant part, enabling the design of foliation-preserving integrators such as projection methods, discrete-gradient schemes, and Lie group–based Runge-Kutta methods. Key contributions include conditions guaranteeing the decomposition (e.g., isometric actions, invariant complements, free and proper actions, and slices) and concrete integrators for both simple foliations and Lie-group foliations (including Lie-Euler and RKMK schemes) with explicit matrix-group examples. The framework advances geometric integration for systems with symmetry or first integrals, providing robust, structure-preserving tools for long-time simulations.

Abstract

Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action of a Lie group has a particularly nice structure, which we study in detail, giving conditions under which all foliate vector fields can be written as the sum of a vector field tangent to the orbits and a vector field invariant under the group action. This allows the application of many techniques of geometric integration, including splitting methods and Lie group integrators.

Paper Structure

This paper contains 5 sections, 10 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Integrators for simple foliations
  3. Lie group foliations
  4. Integrators for Lie group foliate vector fields
  5. Conclusions

Key Result

Theorem 1

molino${\mathfrak{X}}_F$ and $\mathfrak{X}_{\rm tan}$ form Lie algebras. $\mathfrak{X}_{\rm tan}$ is an ideal in ${\mathfrak{X}}_F$. A vector field $X$ is foliate with respect to $F$ if and only if $[X,Y]\in \mathfrak{X}_{\rm tan}$ for all $Y\in \mathfrak{X}_{\rm tan}$.

Figures (2)

  • Figure 1: Three foliate vector fields. Top: a general foliate vector field, $\dot r = r(1-r^2)$, $\dot\theta = -r\cos\theta$ (Eq. (\ref{['eq:polar1']})). Middle: a system with an integral, $\dot r = 0$, $\dot\theta = -r\cos\theta$. Bottom: a system with a continuous symmetry, $\dot r = r(1-r^2)$, $\dot\theta = -(1+r^2/5)$. All three flows map circles to circles. The dots mark times 0, 0.5, and 1.
  • Figure 2: Foliate vs. nonfoliate integrators. The ODE of Eq. (1) is integrated by the (foliate) Lie-Euler method Eq. (\ref{['eq:lie-euler']}) (top), and the (nonfoliate) Euler method (bottom). The 20 initial conditions lie on a circle of radius 2, and four time steps of 0.1 are shown. In the nonfoliate integrator, the final values do not lie on the reference circle shown. Note that the two methods coincide on $x = 0$, where the tangential component vanishes.

Theorems & Definitions (23)

  • Example 1
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • ...and 13 more