1.Introduction

Fractional derivative is a derivative of fractional (non-integer) order and has a long history ¹. It has attracted much attention from mathematicians, physicists, and engineers in recent decades ^2-18. In Ref. ², the concept of the solution has been presented for a differential equation of fractional order with uncertainty. Ref. ³ concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough, and mathematically rigorous study of their properties and the corresponding differential equations. Also, the authors have investigated the applications of fractional calculus to first-order integral equations with power and power logarithmic kernels, and with special functions in kernels and to Euler-Poisson-Darboux’s type equations and differential equations of fractional order in Refs. ⁴. In Ref. ^5-11, the most comprehensive developments on fractional differential and fractional integral-differential equations involving many different potentially useful operators of fractional calculus are provided. In Ref. ¹², the authors propose a novel fractional-order adaptive filter structures such that the output from the conventional filtered-x least mean square algorithm is passed through a new update equation derived from a cost function based on a posteriori error and optimized using fractional derivatives. In Ref. ^13-15, the steady-state fractional advection-dispersion equation on bounded domains in R^d is discussed, and the fractional differential and integral operators are defined and analyzed. A novel method as the double integral method, for obtaining the solution of the fractional differential equation in the class of second-grade fluids models is proposed ¹⁶. Nonlocal differential and integral operators with fractional order and fractal dimension have been recently introduced in Ref. ¹⁷, and in this paper has been defined the powerful mathematical tools to model complex real-world problems that could not be modeled with classical and nonlocal differential and integral operators with a single order. In Ref. ¹⁸, the local asymptotic stability and the global asymptotic stability for the trivial equilibrium point of the fractional electrical RLC circuit have been discussed.

Fractional derivative is regarded as a powerful tool for studying nonlinear systems ^19-22. In Ref. ¹⁹, the local fractional Burgers’ equation (LFBE) is investigated from the point of view of local fractional conservation laws envisaging a nonlinear local fractional transport equation with a linear non-differentiable diffusion term. Reference ²⁰ is a review of physical models that look very promising for the future development of fractional dynamics, and the Authors suggest a short introduction to fractional calculus as a theory of integration and differentiation of noninteger order. Also, Some applications of integro-differentiations of fractional orders in physics are discussed, and the models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented ²⁰. In Ref. ²¹, the problem of robust control of uncertain fractional-order nonlinear complex systems is investigated and, after establishing a simple linear sliding surface, the sliding mode theory is used to derive a novel robust fractional control law for ensuring the existence of the sliding motion in finite time. In Ref. ²², By use of the Gibbs-Appel approach and the complementary constitutive axioms corresponding to the fractional Kelvin-Zener model of the viscoelastic body, the equations of motion were derived. Riemann and Louville ⁹ first constructed the fractional derivative through the integral,

where n=[α]+1 and [x] is the greatest integer equal to or less than x. Riemann-Louville fractional derivative of a constant is not zero. To cure this problem, Caputo ¹⁴ modified Riemann and Louville’s definition for fractional derivative, which is called Caputo fractional derivative,

where n=[α]+1.

Riemann-Louville fractional derivative and Caputo fractional derivative do not obey the Leibniz rule. A new fractional derivative obeying Leibniz’s rule was introduced by Khalil, Horani, Yousef, and Sababheh ²³. It is called a conformable fractional derivative and depends just on the basic limit definition of the derivative. Namely, for a function f:(0,∞)→R the conformable fractional derivative of order α (0<α≤1) of f at x > 0 is defined by

and the conformable fractional derivative at 0 is defined as (∂xa,KHYSf)(0)=limx→0(∂xaf)(x). Some applications of the conformable fractional derivative are given in ^23-34.

Riemann-Louville fractional derivative and Caputo fractional derivative have been successfully applied to characterize the constitutive equations of viscoelastic non-Newtonian fluid, by using the fractional derivatives to replace the integer-order derivatives ^35-37. For the conformable fractional derivative, a study on the classical mechanics was given in ²⁹, and a study on the motion in the viscoelastic medium was given in ³⁰, where the fractional velocity was defined as

The fractional velocity defined in Eq. (4) can be written as

Eq. (5) is not invariant under the translation in time t → τ while it is invariant under the fractional translation in time, tα→tα+τα.

Instead of the fractional velocity with fractional translation in time, we can consider the fractional velocity with fractional translation in space. In this paper, we introduce a new fractional velocity defined by

for x(t')>x(t)>0. We refer to the fractional velocity (6) as fractional velocity with fractional space translation symmetry. The fractional velocity with fractional space translation symmetry is not invariant under the translation in position x→x+a while it is invariant under the fractional translation in position, xα→xα+aα. Using the fractional velocity with fractional space translation symmetry, we define the fractional acceleration with fractional space translation symmetry and construct the fractional Newton equation with fractional space translation symmetry. This paper is organized as follows: In Sec. 2, we discuss the fractional velocity and fractional acceleration. In Sec. 3, we discuss the fractional work, fractional kinetic energy, and fractional potential energy. In Sec. 4, we discuss the fractional Hamiltonian formalism for the fractional mechanics. In Sec. 5, we discuss some examples of the fractional mechanics.

2.Fractional velocity and fractional acceleration: New definitions

Now let us invoke the average velocity when a particle lies on x (t) at time t and lies on x(t´) at time t´ where t´ > t, which is

where the displacement is defined as

e and it is invariant under the ordinary space translation.

First let us consider the case that x(t')>x(t)>0. If we define the fractional displacement as

and define the fractional average velocity as

we know that the fractional average velocity is invariant under the fractional space translation.

The numerator in Eq. (10) can not be regarded as the deformed subtraction because its corresponding deformed addition is not associative. For this reason, we define the fractional addition and fractional subtraction for the case of x > 0 y > 0 as

The fractional addition is commutative and associative and preserves the dimension.

The fractional displacement is then written as

and the fractional average velocity

Here we have a problem. The position x(t) can take a negative value. In that case, the fractional addition or fractional subtraction becomes a complex number. To cure this problem, we should modify the definitions of fractional addition and fractional subtraction so that they may hold for any x,y∈R.

For x > 0 y > 0, the fractional addition and fractional subtraction are defined as

For x > 0, y < 0, the fractional addition and fractional subtraction are defined as

For x < 0, y > 0, the fractional addition and fractional subtraction are defined as

For x < 0, y < 0, the fractional addition and fractional subtraction are defined as

Thus we can write the fractional addition and fractional subtraction as

Besides, we have the relation x⊖αy=x⊕α(-y). For the fractional addition and fractional subtraction, we have the following properties:

1. Distributivity

2. Expansion

From the definition of fractional average velocity, the fractional instantaneous velocity (shortly fractional velocity) is defined by

And

Equation (22) and Eq. (23) are unified as follows:

Or

which is derived in Appendix A.

The fractional average acceleration is defined as

hence the fractional instantaneous acceleration ( shortly fractional acceleration) is given by

The fractional version of Newton’s law is then

Equation (28) is invariant under the following transformation for the fractional velocity,

which is the same as the ordinary case. But, Eq. (28) is not invariant under the ordinary Galilei transformation

Instead, Eq. (28) is invariant under the fractional Galilei transformation, which is given by

for x,x',u>0. Now we will refer to the system whose coordinate x´ is given by Eq. (31) as a fractional inertial frame. For the fractional inertial frame, the fractional version of Newton’s law remains invariant, which is called a fractional Galilei relativity.

3.Fractional work, fractional kinetic energy, and fractional potential energy

In ordinary mechanics, work w is the product of the force and displacement. When a force F is acted on a body with a mass m, and this body moves from x to x´ (x'>x) in a same direction as the force, the work is given by

When the force varies during motion from b _i to b _f , the work is given by

For the conserved force, we have the potential energy F=-∂xV; thus, the work reduces to

For the fractional case, we define the fractional work W as the product of the force and fractional displacement,

If we set x'=x+Δx, we have

Thus, the fractional work reads

If we define the fractional potential energy as

where

we have

Now let us find the fractional kinetic energy from the definition of the fractional work, which is given by

where the fractional kinetic energy is defined as

Thus, we have fractional energy conservation,

4.Fractional Hamiltonian formalism in the fractional mechanics

The fractional classical mechanics is also constructed by the fractional Poisson bracket defined as follows:

Wich gives

If we introduce the fractional Hamiltonian as

We have the fractional Hamilton´s equation of motion as

From the first relation of Eq. (47), the fractional momentum is given by

Inserting Eq. (48) into Eq. (47), we have the same equation like Eq. (28).

Now let us find the geometrical meaning of the fractional Poisson bracket. The fractional Poisson bracket is related to the mechanics in a curved space. In one dimensional curved space, the metric is given by

In a curved space, the Poisson bracket is deformed as

This is the same as Eq. (45) if we take

Or

Thus, our model is related to the mechanics in a one-dimensional curved space with a metric of the form (51). The fractional momentum defined by Eq. (48) is not related to the ordinary translation but the fractional translation T\al(Δx):x→x⊕αΔx. Indeed we can easily check that the fractional momentum is invariant under the fractional translation. This translation obeys the product rule, Tα(Δx)Tα(Δx')=Tα(Δx⊕αΔx'). Thus, the fractional momentum is the conserved quantity for the fractional translation.

6.Conclusion

In this paper, we introduced a new fractional derivative to define a new fractional velocity with fractional space translation symmetry based on the fractional addition rule. We used the fractional addition rule to introduce the concept of fractional displacement, fractional velocity, and fractional acceleration and constructed the fractional version of Newton’s law. We showed that the fractional version of Newton’s law is invariant under the fractional Galilei transformation. We defined the fractional work through the fractional displacement and constructed fractional kinetic energy and fractional potential energy. We also derived the conservation of fractional mechanical energy. We discussed fractional Hamiltonian formalism for the fractional mechanics with the help of the fractional Poisson bracket. We found that the fractional mechanics with fractional space translation symmetry is related to classical mechanics in a curved space. We discussed some examples of the fractional mechanics such as uniform fractional velocity motion, uniform fractional acceleration motion, and fractional harmonic oscillator problem.