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1. Introduction
There are several ways to replace the Newtonian mechanics with a deformed version, which is performed by adopting a deformed derivative instead of the Newton derivative in defining the velocity. For example, q-derivative [1-2] gives the q-deformed mechanics [3-4], fractional derivative [5-10] gives the fractional mechanics [11-14]
Let us consider the deformation
where D t A is the deformed time derivative depending on the deformation parameter A. Then the definition of the velocity is deformed as
Here the deformed velocity reduces to the ordinary Newton velocity when the special value of A is taken (q = 1 in the q-deformed mechanics and α = 1 in the fractional mechanics). This deformed mechanics deserves study as a kind of effective theory when we deal with the dynamics of complicated dynamical models.
As another example of the deformed derivative, we can consider Dunkl derivative which has been widely used in many works in various field of physics including quantum system [15-31]. Recently, the authors of Ref. 29 used the Dunkl derivative to discuss the one-dimensional quantum mechanical model, which is called a Wigner-Dunkl quantum mechanics. Here, the momentum operator is expressed in terms of the Dunkl derivative instead of the ordinary derivative;
where we set ħ = 1, and the Dunkl derivative is defined as
Here, ℛ is the spatial parity (or reflection) operator obeying ℛ:x → -x. Then the Heisenberg relation is deformed as
which is called a Wigner algebra, and the Wigner parameter v is assumed to be real. In Ref. [29] the time derivative was not replaced by the Dunkl derivative.
If we consider the undeformed quantum theory in 1+1 dimension, time and Hamiltonian can be regarded as the quantum operators obeying
The relation (6) can be deformed through Dunkl derivative concerning as follows:
where Ƭ is the temporal parity (or time-reversal) operator obeying Ƭ: t→-t, i.e.,
Then, the time realization is given by
where the Dunkl time-derivative is defined as
Dunkl time derivative arises in the time-dependent Schrödinger equation for Wigner Dunkl quantum mechanics in the form
Another example of Dunkl time derivative is the application of it to the Dunkl electrodynamics [30]. Here the authors introduced the Dunkl field strength tensor of the form
where 0 means time-component and A a means the Dunkl-deformed electromagnetic 4-potential. Using the Dunkl field strength tensor, they discussed Dunkl-Maxwell theory.
The introduction of the Dunkl derivative concerning time in the quantum theory is related to the Dunkl derivative with respect to time in the classical theory. In this paper, we will introduce the Dunkl derivative with respect to time in the classical Newton mechanics when we define the velocity or acceleration. This gives a new deformed mechanics called a Wigner-Dunkl-Newton (WDN) mechanics. This paper is organized as follows. In Sec. 2, we discuss WDN equation of motion. In Sec. 3, we discuss Hamiltonian formalism in WDN mechanics. In Sec. 4, we discuss the v-deformed functions. In Sec. 5, we discuss some mechanical examples.
2. WDN equation of motion
In this section, we will introduce the Dunkl time derivative so that it may deform the ordinary Newton mechanics. Now let us introduce the WDN velocity with the help of Dunkl derivative with respect to time as
The WDN velocity is the same as the ordinary velocity when x(t) is even. But, for odd x(t), we have v v (t) = (d/dx)x(t) + (2v/t)x(t). The WDN acceleration is also obtained by acting the Dunkl derivative with respect to time on the WDN velocity,
This can also be written as
or
where we used
The WDN velocity and WDN acceleration depend on the temporal parity of x(t). With WDN velocity and WDN acceleration, the WDN equation of motion reads
where F is a force.
When a moving observer (x’, t’) moves with the uniform WDN velocity u v relative to a fixed observer (x, t), the acceleration and velocity for a moving observer, a v ’ and v v ’ are related to those for a fixed observer, a v and v v as follows:
The last equation gives the Wigner-Dunkl-Galilei (WDG) transformation.
For the time -reversal (temporal parity), any function can be decomposed into the function with even temporal parity and the function with odd temporal parity, i.e., any function F is given by
wheret
Thus, the WDN equation of motion for odd and even part reads
3. Hamiltonian formalism in WDN mechanics
In the WDN mechanics, the work is not well defined because we have no information for the inverse of the Dunkl derivative (Dunkl integral). Nevertheless, we can obtain the conserved Hamiltonian by introducing the deformed Poisson bracket.
From the time-dependent Schrödinger equation for Wigner Dunkl quantum mechanics, we know the Hamiltonian for classical variables x, p is given by
From the deformed commutator relation (5), we can define the deformed Poisson bracket (DPB) as
Indeed, the Eq. (27) gives the relation
From the time-dependent Schrödinger equation for Wigner Dunkl quantum mechanics, the time evolution of some classical quantity A(x, p) is defined as
which gives the Dunkl-Wigner-Hamilton equation
Thus, WDN equation of motion reads
The evolution of the Hamiltonian is
which implies that the Hamiltonian is constant, i.e., a conserved quantity. From now on we define the force corresponding to the conserved Hamiltonian as
where we call V a WDN potential energy in the WDN mechanics. Like the ordinary Newton mechanics, if there does not exist potential energy obeying the Eq. (34), we have the WDN equation of motion,
4. The v-deformed functions
In this section, we discuss the v-exponential function, v-deformed hyperbolic functions, and v-deformed trigonometric functions. First, consider the following v-deformed differential equation
We will denote the solution of the above equation by
which we call v-exponential function.
Considering parity, we can set the solution of the Eq. (36) as
where y e (t) is the even function obeying Ty e (t) = y e (t), while y o (t) is the odd function obeying Ty o (t) = y o (t). Inserting the Eq. (38) into the Eq. (36) and splitting the Eq. (36) into the even part and odd part we get
From the parity of y e (t) and y o (t), we can set
Inserting the Eq. (40) into the Eq. (39), we get the following recurrence relations
Inserting the Eq. (42) into the Eq. (41), we have
which gives
Thus, we have the following solution of the Eq. (36):
where v-deformed hyperbolic functions are defined as
and
and
Here, the parity relation for the
It can be easily checked that the v-deformed hyperbolic functions reduce to cosh(at) and sinh(at) in the limit v→0. Acting the v-derivative on the v-exponential function and the v-deformed hyperbolic functions, we have
If we replace t → it in the Eq. (46), we have the v-deformed Euler relation
where
One can also express the v-deformed trigonometric functions as
Figure 1 shows the plot of y = cos v for v = 0 (Gray) v = 0.2, (Brown), and for v = -0.2 (Pink). Figure 2 shows the plot of y = sin v (t) for v = 0 (Gray), v = 0.2 (Brown), and for v = -0.2 (Pink).
Acting the v-derivative on the v-deformed trigonometric functions, we get the following relations
5. Some examples
Let us discuss some examples for the WDN mechanics in one dimension.
5.1 Particle at rest
Let us consider that a particle is at rest, and its position is x(0). In ordinary Newton mechanics, we have v(t)=0 which gives x(t)=x(0). In WDN mechanics, this case is replaced with
or
Let us set
Inserting the Eq. (64) into the Eq. (63) and splitting into the even part and the odd part we get
Solving these with the initial position x(0) we get
which gives the same result as the ordinary Newton mechanics,
5.2. Uniform WDN velocity
Let us consider that a particle moves with the uniform WDN velocity, u v = const, and its initial position is x(0). In WDN mechanics, we have
or
Solving this equation, we have
From the Eq. (71), we get
In the Eq. (72), we set
Inserting the Eq. (74) into the Eq. (72) we get
Thus, we have
Because u v is even, we have (1-T)u v . Thus, the WDN acceleration becomes zero.
5.3. Uniform WDN acceleration
Let us consider that a particle moves with the uniform WDN acceleration, a v = const, and its initial position is x(0) and its initial velocity v(0). In WDN mechanics, we have
or
Solving this equation, we have
Or
which gives
Thus, we have
5.4. Resisted motion with linear damping
Let us consider that a particle moves in the viscous medium with the resistance proportional to the WDN velocity, and its initial position is x(0), and its initial WDN velocity v(0). In WDN mechanics, we have
or
Solving this equation, we have
6. Conclusion
From the introduction of the Dunkl derivative concerning time in the quantum theory [15], we proposed a new deformed mechanics called WDN mechanics, where the WDN velocity and WDN acceleration are defined by the Dunkl derivative for time. We discussed Hamiltonian formalism in WDN mechanics. For the Dunkl time derivative, we found some deformed elementary functions such as the v-deformed exponential functions, v-deformed hyperbolic functions, and v-deformed trigonometric functions. Using these functions, we solved some problems in one-dimensional WDN mechanics. Some problems remain unsolved for WDN mechanics. For example, the work is not well defined for WDN mechanics. For this reason, we obtained the conserved Hamiltonian from the deformed Poisson bracket. We think that these problems and their related topics will become clear soon.
