# Fractal Nambu Mechanics: Extending Dynamics with Fractal Calculus

Alireza Khalili Golmankhaneh1, Cemil Tunç2, Davron Aslonqulovich Juraev 3,4
1 Department of Physics, Urmia Branch,
Islamic Azad University, Urmia 63896, Iran
<EMAIL_ADDRESS>
2 Department of Mathematics, Faculty of Sciences,
Van Yuzuncu Yil University, 65080-Campus, Van-Turkey
<EMAIL_ADDRESS>
3 Department of Scientific Research, Innovation and Training of Scientific and
Pedagogical Staff,
University of Economics and Pedagogy, Karshi, 180100, Uzbekistan
4 Department of Mathematics, Anand International College of Engineering,
Jaipur, 303012, India
<EMAIL_ADDRESS>
###### Abstract

In this paper, we extend the principles of Nambu mechanics by incorporating
fractal calculus. This extension introduces Hamiltonian and Lagrangian
mechanics that incorporate fractal derivatives. By doing so, we broaden the
scope of our analysis to encompass the dynamics of fractal systems, enabling
us to capture their intricate and self-similar properties. This novel approach
opens up new avenues for understanding and modeling complex fractal
structures, thereby advancing our comprehension of these intricate phenomena.

††footnotetext:  MSC2020:28A80††Corresponding author Alireza Khalili
Golmankhaneh

##  1 Introduction

Fractal geometry explains shapes with self-similar properties that have
fractional and complex numbers dimensions whose topological dimension is less
than their fractal dimension [44, 39]. Fractals have different measures than
Euclidean shapes. Fractals are seen in nature everywhere like clouds in the
human body and the path of electrons and structures of materials in fractal
forms show new results [8, 41]. Their properties have been studied with the
help of mathematical analysis [22]. Fractals are explained in a mathematically
rigorous manner, with an emphasis on examples and fundamental concepts such as
Hausdorff dimension, self-similar sets, and Brownian motion [11]. Fractal
structures are seen and used in many branches of physical science and art
engineering [38, 10, 35, 34, 19, 21, 64]. Fractal space-time was considered in
the different processes to obtain more general models to explain experimental
data [51, 49, 67, 60]. Different techniques were used to analyze fractals,
including probabilistic techniques, measure theory, fractional calculus,
harmonic analysis, and fractional spaces [3, 65, 63, 58, 18, 14, 36, 68, 12,
15, 13, 25]. The $\alpha$-fractal function with variable parameters was
investigated, focusing on the Weyl-Marchaud variable order fractional
derivative under specific scaling factor conditions [56, 1, 2, 66]. The
extension of Darcy’s law from conventional calculus to fractal calculus in
order to quantify fluid flow in subterranean heterogeneous reservoirs was
investigated [20, 57]

A generalization of Hamiltonian mechanics, Multi-Hamiltonian Nambu mechanics
has some applications [47, 33, 32]. The flows created by a Hamiltonian over a
Poisson manifold form the basis of Nambu mechanics. The trajectory in phase
space is the intersection of the $N-1$ hypersurfaces specified by the
invariants [47, 46]. The flow is perpendicular to all $N-1$ gradients of these
Hamiltonians, parallel to the generalized cross product described by the
corresponding Nambu bracket, and lagrangian to all $N-1$ gradients of these
Hamiltonians. Nambu-Hamilton equations of motion, incorporating several
Hamiltonians, describe the dynamics of Nambu mechanics. As a consistency
condition, the basic identity for the Nambu bracket-which is comparable to the
Jacobi identity-is presented [62, 16, 42]. The Hamiltonian embedding for
Nambu’s novel mechanical equations is given. Using Dirac’s singular formalism,
the embedding is investigated in both classical and quantization forms. The
embedding Hamiltonian’s transformations are contrasted with Nambu’s canonical
transformations [9, 24]. Nambu-Poisson and Nambu-Jacobi brackets are defined
inductively [31]. In Nambu’s generalized mechanics, the action integral is
defined on a world sheet and there is an additional degree of freedom in phase
space [40, 43]. The Nambu mechanics can, in theory, describe physical systems
for which the Dirac mechanics is inappropriate since Nambu’s mechanics is
independent of Dirac’s mechanics, at least in the odd instances [17]. The
conditions for a Hamiltonian system to possess $n$-linear Nambu brackets is
investigated. The system should only have one degree of freedom for $n=3$, but
the criterion still holds for higher orders ($n\geq 4$). Levi-Civita tensors
produce the canonical Nambu bracket, which is the only fundamental solution
locally. [16]. A geometric formulation for a Hamiltonian system-like
generalized form of Nambu mechanics is provided. It introduces a nondegenerate
3-form connected to a 3n-dimensional phase space to model temporal evolution.
A Poisson bracket of two forms is additionally included, enabling the
description of connected rigid bodies and non-integrable fluid flow [52].
Nambu mechanics is discussed using numerous instances of mechanical systems
with the Nambu bracket. The relationship between a mechanical system’s
superintegrability and the existence of the Nambu bracket is the main idea
examined. The Nambu bracket’s uses in field theory are presented [59]. Using
fractional-order differential forms and exterior derivatives, the creation of
a fractional generalization of Nambu mechanics is shown. The generalized
Pfaffian equations and offers a thorough analysis of one such example [5, 69,
26]. A fractal method to quantum chromodynamics is used to produce the Nambu-
Jona-Lasino model, which includes a running coupling. Tsallis non-extensive
statistical distributions in high-energy collisions are explained by the
modified model [45].
The fractal analysis has been explored by multiple researchers through various
approaches, including harmonic analysis [37], measure theory [23, 36, 25, 15,
12, 68], probabilistic methods [7], fractional space [61], fractional calculus
[63, 30], and non-standard methods [48]. Through an expanded framework that
contains differential equations, fractal calculus expands classical calculus.
These equations’ solutions provide functions with fractal support, such as
fractal sets and curves [55, 54, 53, 6, 27, 4]. The usage of fractal versions
of Newton, Lagrange, Hamilton, and Appell’s mechanics is suggested. Following
the definition of fractal velocity and acceleration, the Langevin equation on
fractal curves is obtained. Hamilton’s mechanics on fractal curves are created
via the Legendre transformation, allowing for the simulation of non-
conservative systems with fractional dimensions [29]. Nonstandard analysis
incorporating hyperreal and hyperinteger numbers is used to define left and
right limits and derivatives on fractal curves. The use of nonstandard
analysis is used to define the fractal integral and differential forms [28].
In this study, the generalization of Nambu mechanics is achieved by utilizing
fractal derivatives and fractal differential forms on fractal manifolds.
Through this approach, we extend the framework of Nambu mechanics to encompass
fractal geometries and dynamics.
The following is the paper’s outline:
We present a concise summary of fractal calculus in Section 2, outlining its
key concepts and principles. Moving on to Section 3, we introduce fractal
Nambu mechanics, explaining its fundamental ideas and framework. In Section 4,
we demonstrate the application of fractal differential forms in deriving Nambu
mechanics. Additionally, in Section 6, we showcase the practical
implementation and usefulness of this formalism through relevant examples and
applications. Finally, in Section 7, we provide a comprehensive conclusion
that summarizes the main findings and contributions of our work in the context
of fractal Nambu mechanics.

##  2 Basic Tools

Here, we provide a summary of the fractal calculus for middle-$\epsilon$
Cantor set $F\subset[c_{1},c_{2}]\subset\mathbb{R}$ [55, 27].

######  Definition 1.

The indicator function of $F$ is defined as

$f(F,I)=\begin{cases}1,&\text{if }F\cap I\neq\emptyset,\\\
0,&\text{otherwise},\end{cases}$

where $I=[c_{1},c_{2}]\subset\mathbb{R}$.

######  Definition 2.

The coarse-grained measure of $F\cap[c_{1},c_{2}]$ is defined as

$\mu_{\delta}^{\alpha}(F,c_{1},c_{2})=\inf_{|\mathcal{P}|\leq\delta}\sum_{i=0}^{n-1}\Gamma(\alpha+1)(x_{i+1}-x_{i})^{\alpha}f(F,[x_{i},x_{i+1}]),$
(1)

where $|\mathcal{P}|=\max_{0\leq i\leq n-1}(x_{i+1}-x_{i})$, and $0<\alpha\leq
1$.

######  Definition 3.

The measure function of $F$ is defined as

$\mu^{\alpha}(F,c_{1},c_{2})=\lim_{\delta\rightarrow
0}\mu_{\delta}^{\alpha}(F,c_{1},c_{2}).$

######  Definition 4.

The fractal dimension of $F\cap[c_{1},c_{2}]$ is defined as

$\displaystyle\dim_{\mu}(F\cap[c_{1},c_{2}])$
$\displaystyle=\inf\\{\alpha:\mu^{\alpha}(F,c_{1},c_{2})=0\\}$
$\displaystyle=\sup\\{\alpha:\mu^{\alpha}(F,c_{1},c_{2})=\infty\\}.$

######  Definition 5.

The integral staircase function of $F$ is defined as

$S_{F}^{\alpha}(x)=\begin{cases}\mu^{\alpha}(F,c_{0},x),&\text{if }x\geq
c_{0},\\\ -\mu^{\alpha}(F,x,c_{0}),&\text{otherwise},\end{cases}$

where $c_{0}\in\mathbb{R}$ is a fixed number.

######  Definition 6.

For a function $h$ on an $\alpha$-perfect fractal set $F$, the
$F^{\alpha}$-derivative of $h$ at $x$ is defined as

$D_{F}^{\alpha}h(x)=\begin{cases}\underset{y\rightarrow
x}{F_{-}\text{lim}}\frac{h(y)-h(x)}{S_{F}^{\alpha}(y)-S_{F}^{\alpha}(x)},&\text{if
}x\in F,\\\ 0,&\text{otherwise},\end{cases}$

if the fractal limit $F_{-}\text{lim}$ exists.

######  Definition 7.

The $F^{\alpha}$-integral of a bounded function $h(x)$, where $h\in B(F)$
(i.e., $h$ is a bounded function of $F$), is defined as

$\displaystyle\int_{a}^{b}h(x)\,d_{F}^{\alpha}x$
$\displaystyle=\sup_{\mathcal{P}_{[c_{1},c_{2}]}}\sum_{i=0}^{n-1}\inf_{x\in
F\cap I}h(x)\,(S_{F}^{\alpha}(x_{i+1})-S_{F}^{\alpha}(x_{i}))$
$\displaystyle=\inf_{\mathcal{P}_{[c_{1},c_{2}]}}\sum_{i=0}^{n-1}\sup_{x\in
F\cap I}h(x)\,(S_{F}^{\alpha}(x_{i+1})-S_{F}^{\alpha}(x_{i})),$

where $x\in F$, and the infimum or supremum is taken over all subdivisions
$\mathcal{P}_{[c_{1},c_{2}]}$.

######  Definition 8.

The fractal total derivative of a function $f:F^{n}\rightarrow R^{n}$ can be
defined when $d_{F}^{\alpha}x_{1},d_{F}^{\alpha}x_{2},...,d_{F}^{\alpha}x_{n}$
represent infinitesimal increments along the coordinate directions. It is
denoted as $d_{F}^{\alpha}f$ and can be calculated as:

$d_{F}^{\alpha}f=\sum_{i=1}^{n}(D_{F,x_{i}}^{\alpha}f)d_{F}^{\alpha}x_{i}=\sum_{i=1}^{n}(\partial^{\alpha}_{F,x_{i}}f)d_{F}^{\alpha}x_{i}.$
(2)

Here, $(D_{F,x_{i}}^{\alpha}f)$ or $(\partial^{\alpha}{F,x{i}}f)$ represents
the fractal partial derivative of $f$ with respect to $x_{i}$, and
$d_{F}^{\alpha}x_{i}$ represents the infinitesimal increment in the $i$-th
coordinate direction [28].

##  3 Fractal Nambu mechanics

Nambu mechanics is a mathematical framework for explaining the dynamics of
physical systems including several variables. Fractal Nambu mechanics is a
modified form of Nambu mechanics. The idea of fractals is included in Fractal
Nambu mechanics, enabling the characterization and study of complex systems
with complicated and self-similar structures. The indicator function is used
in Fractal Nambu mechanics to determine whether a fractal set is present or
absent within a specified interval. The coarse-grained measure, which takes
into account various scales of resolution, quantifies the size or extent of
the fractal set within the interval. As the resolution becomes closer to zero,
the measure function offers a limit that captures the fundamental
characteristics of the fractal set. The complexity or self-similarity of the
fractal set is gauged by the fractal dimension. The integral staircase
function, which represents the total measure of the fractal set up to a
specific point, is also introduced by Fractal Nambu mechanics. The fractal
derivative and integral operators are defined using this function. The fractal
integral computes the cumulative impact of a function on the fractal set,
whereas the fractal derivative determines the rate of change of a function on
the fractal set. Fractal Nambu mechanics, which incorporates fractal ideas
into Nambu mechanics, offers a potent mathematical framework for investigating
and comprehending complicated systems that exhibit fractal characteristics. It
provides a deeper understanding of the dynamics and characteristics of such
systems, allowing for a more thorough study and modeling strategy.
We present the mathematical framework of fractal Nambu mechanics as follows:

######  Definition 9.

A fractal manifold denoted as $M^{\alpha}$, characterized by fractal
dimensions, is classified as a Fractal Nambu-Poisson manifold when a mapping
denoted by

$\\{,...,\\}:[C^{\infty}(F)]^{\otimes^{n}}\mapsto C^{\infty}(F),$ (3)

exists. Here, $C^{\infty}(F)$ represents the space of smooth functions defined
on fractal sets. This mapping can be referred to as the fractal Nambu-Poisson
bracket of order $n$, defined for any $f_{1},f_{2},...,f_{2n-1}\in
C^{\infty}(F)$ [55, 27].

The fractal Nambu-Poisson bracket exhibits specific properties, which include
the following:

  * •

Skew-symmetry:

$\\{f_{1},f_{2},...,f_{n}\\}=(-1)^{\epsilon(\sigma)}\\{f_{\sigma(1)},f_{\sigma(2)},...,f_{\sigma(n)}\\}.$
(4)

Here, $\sigma$ represents a permutation in the symmetric group $S_{n}$, which
consists of all possible arrangements of $n$ elements. The function
$\epsilon(\sigma)$ determines the parity of the permutation, determining
whether it is even or odd. The skew-symmetry property ensures that the
resulting bracket remains consistent even when the functions’ order is
altered, accounting for the sign change based on the parity of the permutation
used.

  * •

Jacobi identity:

$\displaystyle\\{\\{f_{1},f_{2},f_{3},...,f_{n}\\},f_{n+1},...,$
$\displaystyle
f_{2n-1}\\}+\\{f_{n},\\{f_{1},f_{2},...,f_{n+1}\\},f_{n+2},...,f_{2n-1}\\}=\\{f_{1},...,f_{n-1},\\{f_{n},...,f_{2n-1}\\}\\},$
(5)

The fractal Nambu-Poisson bracket obeys the Jacobi identity, which is a
fundamental property of Poisson brackets. It states that the bracket of the
bracket of $n$ functions with $n$ additional functions, along with the bracket
of the first function with the bracket of the remaining $2n-1$ functions, is
equal to the bracket of the bracket of the first $n-1$ functions with the last
$2n-1$ functions.

  * •

Leibniz Rule:

$\\{f_{1},f_{2},f_{3},...,f_{n+1}\\}=f_{1}\\{f_{2},f_{3},...,f_{n+1}\\}+f_{2}\\{f_{1},f_{3},...,f_{n+1}\\}.$
(6)

The fractal Nambu-Poisson bracket satisfies the Leibniz rule, also known as
the generalized chain rule. This rule governs how the bracket operates on the
product of functions, ensuring that the bracket operation can be applied to
each function individually while preserving the appropriate sign and
structure.

By satisfying these properties, the fractal Nambu-Poisson bracket of order $n$
provides a fundamental tool for describing the dynamics and relationships of
smooth functions defined on Fractal Nambu-Poisson manifolds.

###  3.1 Fractal Nambu equations

In this subsection, the fractal Nambu equations are formulated using the
fractal Nambu-Poisson bracket, as shown below:

$\displaystyle\frac{d_{F}^{\alpha}x^{i}}{d_{F}^{\alpha}t}$ $\displaystyle=$
$\displaystyle\\{x^{i},H_{1},H_{2},...,H_{n-1}\\}$ (7) $\displaystyle=$
$\displaystyle\sum_{jk...l}\epsilon_{ijk...l}D_{F,x_{j}}^{\alpha}H_{1}D_{F,x_{k}}^{\alpha}H_{2}...D_{F,x_{l}}^{\alpha}H_{n-1}$
$\displaystyle=$
$\displaystyle\frac{\partial^{\alpha}_{F}(x^{i},H_{1},H_{2},...,H_{n-1})}{\partial^{\alpha}_{F}(x^{1},x^{2},...,x^{n})}.$

In this formulation, the local coordinates on the fractal Nambu-Poisson
manifold are represented by $x^{i}$, and the corresponding fractal Nambu-
Hamiltonians are denoted as $H_{1},H_{2},...,H_{n-1}$. The Levi-Civita tensor,
denoted as $\epsilon_{ijk...l}$, is involved in the equations.
It is important to note that while traditional Hamiltonian mechanics describes
the state space using a pair of variables, $x$ and $p$, Nambu mechanics
extends this framework to include $n$ variables represented as $x^{i}$.

######  Example 1.

Consider a triple dynamical variable $\vec{u}=(x,y,z)$ that spans a three-
dimensional phase space. Let $H$ and $G$ be Hamiltonians that determine the
motion of points in the phase space. The fractal Nambu-Hamiltonians are given
by the following equations

$\displaystyle\frac{d_{F}^{\alpha}x}{d_{F}^{\alpha}t}$
$\displaystyle=\frac{\partial^{\alpha}_{F}(H,G)}{\partial^{\alpha}_{F}(y,z)},$
$\displaystyle\frac{d_{F}^{\alpha}y}{d_{F}^{\alpha}t}$
$\displaystyle=\frac{\partial^{\alpha}_{F}(H,G)}{\partial^{\alpha}_{F}(z,x)},$
$\displaystyle\frac{d_{F}^{\alpha}z}{d_{F}^{\alpha}t}$
$\displaystyle=\frac{\partial^{\alpha}_{F}(H,G)}{\partial^{\alpha}_{F}(x,y)}.$
(8)

Alternatively, in vector notation, we have:

$\frac{d_{F}^{\alpha}\vec{u}}{d_{F}^{\alpha}t}=\vec{\nabla}^{\alpha}H\times\vec{\nabla}^{\alpha}G,$
(9)

where

$\vec{\nabla}^{\alpha}=D_{F,x}^{\alpha}\hat{e}_{i}+D_{F,y}^{\alpha}\hat{e}_{j}+D_{F,z}^{\alpha}\hat{e}_{k}.$
(10)

For any function $N(x,y,z)$, we have the following expression:

$\displaystyle\frac{d_{F}^{\alpha}N}{d_{F}^{\alpha}t}$
$\displaystyle=\frac{\partial^{\alpha}_{F}(N,H,G)}{\partial^{\alpha}_{F}(x,y,z)}$
$\displaystyle=\vec{\nabla}^{\alpha}N.(\vec{\nabla}^{\alpha}H\times\vec{\nabla}^{\alpha}G).$
(11)

We can refer to the right-hand side of Eq. (1) as the fractal Nambu-Poisson
bracket, denoted by $[M,H,G]^{\alpha}$. The orbit of the system in the fractal
phase space is determined by the intersection of two surfaces, namely
$H=const.$ and $G=const.$. By using Eq. (9), we can write:

$\vec{\nabla}^{\alpha}.(\vec{\nabla}^{\alpha}H\times\vec{\nabla}^{\alpha}G)=0.$
(12)

This may called the fractal Liouville theorem in the fractal Nambu phase
space.

######  Definition 10.

By selecting an arbitrary $H_{r}$ as a generalized Hamiltonian, we can define
$J_{ij}$ as follows:

$J_{ij}=\epsilon_{ijk...l}D_{F,x_{k}}^{\alpha}H_{1}...D_{F,x_{z}}^{\alpha}H_{r-1}D_{F,x_{c}}^{\alpha}H_{r+1}...D_{F,x_{l}}^{\alpha}H_{n-1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ H=H_{r}.$ (13)

In this definition, $J_{ij}$ satisfies the following conditions:

$\displaystyle J_{ij}$ $\displaystyle=-J_{ji},$ (14) $\displaystyle
J_{il}D_{F,x_{l}}^{\alpha}J_{jk}+J_{jl}D_{F,x_{l}}^{\alpha}J_{ki}+J_{kl}D_{F,x_{l}}^{\alpha}J_{ij}$
$\displaystyle=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (i,j,k,l=1,2,...,n).$ (15)

######  Definition 11.

A mapping from $(x,y,z)$ to $(x^{\prime},y^{\prime},z^{\prime})$ is considered
a fractal canonical transform if it satisfies the condition:

$[x^{\prime},y^{\prime},z^{\prime}]=\frac{\partial^{\alpha}_{F}(x^{\prime},y^{\prime},z^{\prime})}{\partial^{\alpha}_{F}(x,y,z)}=1.$
(16)

Alternatively, it can be expressed as:

$\frac{\partial^{\alpha}_{F}(H^{\prime},G^{\prime})}{\partial^{\alpha}_{F}(H,G)}=1,$
(17)

where $H^{\prime}=h_{1}(H,G)$ and $G^{\prime}=h_{2}(H,G)$. In this definition,
a fractal canonical transform refers to a mapping that preserves the fractal
Nambu structure, ensuring that the fractal Nambu-Poisson brackets remain
invariant under the transformation.

##  4 Fractal differential form and fractal Nambu mechanics

In this section, we introduce the concept of fractal differential forms to
derive Nambu’s equations.

######  Definition 12.

The fractal Poincaré-Cartan 1-form is defined as follows:

$\displaystyle\Omega^{1}=pd_{F}^{\alpha}q-H(p,q)d_{F}^{\alpha}t.$ (18)

######  Definition 13.

The fractal Poincaré-Cartan 2-form is defined as follows:

$\Omega^{2}=pd_{F}^{\alpha}q\wedge
d_{F}^{\alpha}r-H_{1}d_{F}^{\alpha}H_{2}\wedge d_{F}^{\alpha}t.$ (19)

In these definitions, the variables $p$, $q$, and $r$ represent a triplet of
dynamical variables, and $H_{1}$ and $H_{2}$ denote two Hamiltonians. The
symbol $\wedge$ represents the wedge product, a mathematical operation used in
multivariable calculus. These fractal differential forms provide a
mathematical framework for describing the dynamics and relationships between
the variables involved in the Nambu system.

The Hamilton and Nambu equations can be derived by taking the exterior
differentials of $\Omega^{1}$ and $\Omega^{2}$, respectively. The resulting
equations are as follows:
For $\Omega^{1}$:

$\displaystyle
d_{F}^{\alpha}\Omega^{1}=\left(d_{F}^{\alpha}p+\frac{\partial^{\alpha}_{F}H}{\partial^{\alpha}_{F}q}d_{F}^{\alpha}t\right)\wedge\bigg{(}d_{F}^{\alpha}q-\frac{\partial^{\alpha}_{F}H}{\partial^{\alpha}_{F}p}d_{F}^{\alpha}t\bigg{)}=0.$
(20)

This equation leads to Hamilton’s equations:

$\frac{d_{F}^{\alpha}q}{d_{F}^{\alpha}t}=\frac{\partial^{\alpha}_{F}H}{\partial^{\alpha}_{F}p},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d_{F}^{\alpha}p}{d_{F}^{\alpha}t}=-\frac{\partial^{\alpha}_{F}H}{\partial^{\alpha}_{F}q}.$
(21)

For $\Omega^{2}$:

$\displaystyle
d_{F}^{\alpha}\Omega^{2}=\left(d_{F}^{\alpha}q-\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(p,r)}d_{F}^{\alpha}t\right)\wedge\left(d_{F}^{\alpha}p-\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(r,q)}d_{F}^{\alpha}t\right)$
$\displaystyle\wedge\left(d_{F}^{\alpha}r-\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(q,p)}d_{F}^{\alpha}t\right)=0.$
(22)

This equation leads to the Nambu’s equations:

$\displaystyle\frac{d_{F}^{\alpha}q}{d_{F}^{\alpha}t}$
$\displaystyle=\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(p,r)}$
$\displaystyle\frac{d_{F}^{\alpha}p}{d_{F}^{\alpha}t}$
$\displaystyle=\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(r,q)}$
$\displaystyle\frac{d_{F}^{\alpha}r}{d_{F}^{\alpha}t}$
$\displaystyle=\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(q,p)}.$
(23)

These equations describe the dynamics of the system in terms of the fractal
derivatives and partial derivatives with respect to the variables involved.
The Hamilton equations correspond to the case when $H_{1}$ and $H_{2}$ are the
Hamiltonians, while the Nambu equations generalize the dynamics to more
complex systems involving multiple variables and Hamiltonians.

##  5 Fractal Lagrangians in Nambu mechanics

We suggest a postulate that states the existence of Lagrangians matching to
each Hamiltonian in order to achieve genuine Lagrangians in Nambu mechanics.
We assume that the system contains an equal number of Lagrangians and
Hamiltonians. Furthermore, we presume that each Lagrangian is consistent with
the least action principle. In the most straightforward case, we suppose the
presence of two Lagrangians, designated as $L_{1}(q(t),D^{\alpha}_{F}q)$ and
$L_{2}(q(t),D^{\alpha}_{F}q)$, which are equivalent to the system’s two
Hamiltonians. With the help of this postulation, we can provide a thorough
framework for figuring out the Lagrangians in Nambu mechanics [50].

$\displaystyle\delta\int L_{1}(q(t),D^{\alpha}_{F}q)d_{F}^{\alpha}t$
$\displaystyle=0$ $\displaystyle\delta\int
L_{2}(q(t),D^{\alpha}_{F}q)d_{F}^{\alpha}t$ $\displaystyle=0$ (24)

Thus we have the two Euler-Lagrange equations for the two Lagrangians:

$\displaystyle\frac{\partial^{\alpha}_{F}L_{1}}{\partial^{\alpha}_{F}t}-\frac{d_{F}^{\alpha}}{d_{F}^{\alpha}t}\frac{\partial^{\alpha}_{F}L_{1}}{\partial^{\alpha}_{F}D^{\alpha}_{F}q}$
$\displaystyle=0$ (25)
$\displaystyle\frac{\partial^{\alpha}_{F}L_{2}}{\partial^{\alpha}_{F}t}-\frac{d_{F}^{\alpha}}{d_{F}^{\alpha}t}\frac{\partial^{\alpha}_{F}L_{2}}{\partial^{\alpha}_{F}D^{\alpha}_{F}q}$
$\displaystyle=0.$ (26)

Next, we define the first canonical momentum, denoted as $p$, and the second
canonical momentum, denoted as $r$. The expressions for these momenta in terms
of the Lagrangians are as follows:

$\displaystyle p$
$\displaystyle=\frac{\partial^{\alpha}_{F}L_{1}}{\partial^{\alpha}_{F}D^{\alpha}_{F}q}$
(27) $\displaystyle r$
$\displaystyle=\frac{\partial^{\alpha}_{F}L_{2}}{\partial^{\alpha}_{F}D^{\alpha}_{F}q}.$
(28)

These momenta represent important quantities in Nambu mechanics, providing a
measure of the conjugate momenta associated with the respective Lagrangians.
Using Eq.(25) we have

$\displaystyle D^{\alpha}_{F}p$
$\displaystyle=\frac{\partial^{\alpha}_{F}L_{1}}{\partial^{\alpha}_{F}q}$ (29)
$\displaystyle D^{\alpha}_{F}r$
$\displaystyle=\frac{\partial^{\alpha}_{F}L_{2}}{\partial^{\alpha}_{F}q}.$
(30)

Now we define the two Hamiltonians $H_{1}$ and $H_{2}$ through fractal
exterior differentials as follows:

$d_{F}^{\alpha}H_{1}\wedge
d_{F}^{\alpha}H_{2}=\frac{1}{D^{\alpha}_{F}q}d_{F}^{\alpha}(pD^{\alpha}_{F}q-L_{1})\wedge
d_{F}^{\alpha}(rD^{\alpha}_{F}q-L_{2}),$ (31)

where $H_{1}$ and $H_{2}$ are expressed in terms of $q,p$, and $r$. Then the
left-hand side of Eq.(31) is equal

$\displaystyle d_{F}^{\alpha}H_{1}\wedge d_{F}^{\alpha}H_{2}$
$\displaystyle=\left(\frac{\partial^{\alpha}_{F}H_{1}}{\partial^{\alpha}_{F}q}d_{F}^{\alpha}q+\frac{\partial^{\alpha}_{F}H_{1}}{\partial^{\alpha}_{F}p}d_{F}^{\alpha}p+\frac{\partial^{\alpha}_{F}H_{1}}{\partial^{\alpha}_{F}r}d_{F}^{\alpha}r\right)$
$\displaystyle\wedge\left(\frac{\partial^{\alpha}_{F}H_{2}}{\partial^{\alpha}_{F}q}d_{F}^{\alpha}q+\frac{\partial^{\alpha}_{F}H_{2}}{\partial^{\alpha}_{F}p}d_{F}^{\alpha}p+\frac{\partial^{\alpha}_{F}H_{2}}{\partial^{\alpha}_{F}r}d_{F}^{\alpha}r\right)$
$\displaystyle=\bigg{(}\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(p,r)}d_{F}^{\alpha}p\wedge
d_{F}^{\alpha}r+\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(r,q)}d_{F}^{\alpha}r\wedge
d_{F}^{\alpha}q$
$\displaystyle+\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(q,p)}d_{F}^{\alpha}q\wedge
d_{F}^{\alpha}p\bigg{)},$ (32)

and the right-hand side of Eq.(31) is equal to, by virtue of Eqs. (27) and
Eqs. (29) ,

$\displaystyle\frac{1}{D^{\alpha}_{F}q}d_{F}^{\alpha}(pD^{\alpha}_{F}q-L_{1})\wedge
d_{F}^{\alpha}(rD^{\alpha}_{F}q-L_{2})$
$\displaystyle=\frac{1}{D^{\alpha}_{F}q}\left[D^{\alpha}_{F}qd_{F}^{\alpha}p+pd_{F}^{\alpha}D^{\alpha}_{F}q-\left(\frac{\partial^{\alpha}_{F}L_{1}}{\partial^{\alpha}_{F}q}d_{F}^{\alpha}q+\frac{\partial^{\alpha}_{F}L_{1}}{\partial^{\alpha}_{F}D^{\alpha}_{F}q}d_{F}^{\alpha}D^{\alpha}_{F}q\right)\right]$
$\displaystyle\wedge\left[D^{\alpha}_{F}qd_{F}^{\alpha}r+rd_{F}^{\alpha}D^{\alpha}_{F}q-\left(\frac{\partial^{\alpha}_{F}L_{2}}{\partial^{\alpha}_{F}q}d_{F}^{\alpha}q+\frac{\partial^{\alpha}_{F}L_{2}}{\partial^{\alpha}_{F}D^{\alpha}_{F}q}d_{F}^{\alpha}D^{\alpha}_{F}q\right)\right]$
$\displaystyle=D^{\alpha}_{F}qd_{F}^{\alpha}p\wedge
d_{F}^{\alpha}r+D^{\alpha}_{F}pd_{F}^{\alpha}r\wedge
d_{F}^{\alpha}q+D^{\alpha}_{F}rd_{F}^{\alpha}q\wedge d_{F}^{\alpha}p.$ (33)

Comparing Eqs.(5) and (5), we obtain Eq.(4) that establish the validity of our
postulate regarding the existence of two Lagrangians and the corresponding
definitions of the Hamiltonians.

##  6 Applications of Fractal Nambu Mechanics

In this section, we give some applications in classical mechanics.

######  Example 2.

Consider the Euler asymmetric top as an example. The Euler equation for this
system can be derived as follows. Let’s define the following Hamiltonians:

$H_{1}=\frac{L_{1}^{2}}{2I_{1}}+\frac{L_{2}^{2}}{2I_{2}}+\frac{L_{3}^{2}}{2I_{3}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
H_{2}=\frac{1}{2}(L_{1}^{2}+L_{2}^{2}+L_{3}^{2}),$ (34)

where $I_{i},\leavevmode\nobreak\ i=1,2,3$ represent the inertia, and
$L_{i},\leavevmode\nobreak\ i=1,2,3$ represent the components of the angular
momenta. Using Eq.(1), we can obtain fractal Euler equations as:

$\frac{d_{F}^{\alpha}L_{i}}{d_{F}^{\alpha}t}=\frac{\partial^{\alpha}_{F}(H_{1},H_{2})}{\partial^{\alpha}_{F}(x,y,z)}=\epsilon_{ijk}(D_{F,x_{j}}^{\alpha}H_{1}D_{F,x_{k}}^{\alpha}H_{2}).$
(35)

Alternatively, we can express it as:

$\frac{d_{F}^{\alpha}L_{1}}{d_{F}^{\alpha}t}=\frac{I_{3}-I_{2}}{I_{2}I_{3}}L_{2}L_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d_{F}^{\alpha}L_{2}}{d_{F}^{\alpha}t}=\frac{I_{1}-I_{3}}{I_{1}I_{3}}L_{2}L_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d_{F}^{\alpha}L_{3}}{d_{F}^{\alpha}t}=\frac{I_{2}-I_{1}}{I_{1}I_{2}}L_{1}L_{2}.$
(36)

These equations describe the fractal Euler equations for the Euler asymmetric
top system, where the fractal time derivatives of the angular momenta $L_{i}$
are expressed in terms of the inertia $I_{i}$ and the angular momenta
themselves.

######  Example 3.

The fractal Nahm system, which serves as a model for static monopoles, can be
obtained using fractal Nambu’s equation. Consider the Hamiltonians
$H_{1}=a_{1}x_{1}^{2}-a_{2}x_{2}^{2}$ and
$H_{2}=a_{1}x_{1}^{2}-a_{3}x_{3}^{2}$ where $a_{1},\leavevmode\nobreak\
a_{2},\leavevmode\nobreak\ a_{3}$ are constant. Applying fractal Nambu’s
equation, we have:

$\frac{d_{F}^{\alpha}x_{i}}{d_{F}^{\alpha}t}=\\{H_{1},H_{2},x_{i}\\}.$ (37)

Alternatively, we can express it as:

$\frac{d_{F}^{\alpha}x_{1}}{d_{F}^{\alpha}t}=a_{2}a_{3}x_{2}x_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d_{F}^{\alpha}x_{2}}{d_{F}^{\alpha}t}=a_{1}a_{3}x_{1}x_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d_{F}^{\alpha}x_{3}}{d_{F}^{\alpha}t}=a_{1}a_{2}x_{1}x_{2}.$ (38)

Figure 1: Plot of $x_{1}(t)$ exhibiting various fractal dimensions. Figure 2:
Plot of $x_{2}(t)$ exhibiting various fractal dimensions Figure 3: Plot of
$x_{3}(t)$ for various fractal dimensions

In Figures 1, 2, and 3, we showcase the solutions to Eq.(38). Here, we apply
the condition $S_{F}^{\alpha}(t)\leq t^{\alpha}$ with specific values assigned
to the parameters, namely $a_{1}=0.40452$, $a_{2}=-0.222486$, and
$a_{3}=0.494413$.

These equations describe the fractal Nahm system, providing the time
derivatives of the variables $x_{i}$ as products of the corresponding
variables themselves. This model captures the dynamics of static monopoles.

######  Example 4.

Consider a 2-dimensional fractal harmonic oscillator described by the
following Hamiltonians:

$\displaystyle H_{1}$ $\displaystyle=p_{1}^{2}+x_{1}^{2}=C_{1}$ $\displaystyle
H_{2}$ $\displaystyle=p_{2}^{2}+x_{2}^{2}=C_{2}$ $\displaystyle H_{3}$
$\displaystyle=x_{1}p_{2}-x_{2}p_{1}=C_{3}$ $\displaystyle H_{4}$
$\displaystyle=p_{1}p_{2}+x_{1}x_{2}=C_{4}.$ (39)

Here, $C_{i},i=1,2,3,4$, are constants. The corresponding fractal Nambu’s
equation is given by:

$\\{H_{1},H_{2},H_{3},f\\}=-\frac{1}{2C_{4}}\frac{\partial^{\alpha}_{F}(H_{1},H_{2},H_{2},H_{3},f)}{\partial^{\alpha}_{F}(p_{1},p_{2},x_{1},x_{2})}.$
(40)

In this equation, the curly brackets represent the fractal Nambu-Poisson
bracket, and $f$ is an arbitrary function. The equation captures the dynamics
of the system and highlights the interplay between the Hamiltonians and the
additional function $f$. The fractal Nambu’s equation provides a powerful tool
for studying the behavior of the 2-dimensional fractal harmonic oscillator
system.

##  7 Conclusion

In conclusion, Fractal Nambu mechanics is a mathematical framework that
extends Nambu mechanics to incorporate fractal geometries and dynamics.
Fractals, which are shapes with self-similar properties and fractional
dimensions, are found in various natural and artificial systems. By utilizing
fractal derivatives, fractal differential forms, and the fractal Nambu-Poisson
bracket, Fractal Nambu mechanics provides a powerful tool for describing the
dynamics of complex systems with fractal characteristics. The key features of
Fractal Nambu mechanics include the formulation of fractal Nambu equations,
which generalize the Hamilton and Nambu equations to systems with multiple
variables and Hamiltonians. The fractal Nambu-Poisson bracket satisfies
properties such as skew-symmetry, the Jacobi identity, and the Leibniz rule,
ensuring consistency and preserving the Nambu structure. Fractal canonical
transforms, which preserve the fractal Nambu structure, allow for the study of
different coordinate systems. The introduction of fractal differential forms,
such as the fractal Poincaré-Cartan 1-form and 2-form, enables the derivation
of Nambu’s equations and provides a mathematical framework for describing the
dynamics and relationships between variables in the Nambu system. Fractal
Nambu mechanics finds applications in various areas of classical mechanics,
allowing for the study of complex systems with fractal properties. Examples
include the Euler asymmetric top, where the fractal Euler equations can be
derived, and the study of dynamical systems in fractal phase spaces. Overall,
Fractal Nambu mechanics provides a comprehensive and powerful framework for
understanding and modeling complex systems with fractal characteristics,
offering new insights and approaches to studying the dynamics of fractal
structures.

## References

  * [1] A Agathiyan, A Gowrisankar, and Nur Aisyah Abdul Fataf.  On the integral transform of fractal interpolation functions.  Mathematics and Computers in Simulation, 2023.
  * [2] A Agathiyan, A Gowrisankar, Nur Aisyah Abdul Fataf, and Jinde Cao.  Remarks on the integral transform of non-linear fractal interpolation functions.  Chaos, Solitons & Fractals, 173:113749, 2023.
  * [3] Alexander S. Balankin.  A continuum framework for mechanics of fractal materials i: from fractional space to continuum with fractal metric.  Eur. Phys. J. B, 88(4), April 2015.
  * [4] Alexander S Balankin and Baltasar Mena.  Vector differential operators in a fractional dimensional space, on fractals, and in fractal continua.  Chaos Solit. Fractals, 168:113203, 2023.
  * [5] Dumitru Baleanu, Alireza K Golmankhaneh, and Ali K Golmankhaneh.  Fractional nambu mechanics.  International Journal of Theoretical Physics, 48:1044–1052, 2009.
  * [6] Rawid Banchuin.  Nonlocal fractal calculus based analyses of electrical circuits on fractal set.  COMPEL - Int. J. Comput. Math. Electr. Electron. Eng., 41(1):528–549, 2022.
  * [7] Martin T. Barlow and Edwin A. Perkins.  Brownian motion on the sierpinski gasket.  Probab. Theory Rel., 79(4):543–623, nov 1988.
  * [8] Michael F Barnsley.  Fractals Everywhere.  Academic Press, New York, 2014.
  * [9] F Bayen and M Flato.  Remarks concerning nambu’s generalized mechanics.  Physical Review D, 11(10):3049, 1975.
  * [10] Luiz Bevilacqua and Marcelo M Barros.  The inverse problem for fractal curves solved with the dynamical approach method.  Chaos Solit. Fractals, 168:113113, 2023.
  * [11] Christopher J Bishop and Yuval Peres.  Fractals in probability and analysis, volume 162.  Cambridge University Press, Cambridge, 2017.
  * [12] D Bongiorno, G Corrao, et al.  The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line.  In Bollettino di Matematica pura e applicata vol. IV, volume 4, pages 5–16. Aracne, Boston, 2011.
  * [13] Donatella Bongiorno.  Derivatives not first return integrable on a fractal set.  Ricerche di Matematica, 67(2):597–604, 2018.
  * [14] Donatella Bongiorno.  Derivation and integration on a fractal subset of the real line.  In Dr. Sid-Ali Ouadfeul, editor, Fractal Analysis - Applications and Updates, chapter 7. IntechOpen, Rijeka, 2023.
  * [15] Donatella Bongiorno and Giuseppa Corrao.  On the fundamental theorem of calculus for fractal sets.  Fractals, 23(02):1550008, 2015.
  * [16] Cristel Chandre and Atsushi Horikoshi.  Classical nambu brackets in higher dimensions.  J. Math. Phys., 64(5), 2023.
  * [17] Isaac Cohen and Andrés J Kálnay.  On nambu’s generalized hamiltonian mechanics.  International Journal of Theoretical Physics, 12:61–67, 1975.
  * [18] M. Czachor.  Waves along fractal coastlines: From fractal arithmetic to wave equations.  Acta Phys. Pol. B, 50(4):813, 2019.
  * [19] M Czachor.  Contra bellum: Bell’s theorem as a confusion of languages.  Acta Phys. Pol, 143(6):S158–S158, 2023.
  * [20] Lucero Damián Adame, Claudia del Carmen Gutiérrez-Torres, Bernardo Figueroa-Espinoza, Juan Gabriel Barbosa-Saldaña, and José Alfredo Jiménez-Bernal.  A mechanical picture of fractal darcy’s law.  Fractal and Fractional, 7(9):639, 2023.
  * [21] Airton Deppman and Eugenio Megías.  Tsallis statistics and QCD thermodynamics.  In EPJ Web of Conferences, volume 270, page 00033, 2022.
  * [22] MS El Naschie.  On certain infinite dimensional cantor sets and the schrödinger wave.  Chaos Solit. Fractals, 3(1):89–98, 1993.
  * [23] U Freiberg and Martina Zähle.  Harmonic calculus on fractals-a measure geometric approach I.  Potential Anal., 16(3):265–277, 2002.
  * [24] Philippe Gautheron.  Some remarks concerning nambu mechanics.  Letters in Mathematical Physics, 37:103–116, 1996.
  * [25] Massimiliano Giona.  Fractal calculus on [0, 1].  Chaos Solit. Fractals, 5(6):987–1000, 1995.
  * [26] Ali Khalili Golmankhaneh, AlirezaKhalili Golmankhaneh, Dumitru Baleanu, and Mihaela Cristina Baleanu.  Fractional odd-dimensional mechanics.  Adv. Differ. Equ., 2011:1–12, 2011.
  * [27] Alireza Khalili Golmankhaneh.  Fractal Calculus and its Applications.  World Scientific, Singapore, 2022.
  * [28] Alireza Khalili Golmankhaneh, Kerri Welch, Cristina Serpa, and Palle E T Jørgensen.  Non-standard analysis for fractal calculus.  The Journal of Analysis, 31:1895–1916, 2023.
  * [29] Alireza Khalili Golmankhaneh, Kerri Welch, Cemil Tunç, and Yusif S Gasimov.  Classical mechanics on fractal curves.  Eur. Phys. J. Spec. Top., pages 1–9, 2023.
  * [30] Arulprakash Gowrisankar, Alireza Khalili Golmankhaneh, and Cristina Serpa.  Fractal calculus on fractal interpolation functions.  Fractal Fract., 5(4):157, 2021.
  * [31] Janusz Grabowski and Giuseppe Marmo.  Remarks on nambu-poisson and nambu-jacobi brackets.  J. Phys. A Math., 32(23):4239, 1999.
  * [32] Partha Guha.  Applications of nambu mechanics to systems of hydrodynamical type.  J. Math. Phys., 43(8):4035–4040, 2002.
  * [33] Partha Guha.  Applications of nambu mechanics to systems of hydrodynamical type ii.  J. Nonlinear Math. Phys., 11(2):223–232, 2004.
  * [34] Ben Hambly and Peter Koepernik.  Dimension results and local times for superdiffusions on fractals.  Stoch. Process. their Appl., 158:377–417, 2023.
  * [35] Yousef Heider, Franz Bamer, Firaz Ebrahem, and Bernd Markert.  Self-organized criticality in fracture models at different scales.  Examples and Counterexamples, 2:100054, 2022.
  * [36] Huikun Jiang and Weiyi Su.  Some fundamental results of calculus on fractal sets.  Commun. Nonlinear Sci. Numer. Simul., 3(1):22–26, 1998.
  * [37] Jun Kigami.  Analysis on Fractals.  Cambridge University Press, Cambridge, jun 2001.
  * [38] MA Kotov, VM Volchuk, DM Zeziukov, and TM Pavlenko.  Application of 3d fractal modeling for predicting concrete strength characteristics.  In AIP Conference Proceedings, volume 2526. AIP Publishing, 2023.
  * [39] Michel L. Lapidus, Goran Radunović, and Darko Žubrinić.  Fractal Zeta Functions and Fractal Drums.  Springer International Publishing, New York, 2017.
  * [40] CC Lassig and Girish C Joshi.  Constrained systems described by nambu mechanics.  Lett. Math. Phys ., 41:59–63, 1997.
  * [41] O. Linton.  Fractals: On the Edge of Chaos.  Wooden Books. Bloomsbury, New York, 2021.
  * [42] Jaume Llibre, Rafael Ramírez, and Valentín Ramírez.  Generalized Cartesian–Nambu vector fields.  In Dynamics through First-Order Differential Equations in the Configuration Space, pages 177–283. Springer, New York, 2023.
  * [43] Nugzar Makhaldiani.  Nambu-poisson dynamics with some applications.  arXiv preprint arXiv:1201.5105, 2011.
  * [44] Benoit B Mandelbrot.  The Fractal Geometry of Nature.  WH freeman, New York, 1982.
  * [45] E Megias, MJ Teixeira, VS Timóteo, and A Deppman.  Nambu–jona-lasinio model with a fractal inspired coupling.  arXiv preprint arXiv:2203.11080, 2022.
  * [46] Yoichiro Nambu.  On lagrangian and hamiltonian formalism.  Progress of Theoretical Physics, 7(2):131–170, 1952.
  * [47] Yoichiro Nambu.  Generalized hamiltonian dynamics.  Physical Review D, 7(8):2405, 1973.
  * [48] L. Nottale and J. Schneider.  Fractals and nonstandard analysis.  J. Math. Phys., 25(5):1296–1300, may 1984.
  * [49] Laurent Nottale.  Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity.  World Scientific, Singapore, 1993.
  * [50] Tsuyoshi Ogawa and Toshiaki Sagae.  Nambu mechanics in the lagrangian formalism.  International Journal of Theoretical Physics, 39:2875–2890, 2000.
  * [51] GN Ord.  Fractal space-time: a geometric analogue of relativistic quantum mechanics.  J. Phys. A Math., 16(9):1869, 1983.
  * [52] Sagar A Pandit and Anil D Gangal.  On generalized nambu mechanics.  J. Phys. A Math., 31(12):2899, 1998.
  * [53] A. Parvate and A.D. Gangal.  Calculus on fractal subsets of real line-II: Conjugacy with ordinary calculus.  Fractals, 19(03):271–290, sep 2011.
  * [54] A. Parvate, S. Satin, and A.D. Gangal.  Calculus on fractal curves in $\mathbb{R}^{n}$.  Fractals, 19(01):15–27, mar 2011.
  * [55] Abhay Parvate and Anil D Gangal.  Calculus on fractal subsets of real line-I: Formulation.  Fractals, 17(01):53–81, 2009.
  * [56] TMC Priyanka, C Serpa, and A Gowrisankar.  $\alpha$-fractal function with variable parameters: An explicit representation.  Fractals, 2023.
  * [57] Didier Samayoa, Ernesto Pineda León, Lucero Damián Adame, Eduardo Reyes de Luna, and Andriy Kryvko.  The hausdorff dimension and capillary imbibition.  Fractal and Fractional, 6(6):332, 2022.
  * [58] Trifce. Sandev.  Fractional Equations and Models: Theory and Applications.  Springer, New York, 2020.
  * [59] AO Shishanin.  Nambu mechanics and its applications.  In IOP Conference Series: Materials Science and Engineering, volume 468, page 012029. IOP Publishing, 2018.
  * [60] M F Shlesinger.  Fractal time in condensed matter.  Annu. Rev. Phys. Chem., 39(1):269–290, oct 1988.
  * [61] Frank H Stillinger.  Axiomatic basis for spaces with noninteger dimension.  J. Math. Phys., 18(6):1224–1234, 1977.
  * [62] Leon Takhtajan.  On foundation of the generalized nambu mechanics.  Commun. Math. Phys., 160:295–315, 1994.
  * [63] Vasily E. Tarasov.  Fractional Dynamics.  Springer Berlin Heidelberg, New York, 2010.
  * [64] Walid Tarraf, Diogo Queiros-Condé, Patrick Ribeiro, and Rafik Absi.  Fractal geometric model for statistical intermittency phenomenon.  Entropy, 25(5):749, 2023.
  * [65] Vladimir V Uchaikin.  Fractional Derivatives for Physicists and Engineers, volume 2.  Springer, New York, 2013.
  * [66] R Valarmathi and A Gowrisankar.  Variable order fractional calculus on $\alpha$-fractal functions.  The Journal of Analysis, 31:2799–2815, 2023.
  * [67] Susie Vrobel.  Fractal Time.  World Scientific, Singapore, jan 2011.
  * [68] W.D. Withers.  Fundamental theorems of calculus for hausdorff measures on the real line.  J. Math. Anal. Appl., 129(2):581–595, feb 1988.
  * [69] Yan-Li Xu and Shao-Kai Luo.  Fractional nambu dynamics.  Acta Mechanica, 226(11):3781–3793, 2015.