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The Fractal Lie Derivative: Theory and Applications

Alireza Khalili Golmankhaneh, Elham Hashemzadeh, Carlo Cattani, Donal O'Regan, Palle E. T. Jørgensen

TL;DR

The paper develops a Lie-theoretic framework for fractal calculus to analyze symmetries and conservation laws in fractal dynamical systems. By introducing $d_{F}^{\alpha}$, $D_{F}^{\alpha}$, the $k$-fractal jet space, and fractal prolongations, it extends Noether's theorem to fractal Lagrangian mechanics, deriving conserved currents $J$ with $D_{F}^{\alpha} J = 0$ and fractal Euler–Lagrange equations. It then builds a comprehensive differential-geometry toolkit for fractal manifolds—covering fractal differential forms, wedge products, the exterior derivative, Lie derivatives, and jet-space methods—and applies symmetry analysis to nonlinear and linear fractal differential equations, yielding explicit infinitesimals and generators for representative cases, including time-translation, scaling, and oscillatory symmetries. The work provides a rigorous, extensible framework for modeling and solving fractal-dynamics problems, with potential applications in physics and complex fractal media.

Abstract

This paper presents a new Lie theoretic approach to fractal calculus, which in turn yields such new results as a Fractal Noether's Theorem, a setting for fractal differential forms, for vector fields, and Lie derivatives, as well as k-fractal jet space, and algorithms for k-th fractal prolongation. The symmetries of the fractal nonlinear \(n\)-th \(α\)-order differential equation are examined, followed by a discussion of the symmetries of the fractal linear \(n\)-th \(α\)-order differential equation. Additionally, the symmetries of the fractal linear first \(α\)-order differential equation are derived. Several examples are provided to illustrate and highlight the details of these concepts.

The Fractal Lie Derivative: Theory and Applications

TL;DR

The paper develops a Lie-theoretic framework for fractal calculus to analyze symmetries and conservation laws in fractal dynamical systems. By introducing , , the -fractal jet space, and fractal prolongations, it extends Noether's theorem to fractal Lagrangian mechanics, deriving conserved currents with and fractal Euler–Lagrange equations. It then builds a comprehensive differential-geometry toolkit for fractal manifolds—covering fractal differential forms, wedge products, the exterior derivative, Lie derivatives, and jet-space methods—and applies symmetry analysis to nonlinear and linear fractal differential equations, yielding explicit infinitesimals and generators for representative cases, including time-translation, scaling, and oscillatory symmetries. The work provides a rigorous, extensible framework for modeling and solving fractal-dynamics problems, with potential applications in physics and complex fractal media.

Abstract

This paper presents a new Lie theoretic approach to fractal calculus, which in turn yields such new results as a Fractal Noether's Theorem, a setting for fractal differential forms, for vector fields, and Lie derivatives, as well as k-fractal jet space, and algorithms for k-th fractal prolongation. The symmetries of the fractal nonlinear -th -order differential equation are examined, followed by a discussion of the symmetries of the fractal linear -th -order differential equation. Additionally, the symmetries of the fractal linear first -order differential equation are derived. Several examples are provided to illustrate and highlight the details of these concepts.

Paper Structure

This paper contains 8 sections, 1 theorem, 100 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Fundamental Definitions in Fractal Calculus
  3. Fractal Noether's Theorem
  4. Fractal Differential Forms, Vector Fields, and Lie Derivatives
  5. Symmetries of the Fractal Nonlinear n-th $\alpha$-Order Differential Equation
  6. Symmetries of the Fractal Linear n-th $\alpha$-Order Differential Equation
  7. Symmetries of the Fractal Linear First $\alpha$-Order Differential Equation
  8. Conclusion

Key Result

Theorem 1

Consider a system with a fractal Lagrangian $L(q_i, D_{F}^{\alpha}q_i, t)$, where $q_i$ are the generalized coordinates, $D_{F}^{\alpha}q_i$ are their fractal time derivatives, and $t$ represents fractal time. The action $S$ of the system is defined by: This Fractal Noether's theorem states that if the action $S$ remains invariant under a fractal continuous transformation given by: where $\epsil

Figures (4)

  • Figure 1: 3D plot of the function $f(x, y) = \sin(2\pi x) \cos(2\pi y)$ evaluated on a Cantor set $\times$ Cantor set. The plot visualizes the structure of $f(x, y)$ over the fractal space formed by the Cantor set.
  • Figure 2: The figure shows the projection of the vector field, with the x-component of the field represented as $U = \sin(2\pi X)$ and the y-component as $V = \cos(2\pi Y)$, where $X$ and $Y$ are points from the Cantor set.
  • Figure 3: Plot of the 0-fractal form Eq.\ref{['4312']} on the Cantor set. The function values are represented as individual points corresponding to the Cantor set elements.
  • Figure 4: Visualization of the base solution and symmetry transformations of the Cantor set's integral staircase function $S_{F}^{\alpha}(x)$. The figure illustrates the base solution alongside various symmetry transformations, including time translation, scaling, and oscillatory symmetries.

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 32 more