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Euler--Poincaré reduction and the Kelvin--Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

Yusuke Ono, Simone Fiori, Linyu Peng

TL;DR

The paper develops a discrete Euler–Poincaré reduction framework for Lagrangian systems on Lie groups with advected parameters and additional dynamics, using a group difference map (such as the Cayley transform or matrix exponential) to update on the group while preserving geometric structure. It extends the Kelvin–Noether theorem to both continuous and discrete settings in this context and formulates the corresponding discrete Euler–Poincaré equations with extra dynamics. The methodology is applied to underwater-vehicle dynamics, yielding explicit continuous and discrete equations for attitude, translation, and advected parameters, along with energy and Kelvin–Noether quantities that are preserved up to numerical iteration effects. Numerical simulations demonstrate structure preservation and the practical viability of the scheme for control and navigation tasks in underwater robotics, while outlining avenues for extending the framework to infinite-dimensional groups and more complex fluid-structure interactions.

Abstract

The Euler--Poincaré equations, firstly introduced by Henri Poincaré in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various settings, such as semidirect products, advected parameters, and field theory, and have been widely applied to mechanics and physics. In this paper, we introduce the discrete Euler--Poincaré reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics, utilizing the group difference map technique. Specifically, the group difference map is defined using either the Cayley transform or the matrix exponential. The continuous and discrete Kelvin--Noether theorems are extended accordingly, that account for Kelvin--Noether quantities of the corresponding continuous and discrete Euler--Poincaré equations. As an application, we show both continuous and discrete Euler--Poincaré formulations about the dynamics of underwater vehicles, followed by numerical simulations. Numerical results illustrate the scheme's ability to preserve geometric properties over extended time intervals, highlighting its potential for practical applications in the control and navigation of underwater vehicles, as well as in other domains.

Euler--Poincaré reduction and the Kelvin--Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

TL;DR

The paper develops a discrete Euler–Poincaré reduction framework for Lagrangian systems on Lie groups with advected parameters and additional dynamics, using a group difference map (such as the Cayley transform or matrix exponential) to update on the group while preserving geometric structure. It extends the Kelvin–Noether theorem to both continuous and discrete settings in this context and formulates the corresponding discrete Euler–Poincaré equations with extra dynamics. The methodology is applied to underwater-vehicle dynamics, yielding explicit continuous and discrete equations for attitude, translation, and advected parameters, along with energy and Kelvin–Noether quantities that are preserved up to numerical iteration effects. Numerical simulations demonstrate structure preservation and the practical viability of the scheme for control and navigation tasks in underwater robotics, while outlining avenues for extending the framework to infinite-dimensional groups and more complex fluid-structure interactions.

Abstract

The Euler--Poincaré equations, firstly introduced by Henri Poincaré in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various settings, such as semidirect products, advected parameters, and field theory, and have been widely applied to mechanics and physics. In this paper, we introduce the discrete Euler--Poincaré reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics, utilizing the group difference map technique. Specifically, the group difference map is defined using either the Cayley transform or the matrix exponential. The continuous and discrete Kelvin--Noether theorems are extended accordingly, that account for Kelvin--Noether quantities of the corresponding continuous and discrete Euler--Poincaré equations. As an application, we show both continuous and discrete Euler--Poincaré formulations about the dynamics of underwater vehicles, followed by numerical simulations. Numerical results illustrate the scheme's ability to preserve geometric properties over extended time intervals, highlighting its potential for practical applications in the control and navigation of underwater vehicles, as well as in other domains.
Paper Structure (9 sections, 6 theorems, 102 equations, 3 figures, 2 tables)

This paper contains 9 sections, 6 theorems, 102 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The continuous Euler--Poincaré reduction
  3. The Euler--Poincaré equations with advected parameters
  4. The Euler--Poincaré equations with advected parameters and additional dynamics
  5. The continuous Kelvin--Noether theorem
  6. The discrete Euler--Poincaré equations with advected parameters and additional dynamics
  7. Application to the dynamics of underwater vehicles
  8. Numerical simulations
  9. Conclusion

Key Result

Theorem 1

Let $G$ be a Lie group that acts on a dual vector space $V^*$ from the left. Assume that the Lagrangian $L:TG \times V^* \to \mathbb{R}$ is left $G$-invariant. When $a_0 \in V^*$, the corresponding function $L_{a_0}: TG \to \mathbb{R}$, defined by $L_{a_0}(v_g) := L(v_g, a_0)$, is left $G_{a_0}$-inv For a curve $g(t)$ on $G$, let $\xi(t) = g(t)^{-1} \dot{g}(t) \in \mathfrak{g}$ and define a curve

Figures (3)

  • Figure 1: Relative error of the total energy, $\left|\dfrac{E_k - E_0}{E_0}\right|$, over time span $[0, 500]$.
  • Figure 2: Relative error of the Kelvin--Noether quantity, $\left|\dfrac{\mathcal{I}_k - \mathcal{I}_0}{\mathcal{I}_0}\right|$, over time span $[0, 500]$.
  • Figure 3: Left: 3-dimensional visualization of the vehicle's trajectory over time span $[0, 500]$. Right: The trajectory of the vehicle along each axis of the reference frame over time span $[0, 500]$.

Theorems & Definitions (14)

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