Analysis and modeling of fractional Mandelbrot scaling for parameterization-invariant theories for mechanical systems

This study presents a comprehensive analysis and modeling framework that integrates Mandelbrot’s fractal scaling with fractional variational principles to advance the understanding of parameterization-invariant theories for mechanical systems. Using the principle of least action, we derive the Euler–Lagrange equations of motion for two key scenarios that emphasize the preservation of symmetry in mathematical formulations. In the first scenario, Mandelbrot’s fractal scaling laws are incorporated into variational problems by reformulating the action integral using a scaling measure, which allows time to be parameterized accordingly.. This approach yields Euler–Lagrange equations exhibiting intrinsic symmetries that reflect the self-similar and scale-invariant properties characteristic of fractal geometries, thereby extending classical mechanics into fractional-dimensional spaces. The second scenario focuses on the role of fractional calculus in modeling mechanical systems, particularly on symmetry-preserving fractional formulations. Taking a fractional spring pendulum as a case study, we begin by deriving the classical equations of motion from the traditional Lagrangian formulation. Subsequently, the model is extended to fractional domains by employing fractional Euler–Lagrange equations based on both singular and non-singular kernels, such as the Caputo and Atangana–Baleanu derivatives. This extension introduces fractional symmetries that generalize classical system symmetries, thereby enriching the theoretical framework. Numerical simulations illustrate the influence of the fractional order and initial conditions on the dynamic response of the system, confirming that fractional calculus enhances model accuracy while preserving and generalizing essential symmetry properties. The results highlight the significant impact of incorporating fractional dynamics and fractal scaling on understanding complex mechanical behaviors influenced by memory and non-local interactions. Overall, this work demonstrates a novel intersection of fractal geometry, fractional calculus, and nonlinear dynamics, offering a unified parameterization-invariant approach to studying mechanical systems in fractional-dimensional spaces. The findings provide new insights into the fundamental connections between symmetry, scaling laws and fractional dynamics, with potential applications spanning physics, engineering and applied mathematics.