Classical mechanics on fractal curves

Golmankhaneh, Alireza Khalili; Welch, Kerri; Tunç, Cemil; Gasimov, Yusif S.

doi:10.1140/epjs/s11734-023-00775-y

Cited by 5 publications

(2 citation statements)

References 42 publications

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“…Fractal counterparts of Newtonian, Lagrangian, Hamiltonian, and Appellian mechanics were proposed. Fractal α-velocity and α-acceleration were defined, enabling the formulation of the Langevin equation on fractal curves [34]. The fractal Frechet derivative and the fractal generalized Euler-Lagrange equation and the fractal Du Bois-Reymond optimality condition were introduced [35].…”

Section: Introductionmentioning

confidence: 99%

Fractal Schrödinger equation: implications for fractal sets

Khalili Golmankhaneh,

Pellis,

Zingales

2024

J. Phys. A: Math. Theor.

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This paper delves into the world of fractal calculus, investigating its implications for fractal sets. It introduces the Fractal Schrödinger Equation and provides insights into its consequences. The study presents a General Solution for the Time-Dependent Schrödinger Equation, unveiling its core aspects. Exploring quantum mechanics in the context of fractals, the paper analyzes the Probability Density of the Radial Hydrogen Atom, unveiling its behavior within fractal dimensions. The investigation extends to deciphering the intricate Energy Levels of the Hydrogen Atom, uncovering the interplay of quantum mechanics and fractal geometry. Innovatively, the research applies the Fractal Schrödinger Equation to Simple Harmonic Motion, leading to the introduction of the Fractal Probability Density Function for the Harmonic Oscillator. The paper employs a series of illustrative ﬁgures that enhance the comprehension of the ﬁndings. By intertwining quantum mechanics and fractal mathematics, this research paves the way for deeper insights into their relationship.

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Section: Introductionmentioning

confidence: 99%

Fractal Schrödinger equation: implications for fractal sets

Khalili Golmankhaneh,

Pellis,

Zingales

2024

J. Phys. A: Math. Theor.

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“…In Ref. [ 5 ], the authors mainly explore the fractal analogue of classical mechanics such as Newton, Lagrange, Hamilton and Appell’s mechanics via fractal calculus. Further, they obtain the Langevin equation on fractal curves, namely, the Koch-like curves, by defining the fractal -velocity and -acceleration.…”

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confidence: 99%