1. Introduction
Magnetic flux density tends to be pretty impressive. Fermi Walker systems are flexible connections of fluxes, solids, and streams, are full of relationship, model nonlinear equations, and are solidly influenced by magnetic variations. These quasi systems are more generous, enveloping biomaterials, liquids, polymers, gels, and foams, and have an extensive range of associated uniformly motions. Proceeding with these motions desires a hidden characterization of complexities of these profuse Magnetic flux systems [1].
A great hypersensitive magnetic torque calibrates for operation in fluctuated large electromagnetic fields has been importuned, exclusively for direction of formulation scientific electromagnetic densities of original magnetic particles. Thus, densities are of powerful influence from perspective of perception of peculiar magnetic attributes. Also, even with last advancement of crystal magnetic theory, it is not uncomplicated to expand magnetic crystals by magnetic moments.
Uniformly accelerated structures have been obtained in [2] and, more recently, in Refs. [3,4]. In this paper, authors suggest new methods for finding solutions for uniformly quasi accelerated motion in a general curved spacetime. This extends results of [5], where explicit solutions are computed for flat spacetime only. Other approaches rely on convenient use of Frenet equations [6,7].
Magnetic flux density combinations are extensively operated as a magnetic moment component considering numerous varieties of machines like potential generators, turbines and activators. Magnetic hybrids are frequently embittered with meanwhile application of these machines. Melting is a result of the destruction of electrical energy at the elastic magnetic torque. Accordingly, advancement in the capability of elastic magnetic torque is momentous to recover the electrical energy and to bypass to contaminate bordering machines. Hence, preserving electrical energy benefits at preserving the earth’s ordinary reserves and climates. The inferior amorphous-forming capability of hybrids combines cripples with its operational capability at power generators and turbines [8].
Electromagnetic cover, just as an appropriate cover approach to establish a point, has explicit operation forecasts in various areas, being biomedicine, essential sensibility, and act cover [9]. The ultimate significant utilization areas are essentially nosy surgery [10]. The electromagnetic cover system may be managed like a split of the navigation structure to establish a certain-future situation for the pharmaceutical apparatus in the inmate’s frame, implement extreme advantage to scientists [11]. Correlated with alternative navigation designs as optical new navigation, the electromagnetic cover has no radioactivity cripple or no constraint of radiation procedure. Also, an electromagnetic traverse system found on abstain energy of some magnetic flux density field characterized by some azimuth intersection is recommended [12].
The biharmonic particles also play an important role in geometry. Also, a large number of experts have analyzed geometric biharmonic particles and surface conditions, [13,14]. On the other hand, the energy concept has been obtained with some characterizations, [15-19].
Magnetic particles provide an impressive, reliable, and densely parallel synthesized attitude. By utilizing magnetic torque, principal magnetic matter experimenters have established advanced technologies in physics, optics, world sciences, and basic materials. Categorically, magnetic torque has been managed to construct particle interruptions, to assemble an influence fusion magnetic particles, and to apply and shape optical materials [20-25].
Aim of the research work, a general construction method is proposed for magnetic interpretation of new quasi uniformly accelerated potential electric energy and Fermi parallel transportation in cold plasma with illustrations of results. We consider the quasi uniformly accelerated motion with some Lorentz fields. The applications of these models are geometric and physical interpretations of quasi uniformly accelerated potential electric energy.
The construction of our article is as follows: Firstly, we construct the quasi uniformly accelerated motion (QUAM) with Fermi derivative by cold plasma in Heisenberg space. We define new quasi uniformly accelerated potential electric energy of quasi normal magnetic biharmonic particles and some Lorentz fields. Moreover, we design the new relationship between the quasi uniformly accelerated motion and the Fermi parallel transportation in cold plasma. Also, we collect new physical geometric designs for a quasi uniformly accelerated motion of normal magnetic biharmonic particles in Heisenberg space. Finally, we construct quasi uniformly accelerated potential electric energy with respect to its electric field and some quasi curvatures.
2. Backround on quasi frame and Heisenberg space
Heisenberg space have following metric [26]:
g=dx2+dy2+(dz-xdy)2.
Heisenberg space’s basis is given by
f1=∂∂x, f2=∂∂y+x∂∂z, f3=∂∂z.
Let α be a particle with arclength. Also, t, n, and b describe tangent, principal normal, and secondary normal fields, respectively. Then, Frenet equations are given by
∇tt∇tn∇tb=κ-κτ-τtnb,
where κ and τ are curvatures of particles.
A new quasi frame of a particle is presented by [27],
Tq=t,Nq=t×kt×k,Bq=Tq∧Nq,
where k= (0,0,1) is the projection field.
Since the new quasi frame is provided by
∇sTq=ϱ1Nq+ϱ2Bq,
∇sNq=-ϱ1Tq+ϱ3Bq,
∇sBq=-ϱ2Tq-ϱ3Nq,
where the angle ψ is between the quasi field n
q and the vector n and
ϱ1=κcosψ,ϱ2=-κsinψ,ϱ3=ψ'+τ.
3. Quasi uniformly accelerated motion of quasi normal magnetic biharmonic particles
In this section, we characterize the quasi uniformly accelerated motion of moving charged quasi normal magnetic biharmonic particles with unit speed normal magnetic biharmonic particles in Heisenberg space. We obtain necessary and sufficient conditions that have to be satisfied by the particle in terms of the Frenet scalars of the worldline of magnetic curves. We present the following definition of the quasi uniformly accelerated motion.
Let α be a biharmonic particle and B be a magnetic field in the Heisenberg space. We call the curve α as a quasi normal magnetic biharmonic particle if the quasi normal field of the biharmonic particle meets the following Lorentz force equation.
∇sNq=ϕ(Nq)=B×Nq.
Fermi Walker derivative of any field R is defined by
∇sfR=∇sR-hTq,R∇sTq+h(∇sTq,R)Tq,
where ∇ is Levi-Civita connection associated metric h [28].
♣ The particle
X
is quasi uniformly accelerated motion iff
∇sf(∇sX)=0.
• Lorentz force of
T
q
,
N
q
,
B
q
are presented
ϕ(Tq)=(ϱ1cosχ1s+χ2+ωcosφsinχ1s+χ2)f1+(ωcosφcosχ1s+χ2-ϱ1sinχ1s+χ2)f2-ωsinφf3,
ϕ(Nq)=(ϱ3cosφsinχ1s+χ2-ϱ1sinφsinχ1s+χ2)f1+(ϱ3cosφcosχ1s+χ2-ϱ1sinφcosχ1s+χ2)f2-(ϱ3sinφ+ϱ1cosφ)f3,
ϕ(Bq)=-(ϱ3cosχ1s+χ2+ωsinφsinχ1s+χ2)f1+(ϱ3sinχ1s+χ2-ωsinφcosχ1s+χ2)f2-ωcosφf3,
where ω = φ(T
q) · B
q
.
• Magnetic vector field B of normal magnetic biharmonic particle is given by
B=(ϱ3sinφsinχ1s+χ2-ωcosχ1s+χ2+ϱ1cosφsinχ1s+χ2)f1+(ϱ3sinφcosχ1s+χ2+ωsinχ1s+χ2+ϱ1cosφcosχ1s+χ2)f2+(ϱ3cosφ-ϱ1sinφ)f3.
Lemma 1.
♣ Fermi Walker derivative of φ(T
q) is
∇sfϕ(Tq)=((ω'+ϱ1ϱ3)cosφsinχ1s+χ2+(ϱ1'-ϱ3ω)cosχ1s+χ2)f1
+((ω'+ϱ1ϱ3)cosφcosχ1s+χ2-(ϱ1'-ϱ3ω)sinχ1s+χ2)f2
-(ω'+ϱ1ϱ3)sinφf3.
♣ Fermi Walker derivative of φ(N
q) is
∇sfϕ(Nq)=(ϱ3'cosφsinχ1s+χ2-ϱ1'sinφsinχ1s+χ2-ϱ32cosχ1s+χ2)f1
+(ϱ3'cosφcosχ1s+χ2-ϱ1'sinφcosχ1s+χ2+ϱ32sinχ1s+χ2)f2
-(ϱ3'sinφ+ϱ1'cosφ)f3.
♣ Fermi Walker derivative of φ(B
q) is
∇sfϕ(Bq)=-(ω'sinφsinχ1s+χ2+ϱ3'cosχ1s+χ2+ϱ32cosφsinχ1s+χ2)f1
+(ϱ3'sinχ1s+χ2-ϱ32cosφcosχ1s+χ2-ω'sinφcosχ1s+χ2)f2
+(ϱ32sinφ-ω'cosφ)f3.
♣ Fermi Walker derivative of B is
∇sfB=(ϱ3'sinφsinχ1s+χ2-(ω'+ϱ1ϱ3)cosχ1s+χ2+(ϱ1'-ωϱ3)
cosφsinχ1s+χ2)f1+(ϱ3'sinφcosχ1s+χ2+(ω'+ϱ1ϱ3)sinχ1s+χ2
+(ϱ1'-ωϱ3)cosφcosχ1s+χ2)f2+(ϱ3'cosφ-(ϱ1'-ωϱ3)sinφ)f3.
Lemma 2.
♣ φ(T
q) is Fermi Walker parallel iff
((ω'+ϱ1ϱ3)cosφsinχ1s+χ2+(ϱ1'-ϱ3ω)cosχ1s+χ2)=0
((ω'+ϱ1ϱ3)cosφcosχ1s+χ2-(ϱ1'-ϱ3ω)sinχ1s+χ2)=0
(ω'+ϱ1ϱ3)sinφ=0.
♣ φ(N
q) is Fermi Walker parallel iff
(ϖ'cosφsinχ1s+χ2-ϰ1'sinφsinχ1s+χ2-ϰ3ϖcosχ1s+χ2)=0,
(ϖ'cosφcosχ1s+χ2-ϰ1'sinφcosχ1s+χ2+ϰ3ϖsinχ1s+χ2)=0,
(ϖ'sinφ+ϰ1'cosφ)=0.
♣ φ(B
q) is Fermi Walker parallel iff
(ω'sinφsinχ1s+χ2+ϱ3'cosχ1s+χ2+ϱ32cosφsinχ1s+χ2)=0,
(ϱ3'sinχ1s+χ2-ϱ32cosφcosχ1s+χ2-ω'sinφcosχ1s+χ2)=0,
(ϱ32sinφ-ω'cosφ)=0.
♣ B is Fermi Walker parallel iff
(ϱ3'sinφsinχ1s+χ2-(ω'+ϱ1ϱ3)cosχ1s+χ2+(ϱ1'-ωϱ3)cosφsinχ1s+χ2)=0,
(ϱ3'sinφcosχ1s+χ2+(ω'+ϱ1ϱ3)sinχ1s+χ2+(ϱ1'-ωϱ3)cosφcosχ1s+χ2)=0,
(ϱ3'cosφ-(ϱ1'-ωϱ3)sinφ)=0.
Quasi uniformly accelerated motion
In this subsection, we will characterize the uniformly quasi accelerated motion (QUAM) in the Heisenberg space.
♠ The φ(T
q) obeys a quasi uniformly accelerated motion iff
(((ω'+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)+(ϱ12+ϱ2ω)ϱ2)
×cosφsinχ1s+χ2-(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)+ϱ2(ω'+ϱ1ϱ3)
-((ϱ1'-ϱ3ω)ϱ1+ϱ2(ω'+ϱ1ϱ3)))sinφsinχ1s+χ2+((ϱ1'-ϱ3ω)'
-(ϱ12+ϱ2ω)ϱ1-ϱ3(ω'+ϱ1ϱ3)+(ϱ12+ϱ2ω)ϱ1)cosχ1s+χ2)=0,
(((ω'+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)+(ϱ12+ϱ2ω)ϱ2)
×cosφcosχ1s+χ2-((ϱ1'-ϱ3ω)'-(ϱ12+ϱ2ω)ϱ1-ϱ3(ω'+ϱ1ϱ3)
+(ϱ12+ϱ2ω)ϱ1)sinχ1s+χ2-(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)
+ϱ2(ω'+ϱ1ϱ3)-((ϱ1'-ϱ3ω)ϱ1+ϱ2(ω'+ϱ1ϱ3)))sinφcosχ1s+χ2)=0,
(((ω'+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)+(ϱ12+ϱ2ω)ϱ2)sinφ
+(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)+ϱ2(ω'+ϱ1ϱ3)-((ϱ1'-ϱ3ω)ϱ1
+ϱ2(ω'+ϱ1ϱ3)))cosφ)=0,
By Fermi derivative and following equations, we obtain above system.
∇sϕ(Tq)=((ω'+ϱ1ϱ3)cosφsinχ1s+χ2-(ϱ12+ϱ2ω)sinφsinχ1s+χ2+(ϱ1'-ϱ3ω)
×cosχ1s+χ2)f1+((ω'+ϱ1ϱ3)cosφcosχ1s+χ2-(ϱ12+ϱ2ω)sinφcosχ1s+χ2
-(ϱ1'-ϱ3ω)sinχ1s+χ2)f2-((ω'+ϱ1ϱ3)sinφ+(ϱ12+ϱ2ω)cosφ)f3.
It provides us the following equation
∇sf∇sϕ(Tq)=(((ω'+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)+(ϱ12+ϱ2ω)ϱ2)
×cosφsinχ1s+χ2-(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)+ϱ2(ω'+ϱ1ϱ3)
-((ϱ1'-ϱ3ω)ϱ1+ϱ2(ω'+ϱ1ϱ3)))sinφsinχ1s+χ2+((ϱ1'-ϱ3ω)'
-(ϱ12+ϱ2ω)ϱ1-ϱ3(ω'+ϱ1ϱ3)+(ϱ12+ϱ2ω)ϱ1)cosχ1s+χ2)f1'
+(((ω+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)+(ϱ12+ϱ2ω)ϱ2)cosφcosχ1s+χ2
-((ϱ1'-ϱ3ω)'-(ϱ12+ϱ2ω)ϱ1-ϱ3(ω'+ϱ1ϱ3)+(ϱ12+ϱ2ω)ϱ1)sinχ1s+χ2
-(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)+ϱ2(ω'+ϱ1ϱ3)-((ϱ1'-ϱ3ω)ϱ1
+ϱ2(ω'+ϱ1ϱ3)))sinφcosχ1s+χ2)f2-(((ω'+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2
+ϱ3(ϱ1'-ϱ3ω)+(ϱ12+ϱ2ω)ϱ2)sinφ+(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)
+ϱ2(ω'+ϱ1ϱ3)-((ϱ1'-ϱ3ω)ϱ1+ϱ2(ω'+ϱ1ϱ3)))cosφ)f3.
♠ The φ(N
q) obeys a quasi uniformly accelerated motion iff
(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)
-(ϱ12+ϱ32)ϱ1))sinφsinχ1s+χ2-((ϱ1'+ϱ3ϱ2)ϱ1+(ϱ12+ϱ32)'
+ϱ3(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)ϱ1)cosχ1s+χ2+((ϱ3'+ϱ1ϱ2)'
-(ϱ12+ϱ32)ϱ3-(ϱ1'+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))cosφsinχ1s+χ2)=0,
(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)
-(ϱ12+ϱ32)ϱ1))sinφcosχ1s+χ2+((ϱ1'+ϱ3ϱ2)ϱ1+(ϱ12+ϱ32)'
+ϱ3(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)ϱ1)sinχ1s+χ2+((ϱ3'+ϱ1ϱ2)'
-(ϱ12+ϱ32)ϱ3-(ϱ1'+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))cosφcosχ1s+χ2)=0
(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ12+ϱ32)ϱ1))cosφ
-((ϱ3'+ϱ1ϱ2)'-(ϱ12+ϱ32)ϱ3-(ϱ1'+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))sinφ)=0.
Also,
∇sϕ(Nq)=((ϱ3'+ϱ1ϱ2)cosφsinχ1s+χ2-(ϱ1'+ϱ3ϱ2)sinφsinχ1s+χ2
-(ϱ12+ϱ32)cosχ1s+χ2)f1+((ϱ3'+ϱ1ϱ2)cosφcosχ1s+χ2
-(ϱ1'+ϱ3ϱ2)sinφcosχ1s+χ2+(ϱ12+ϱ32)sinχ1s+χ2)f2
-((ϱ3'+ϱ1ϱ2)sinφ+(ϱ1'+ϱ3ϱ2)cosφ)f3.
By Fermi derivative and following equations, we easily obtain above system.
∇sf∇sϕ(Nq)=(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)
-(ϱ12+ϱ32)ϱ1))sinφsinχ1s+χ2-((ϱ1'+ϱ3ϱ2)ϱ1+(ϱ12+ϱ32)'
+ϱ3(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)ϱ1)cosχ1s+χ2+((ϱ3'+ϱ1ϱ2)'
-(ϱ12+ϱ32)ϱ3-(ϱ1'+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))cosφsinχ1s+χ2)f1
+(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)
-(ϱ12+ϱ32)ϱ1))sinφcosχ1s+χ2+((ϱ1'+ϱ3ϱ2)ϱ1+(ϱ12+ϱ32)'
+ϱ3(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)ϱ1)sinχ1s+χ2+((ϱ3'+ϱ1ϱ2)'
-(ϱ12+ϱ32)ϱ3-(ϱ1'+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))cosφcosχ1s+χ2)f2
+(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)
-(ϱ12+ϱ32)ϱ1))cosφ-((ϱ3'+ϱ1ϱ2)'-(ϱ12+ϱ32)ϱ3
-(ϱ1'+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))sinφ)f3.
♠ The φ(B
q) obeys a quasi uniformly accelerated motion iff
((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1-ω')'+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1+(ϱ32
+ωϱ2)ϱ2))sinφsinχ1s+χ2+((ϱ3ϱ1-ω')ϱ1-(ϱ3'+ωϱ1)'+ϱ3(ϱ32
+ωϱ2)-(ϱ3ϱ1-ω')ϱ1)cosχ1s+χ2+(ϱ2(ϱ3ϱ1-ω')-(ϱ32
+ωϱ2)'-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)cosφsinχ1s+χ2)=0,
((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1-ω')'+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1+(ϱ32
+ωϱ2)ϱ2))sinφcosχ1s+χ2-((ϱ3ϱ1-ω')ϱ1-(ϱ3'+ωϱ1)'+ϱ3(ϱ32
+ωϱ2)-(ϱ3ϱ1-ω')ϱ1)sinχ1s+χ2+(ϱ2(ϱ3ϱ1-ω')-(ϱ32+ωϱ2)'
-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)cosφcosχ1s+χ2)=0,
((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1-ω')'+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1
+(ϱ32+ωϱ2)ϱ2))cosφ-(ϱ2(ϱ3ϱ1-ω')-(ϱ32
+ωϱ2)'-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)sinφ)=0.
Then, it is easy to see that
∇sϕ(Bq)=((ϱ3ϱ1-ω')sinφsinχ1s+χ2-(ϱ3'+ωϱ1)cosχ1s+χ2
-(ϱ32+ωϱ2)cosφsinχ1s+χ2)f1+((ϱ3ϱ1-ω')sinφcosχ1s+χ2
+(ϱ3'+ωϱ1)sinχ1s+χ2-(ϱ32+ωϱ2)cosφcosχ1s+χ2)f2
+((ϱ3ϱ1-ω')cosφ+(ϱ32+ωϱ2)sinφ)f3.
In a similar way, we get
∇sf∇sϕ(Bq)=((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1-ω')'+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1
+(ϱ32+ωϱ2)ϱ2))sinφsinχ1s+χ2+((ϱ3ϱ1-ω')ϱ1-(ϱ3'+ωϱ1)'
+ϱ3(ϱ32+ωϱ2)-(ϱ3ϱ1-ω')ϱ1)cosχ1s+χ2+(ϱ2(ϱ3ϱ1-ω')-(ϱ32+ωϱ2)'
-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)cosφsinχ1s+χ2)f1+((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1-ω')'
+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1+(ϱ32+ωϱ2)ϱ2))sinφcosχ1s+χ2-((ϱ3ϱ1-ω')ϱ1
-(ϱ3'+ωϱ1)'+ϱ3(ϱ32+ωϱ2)-(ϱ3ϱ1-ω')ϱ1)sinχ1s+χ2+(ϱ2(ϱ3ϱ1-ω')
-(ϱ32+ωϱ2)'-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)cosφcosχ1s+χ2)f2+((ϱ1(ϱ3'+ωϱ1)
+ϱ3ϱ1-ω')'+ϱ2ϱ32+ωϱ2-ϱ3'+ωϱ1ϱ1+ϱ32+ωϱ2ϱ2cosφ
-(ϱ2(ϱ3ϱ1-ω')-(ϱ32+ωϱ2)'-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)sinφ)f3.
4 Energy flux density in cold plasma
The force acting on an electron in the cold plasma is given by
F=-q(E+v×B)-mv, mq=π,
where v is a velocity of electrons, m is a mass, q is an electric charge, B is a magnetic field and E is an electric field [29-31].
From the force equation, the electric field is given by
E=((ω-πϱ2)cosφsinχ1s+χ2-πsinφsinχ1s+χ2+(ϱ1
-πϱ1)cosχ1s+χ2)f1+((ω-πϱ2)cosφcosχ1s+χ2
-πsinφcosχ1s+χ2-(ϱ1-πϱ1)sinχ1s+χ2)f2
-((ω-πϱ2)sinφ+πcosφ)f3.
♠The Ɛ obeys a quasi uniformly accelerated motion iff
((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ2
+ϱ3((ϱ1-πϱ1)'-ϱ3(ω-πϱ2)))cosφsinχ1s+χ2-((π+ϱ1(ϱ1-πϱ1)
+ϱ2(ω-πϱ2))'+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω-πϱ2)'+ϱ3(ϱ1
-πϱ1)))sinφsinχ1s+χ2+(((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))'-(π+ϱ1(ϱ1
-πϱ1)+ϱ2(ω-πϱ2))ϱ1-ϱ3((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))cosχ1s+χ2)=0,
((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ2
+ϱ3((ϱ1-πϱ1)'-ϱ3(ω-πϱ2)))cosφcosχ1s+χ2-((π+ϱ1(ϱ1-πϱ1)
+ϱ2(ω-πϱ2))'+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω-πϱ2)'+ϱ3(ϱ1
-πϱ1)))sinφcosχ1s+χ2-(((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))'-(π+ϱ1(ϱ1-πϱ1)
+ϱ2(ω-πϱ2))ϱ1-ϱ3((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))sinχ1s+χ2)=0,
((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ2+ϱ3((ϱ1
-πϱ1)'-ϱ3(ω-πϱ2)))sinφ+((π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))'
+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))cosφ)=0.
We instantly calculate
∇sE=(((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))cosφsinχ1s+χ2+((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))
×cosχ1s+χ2-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))sinφsinχ1s+χ2)f1
+(((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))cosφcosχ1s+χ2-(π+ϱ1(ϱ1-πϱ1)
+ϱ2(ω-πϱ2))sinφcosχ1s+χ2-((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))sinχ1s+χ2)f2
-(((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))sinφ+(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))cosφ)f3.
So, we obtain
∇sf∇sE=((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ2
+ϱ3((ϱ1-πϱ1)'-ϱ3(ω-πϱ2)))cosφsinχ1s+χ2-((π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))'
+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))sinφsinχ1s+χ2
+(((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ1-ϱ3((ω-πϱ2)'
+ϱ3(ϱ1-πϱ1)))cosχ1s+χ2)f1+((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'-(π+ϱ1(ϱ1
-πϱ1)+ϱ2(ω-πϱ2))ϱ2+ϱ3((ϱ1-πϱ1)'-ϱ3(ω-πϱ2)))cosφcosχ1s+χ2-((π
+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))'+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω-πϱ2)'+ϱ3(ϱ1
-πϱ1)))sinφcosχ1s+χ2-(((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω
-πϱ2))ϱ1-ϱ3((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))sinχ1s+χ2)f2-((((ω-πϱ2)'+ϱ3(ϱ1
-πϱ1))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ2+ϱ3((ϱ1-πϱ1)'-ϱ3(ω-πϱ2)))sinφ
+((π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))'+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))
+ϱ2((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))cosφ)f3.
♠The B obeys a uniformly quasi accelerated motion iff
(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1+ϱ2(ϱ1'+ϱ3ϱ2
-ωϱ3))sinφsinχ1s+χ2+((ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1-ϱ3(ϱ1'+ϱ3ϱ2-ωϱ3)-ω''
-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1)cosχ1s+χ2+(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'+ϱ3ϱ2
-ωϱ3)'-ϱ3ω'-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)cosφsinχ1s+χ2)=0,
(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1+ϱ2(ϱ1'
+ϱ3ϱ2-ωϱ3))sinφcosχ1s+χ2-((ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1-ϱ3(ϱ1'+ϱ3ϱ2
-ωϱ3)-ω''-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1)sinχ1s+χ2+(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)
+(ϱ1'+ϱ3ϱ2-ωϱ3)'-ϱ3ω'-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)cosφcosχ1s+χ2)=0,
(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1+ϱ2(ϱ1'
+ϱ3ϱ2-ωϱ3))cosφ-(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'+ϱ3ϱ2-ωϱ3)'
-ϱ3ω'-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)sinφ)=0.
Then
∇sB=((ϱ3'+ωϱ1-ϱ1ϱ2)sinφsinχ1s+χ2-ω'cosχ1s+χ2+(ϱ1'+ϱ3ϱ2
-ωϱ3)cosφsinχ1s+χ2)f1+((ϱ3'+ωϱ1-ϱ1ϱ2)sinφcosχ1s+χ2
+ω'sinχ1s+χ2+(ϱ1'+ϱ3ϱ2-ωϱ3)cosφcosχ1s+χ2)f2
+((ϱ3'+ωϱ1-ϱ1ϱ2)cosφ-(ϱ1'+ϱ3ϱ2-ωϱ3)sinφ)f3.
By calculating Fermi Walker derivative
∇sf∇sB=(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1
+ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3))sinφsinχ1s+χ2+((ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1
-ϱ3(ϱ1'+ϱ3ϱ2-ωϱ3)-ω''-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1)cosχ1s+χ2
+(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'+ϱ3ϱ2-ωϱ3)'-ϱ3ω'-(ϱ3'+ωϱ1
-ϱ1ϱ2)ϱ2)cosφsinχ1s+χ2)f1+(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'
-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1+ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3))sinφcosχ1s+χ2
-((ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1-ϱ3(ϱ1'+ϱ3ϱ2-ωϱ3)-ω''-(ϱ3'+ωϱ1
-ϱ1ϱ2)ϱ1)sinχ1s+χ2+(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'+ϱ3ϱ2
-ωϱ3)'-ϱ3ω'-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)cosφcosχ1s+χ2)f2
+(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1+ϱ2(ϱ1'
+ϱ3ϱ2-ωϱ3))cosφ-(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'+ϱ3ϱ2
-ωϱ3)'-ϱ3ω'-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)sinφ)f3.
5 Application to quasi uniformly circular potential electric energy
In this section, we obtain the polar plot for the time variation of potential electrical energy with respect to its electric field and energy flux density in the radial direction.
The quasi uniformly circular potential electric energy of field X in the electric field Ɛ is defined by
LfX=∇sf(∇sX)⋅E.
In previous work, we investigate the special type of magnetic trajectories such that it corresponds to a moving charged particle in an associated magnetic field in Heisenberg space. This study differs from the former studies in the literature since it is considered in the Heisenberg space. We considered uniformly accelerated motion (UAM), unchanged direction motion (UDM), and uniform circular motion (UCM) of normal magnetic biharmonic particles in Heisenberg space, [32].
We further improve an alternative method to find uniformly quasi circular potential electric energy of biharmonic normal magnetic particles in the Heisenberg space. We also give the relationships between physical and geometrical characterizations of uniformly quasi circular potential electric energy. Finally, we illustrate important figures for quasi uniformly circular potential electric energy with respect to its electric field in the radial direction.
• Quasi uniformly circular potential electric energy of φ(T
q) in the electric field Ɛ
Lfϕ(Tq)=(((ω'+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)+(ϱ12
+ϱ2ω)ϱ2)cosφsinχ1s+χ2-(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)+ϱ2(ω'
+ϱ1ϱ3)-((ϱ1'-ϱ3ω)ϱ1+ϱ2(ω'+ϱ1ϱ3)))sinφsinχ1s+χ2+((ϱ1'
-ϱ3ω)'-(ϱ12+ϱ2ω)ϱ1-ϱ3(ω'+ϱ1ϱ3)+(ϱ12+ϱ2ω)ϱ1)cosχ1s+χ2)((ω
-πϱ2)cosφsinχ1s+χ2-πsinφsinχ1s+χ2+(ϱ1-πϱ1)cosχ1s+χ2)+(((ω'
+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)+(ϱ12+ϱ2ω)ϱ2)cosφcosχ1s+χ2
-((ϱ1'-ϱ3ω)'-(ϱ12+ϱ2ω)ϱ1-ϱ3(ω'+ϱ1ϱ3)+(ϱ12+ϱ2ω)ϱ1)sinχ1s+χ2
-(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)+ϱ2(ω'+ϱ1ϱ3)-((ϱ1'-ϱ3ω)ϱ1+ϱ2(ω'
+ϱ1ϱ3)))sinφcosχ1s+χ2)((ω-πϱ2)cosφcosχ1s+χ2-πsinφcosχ1s+χ2
-(ϱ1-πϱ1)sinχ1s+χ2)+(((ω'+ϱ1ϱ3)'-(ϱ12+ϱ2ω)ϱ2+ϱ3(ϱ1'-ϱ3ω)
+(ϱ12+ϱ2ω)ϱ2)sinφ+(((ϱ12+ϱ2ω)'+ϱ1(ϱ1'-ϱ3ω)+ϱ2(ω'+ϱ1ϱ3)-((ϱ1'
-ϱ3ω)ϱ1+ϱ2(ω'+ϱ1ϱ3)))cosφ)((ω-πϱ2)sinφ+πcosφ).
Figure 1 shows the shape of the uniformly quasi circular potential density for φ(T
q) with appropriate values of the ϱ1
, ϱ2
, and ϱ3
.
• Quasi uniformly circular potential electric energy of φ(N
q) in the electric field Ɛ
Lfϕ(Nq)=(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)
-(ϱ12+ϱ32)ϱ1))sinφsinχ1s+χ2-((ϱ1'+ϱ3ϱ2)ϱ1+(ϱ12+ϱ32)'+ϱ3(ϱ3'+ϱ1ϱ2)
-(ϱ1'+ϱ3ϱ2)ϱ1)cosχ1s+χ2+((ϱ3'+ϱ1ϱ2)'-(ϱ12+ϱ32)ϱ3-(ϱ1'+ϱ3ϱ2)ϱ2
+ϱ2(ϱ1'+ϱ3ϱ2))cosφsinχ1s+χ2)((ω-πϱ2)cosφsinχ1s+χ2
-πsinφsinχ1s+χ2+(ϱ1-πϱ1)cosχ1s+χ2)+(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'
+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'+(ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ12+ϱ32)ϱ1))sinφcosχ1s+χ2
+((ϱ1'+ϱ3ϱ2)ϱ1+(ϱ12+ϱ32)'+ϱ3(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)ϱ1)sinχ1s+χ2
+((ϱ3'+ϱ1ϱ2)'-(ϱ12+ϱ32)ϱ3-(ϱ1'+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))cosφ
cosχ1s+χ2)((ω-πϱ2)cosφcosχ1s+χ2-πsinφcosχ1s+χ2-(ϱ1
-πϱ1)sinχ1s+χ2)-(((ϱ12+ϱ32)ϱ1-ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ1'+ϱ3ϱ2)'
+(ϱ2(ϱ3'+ϱ1ϱ2)-(ϱ12+ϱ32)ϱ1))cosφ-((ϱ3'+ϱ1ϱ2)'-(ϱ12+ϱ32)ϱ3-(ϱ1'
+ϱ3ϱ2)ϱ2+ϱ2(ϱ1'+ϱ3ϱ2))sinφ)((ω-πϱ2)sinφ+πcosφ).
Figure 2 shows the shape of the uniformly quasi circular potential density for φ(N
q) with appropriate values of the ϱ1
, ϱ2
, and ϱ3
.
• Quasi uniformly circular potential electric energy of φ(B
q) in the electric field Ɛ
Lfϕ(Bq)=((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1-ω')'+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1
+(ϱ32+ωϱ2)ϱ2))sinφsinχ1s+χ2+((ϱ3ϱ1-ω')ϱ1-(ϱ3'+ωϱ1)'
+ϱ3(ϱ32+ωϱ2)-(ϱ3ϱ1-ω')ϱ1)cosχ1s+χ2+(ϱ2(ϱ3ϱ1-ω')-(ϱ32+
ωϱ2)'-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)cosφsinχ1s+χ2)((ω
-πϱ2)cosφsinχ1s+χ2-πsinφsinχ1s+χ2+(ϱ1-πϱ1)cosχ1s+χ2)
+((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1-ω')'+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1+(ϱ32
+ωϱ2)ϱ2))sinφcosχ1s+χ2-((ϱ3ϱ1-ω')ϱ1-(ϱ3'+ωϱ1)'+ϱ3(ϱ32
+ωϱ2)-(ϱ3ϱ1-ω')ϱ1)sinχ1s+χ2+(ϱ2(ϱ3ϱ1-ω')-(ϱ32+ωϱ2)'-ϱ3(ϱ3'
+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)cosφcosχ1s+χ2)((ω-πϱ2)cosφcosχ1s+χ2
-πsinφcosχ1s+χ2-(ϱ1-πϱ1)sinχ1s+χ2)-((ϱ1(ϱ3'+ωϱ1)+(ϱ3ϱ1
-ω')'+ϱ2(ϱ32+ωϱ2)-((ϱ3'+ωϱ1)ϱ1+(ϱ32+ωϱ2)ϱ2))cosφ-(ϱ2(ϱ3ϱ1-ω')
-(ϱ32+ωϱ2)'-ϱ3(ϱ3'+ωϱ1)-(ϱ3ϱ1-ω')ϱ2)sinφ)((ω-πϱ2)sinφ+πcosφ).
Figure 3 shows the shape of the uniformly quasi circular potential density for φ(B
q) with appropriate values of the ϱ1
, ϱ2
, and ϱ3
.
• Quasi uniformly circular potential electric energy of Ɛ in the electric field Ɛ
Lf(E=((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ2+ϱ3((ϱ1
-πϱ1)'-ϱ3(ω-πϱ2)))cosφsinχ1s+χ2-((π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))'
+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))sinφsinχ1s+χ2
+(((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))ϱ1-ϱ3((ω-πϱ2)'
+ϱ3(ϱ1-πϱ1)))cosχ1s+χ2)((ω-πϱ2)cosφsinχ1s+χ2-πsinφsinχ1s+χ2+(ϱ1
-πϱ1)cosχ1s+χ2)+((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω
-πϱ2))ϱ2+ϱ3((ϱ1-πϱ1)'-ϱ3(ω-πϱ2)))cosφcosχ1s+χ2-((π+ϱ1(ϱ1-πϱ1)
+ϱ2(ω-πϱ2))'+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))
×sinφcosχ1s+χ2-(((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))'-(π+ϱ1(ϱ1-πϱ1)+ϱ2(ω
-πϱ2))ϱ1-ϱ3((ω-πϱ2)'+ϱ3(ϱ1-πϱ1)))sinχ1s+χ2)((ω-πϱ2)cosφcosχ1s+χ2
-πsinφcosχ1s+χ2-(ϱ1-πϱ1)sinχ1s+χ2)+((((ω-πϱ2)'+ϱ3(ϱ1-πϱ1))'
-π+ϱ1ϱ1-πϱ1+ϱ2ω-πϱ2ϱ2+ϱ3ϱ1-πϱ1)'-ϱ3ω-πϱ2sinφ
+((π+ϱ1(ϱ1-πϱ1)+ϱ2(ω-πϱ2))'+ϱ1((ϱ1-πϱ1)'-ϱ3(ω-πϱ2))+ϱ2((ω
-πϱ2)'+ϱ3(ϱ1-πϱ1)))cosφ)((ω-πϱ2)sinφ+πcosφ).
Figure 4 shows the shape of the uniformly quasi circular potential density for Ɛ with appropriate values of the ϱ1
, ϱ2
, and ϱ3
.
• Quasi uniformly circular potential electric energy of B in the electric field Ɛ
Lf(B)=(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1+ϱ2(ϱ1'+ϱ3ϱ2
-ωϱ3))sinφsinχ1s+χ2+((ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1-ϱ3(ϱ1'+ϱ3ϱ2-ωϱ3)-ω''-(ϱ3'
+ωϱ1-ϱ1ϱ2)ϱ1)cosχ1s+χ2+(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'+ϱ3ϱ2-ωϱ3)'-ϱ3ω'
-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)cosφsinχ1s+χ2)((ω-πϱ2)cosφsinχ1s+χ2-πsinφsinχ1s+χ2
+(ϱ1-πϱ1)cosχ1s+χ2)+(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3)-ω'ϱ1
+ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3))sinφcosχ1s+χ2-((ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1-ϱ3(ϱ1'+ϱ3ϱ2
-ωϱ3)-ω''-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ1)sinχ1s+χ2+(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'+ϱ3ϱ2
-ωϱ3)'-ϱ3ω'-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)cosφcosχ1s+χ2)((ω-πϱ2)cosφcosχ1s+χ2
-πsinφcosχ1s+χ2-(ϱ1-πϱ1)sinχ1s+χ2)-(((ϱ3'+ωϱ1-ϱ1ϱ2)'+ϱ1ω'-ϱ2(ϱ1'
+ϱ3ϱ2-ωϱ3)-ω'ϱ1+ϱ2(ϱ1'+ϱ3ϱ2-ωϱ3))cosφ-(ϱ2(ϱ3'+ωϱ1-ϱ1ϱ2)+(ϱ1'
+ϱ3ϱ2-ωϱ3)'-ϱ3ω'-(ϱ3'+ωϱ1-ϱ1ϱ2)ϱ2)sinφ)((ω-πϱ2)sinφ+πcosφ).
Figure 5 shows the shape of the uniformly quasi circular potential density for Ɛ with appropriate values of the ϱ1
, ϱ2
, and ϱ3
.
6. Conclusion
The uniform motion of the physical system, in particular, a mechanism defined along with the particle in a given appropriate spacetime structure, can be described by minimizing the action functional and can further computation be obtained by the principle of the least action [33-35]. The investigation of the uniform motion of the particle is very efficient for the exact comprehension of many physical processes such as vortex filaments, integrable systems, dynamics of Heisenberg chain, soliton equation theory, relativity, sigma models, water wave theory, field theories, fluid dynamics, linear and nonlinear optics, and so on.
We have examined the quasi uniformly accelerated motion (QUAM) with cold plasma in Heisenberg space. Thus we investigate the shape of the uniformly quasi circular potential density for a quasi uniformly accelerated motion of normal magnetic biharmonic particles in Heisenberg space. Finally, we construct quasi uniformly accelerated potential electric energy with respect to its electric field and some quasi curvatures by Figs. 1-5.
In future studies, we will investigate the physical implications of the uniformly quasi circular motion (UQCM) by obtaining different trajectories in de Sitter spacetime, Anti de Sitter spacetime, etc.
Acknowledgment
The authors would like to express their sincere gratitude to the referees for the valuable suggestions to improve the paper.
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