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Generalized Lagrangian Coherent Structures in Finsler Manifolds

Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales

TL;DR

The paper addresses identifying Lagrangian Coherent Structures in manifolds with non-Euclidean geometry, extending the concept from Euclidean settings to both Riemannian and Finsler spaces. It introduces generalized deformation tensors, $C_t(x)$ in the Riemannian case and $\mathcal{C}_t(x)$ in the Finsler setting, whose dominant eigenvalues guide the location of invariant LCS surfaces. The main contributions include existence results for smooth, invariant LCS surfaces defined by level sets of the maximal eigenvalue $\lambda_1(x)$, and the definitions of Hypercomplex LCS, all supported by spectral analysis and, for Finsler spaces, explicit use of the fundamental tensor $G(x)$. The work provides numerical procedures for computing these structures and highlights potential applications in astrophysics, relativistic fluid dynamics, and planetary science, enabling robust analysis of geodesic transport in curved geometries.

Abstract

This paper introduces a novel theoretical framework for identifying Lagrangian Coherent Structures (LCS) in manifolds with non-constant curvature, extending the theory to Finsler manifolds. By leveraging Riemannian and Finsler geometry, we generalize the deformation tensor to account for geodesic stretching in these complex spaces. The main result demonstrates the existence of invariant surfaces acting as LCS, characterized by dominant eigenvalues of the generalized deformation tensor. We discuss potential applications in astrophysics, relativistic fluid dynamics, and planetary science. This work paves the way for exploring LCS in intricate geometrical settings, offering new tools for dynamical system analysis.

Generalized Lagrangian Coherent Structures in Finsler Manifolds

TL;DR

The paper addresses identifying Lagrangian Coherent Structures in manifolds with non-Euclidean geometry, extending the concept from Euclidean settings to both Riemannian and Finsler spaces. It introduces generalized deformation tensors, in the Riemannian case and in the Finsler setting, whose dominant eigenvalues guide the location of invariant LCS surfaces. The main contributions include existence results for smooth, invariant LCS surfaces defined by level sets of the maximal eigenvalue , and the definitions of Hypercomplex LCS, all supported by spectral analysis and, for Finsler spaces, explicit use of the fundamental tensor . The work provides numerical procedures for computing these structures and highlights potential applications in astrophysics, relativistic fluid dynamics, and planetary science, enabling robust analysis of geodesic transport in curved geometries.

Abstract

This paper introduces a novel theoretical framework for identifying Lagrangian Coherent Structures (LCS) in manifolds with non-constant curvature, extending the theory to Finsler manifolds. By leveraging Riemannian and Finsler geometry, we generalize the deformation tensor to account for geodesic stretching in these complex spaces. The main result demonstrates the existence of invariant surfaces acting as LCS, characterized by dominant eigenvalues of the generalized deformation tensor. We discuss potential applications in astrophysics, relativistic fluid dynamics, and planetary science. This work paves the way for exploring LCS in intricate geometrical settings, offering new tools for dynamical system analysis.
Paper Structure (17 sections, 3 theorems, 16 equations)

This paper contains 17 sections, 3 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Mathematical Development
  3. Lagrangian Coherent Structures and Dominant Eigenvalues of the Curvature Deformation Tensor
  4. Definition of Hypercomplex Lagrangian Coherent Structure (LCS)
  5. Hypercomplex Lagrangian Coherent Structures and Dominant Eigenvalues of the Deformation Tensor
  6. Extension of LCS Theory to Finsler Manifolds
  7. Preliminary Definitions
  8. Generalized Deformation Tensor
  9. Existence of LCS in Finsler Manifolds
  10. Results
  11. Conclusions
  12. Appendix: Key Mathematical Formulations
  13. Finsler Manifolds
  14. Deformation Gradient and Tensor
  15. Eigenvalue Decomposition
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $M$ be a compact Riemannian manifold with metric $g$ and non-constant curvature. Let $\phi^t$ be a smooth flow on $M$. Then, there exist material surfaces $\Sigma \subset M$ such that $C_t(x)$ has dominant, non-degenerate eigenvalues along $\Sigma$. These surfaces act as Lagrangian Coherent Stru

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof