Research
Gravitation, Mathematical Physics and Field Theory
New magnetic flux flows with Heisenberg ferromagnetic spin of optical
quasi velocity magnetic flows with flux density
T. Körpinara
R. Cem Demirkolb
Z. Körpinarc
V. Asild
aMuş Alparslan University, Department of
Mathematics, 49250, Muş, Turkey,
bMuş Alparslan University, Department of
Mathematics, 49250, Muş, Turkey,
cMuş Alparslan University, Department of
Administration, 49250, Muş, Turkey,
dFirat University, Department of Mathematics,
23100, Elazıǧ, Turkey.
Abstract
In this article, we first offer the approach of quasi magnetic Lorentz flux of
quasi velocity magnetic flows of particles by the quasi frame in 3D space. We
then obtain new optical conditions of quasi magnetic Lorentz flux by using
directional quasi fields. Moreover, we determine the quasi magnetic Lorentz flux
for quasi vector fields. Additionally, we give new constructions for quasi
curvatures of quasi velocity magnetic flows by considering Heisenberg
ferromagnetic spin. Finally, the magnetic flux surface is demonstrated on a
static and uniform magnetic surface by using the analytical and numerical
results.
Keywords: Quasi directional frame; flows; Heisenberg ferromagnetic spin; geometric magnetic flux density
PACS: 04.20.-q; 03.50.De; 02.40.-k
1.Introduction
The magnetic flux density is an important theoretical model for quantum physics,
quantum magnetics, and quantum optics. Magnetic flux density determines the optical
solitons for magnetic flux and electric flux density across the symmetry. The flux
density has been studied by numerous scientists from different viewpoints. Lorentz
flux is an important subclass of the flux density and it is more related to the
theory of optical propagation. One can also find a correlation between optical and
magnetic soliton solutions of different models by considering the flux density.
An ideal optical fiber has perfect circular symmetry. The polarizations are
completely degenerate. Perturbations and imperfections during the fabrication
process may introduce anisotropies, which are mostly of a linear or Cartesian type.
Bending and stretching optical fibers does also determine linear birefringence.
Rotational effects of polarization are difficult to produce since their production
process has not been fully comprehended for many cases.
In recent times, the advancement of glowing lasers and the utilization of optical
fiber mechanics have increased the importance of flow propagation by curled fluid
flows and space-curved. Exclusively torque forces of the geometric phase of isolated
light anholonomy with some optical fibers have been investigated by numerous
researchers. For example, Smith 1
examined that a torque of the divergence of light is generated along with the
monochromatic optical fiber bundle thanks to the magnetic particle’s flows in a
certain transformer. Another preliminary geometric effect of the torque of magnetic
divergence of light propagating in a magnetic optical fiber detecting a magnetic
trajectory was presented by 2. Ross
improved a totally geometric system to investigate the coiled optical fiber with a
fixed-torsion and investigated its effects with several measurements. Tomita and
Chiao 3 summarized the previous
review of Ross for more general fiber shapes. Also, Chiao and Wu 4 obtained an important theoretical
phase as a result of geometric phase torque. Haldane defined the geometric rotation
of the polarization angle in an ideal cylindrical optical fiber without
birefringence for arbitrary fiber paths in terms of the image of the path in the
tangent vector space 5. Apart from
previous researches, we proposed some new approaches to compute the electromagnetic
phase with an antiferromagnetic chain 6-10.
The geometric phase investigation along the optical fiber investigation is mostly
conducted by observing the action of electromagnetic particles and their features.
Some nonlinear evolution structures are frequently encountered particularly in
genuine-state physics, chemical physics, plasma physics, optical physics, fluid
mechanics, etc. Even though these equations have been heavily used in many
structures, it requires very hard work to obtain the explicit solutions of
approximate systems. Thus, there exists no global or unified approach to demonstrate
the exact solutions of all nonlinear transformation systems 11-23.
The aim of the present paper is to introduce a new geometric interpretationof the
notion of the Heisenberg ferromagnetic spin for quasi flows of magnetic particles
with the quasi-frame in the space. Eventually, we obtain new optical conditions of
quasi magnetic Lorentz flux by using directional quasi fields. Moreover, we
determine the quasi magnetic Lorentz flux for quasi vector fields. Also, we give new
constructions for quasi curvatures of quasi velocity magnetic flows by considering
Heisenberg ferromagnetic spin. Finally, the magnetic flux surface is demonstrated in
a static and uniform magnetic surface by using the analytical and numerical
results.
2.Background on the quasi frame
Let Ψ=Ψ(s) be an arclength parametrized particle in the 3D space (R,⋅), i.e. (Ψs⋅Ψs)=1. The arclength parametrized particle is also called a unit speed
particle. A unit speed particle Ψ is called to be a Serret particle if Ψss≠0. This particle introduces an orthonormal field (t,n,b),
which satisfies the following formulae
∇st∇sn∇sb=0κ0-κ0τ0-τ0tnb.
We define a quasi frame (tq, nq, bq)by only parallel transporting to the tangent vector of the frame along
with particle. The quasi directional frame of a regular particle is given by
tq=t, nq=t×πt×π, bq=t×nq,
where π is a projection vector and can be selected as the following
π=1,0,0.
If the angle between the quasi normal vecctor nq and the normal vector n is choosen as ψ, then the following relation is obtained between the quasi and SF frame
24,25:
tq=t, nq=cosψn+sinψb,bq=-sinψn+cosψb,
Therefore, the quasi frame equations are expressed as
∇stq=ϰ1nq+ϰ2bq,∇snq=-ϰ1tq+ϰ3bq,∇sbq=-ϰ2tq-ϰ3nq,
where the quasi curvatures are
ϰ1=κcosψ, ϰ2=-κsinψ, ϰ3=ψ'+τ.
3.Flows of velocity magnetic particles
In this section, magnetic surfaces with the time evolution of quasi-velocity magnetic
partices in 3D space are described. The time evolution is assumed to be
one-dimensional and embedded in the 3D space. Thus, the fundamental geometric
construction of the flows as surfaces can naturally be induced by the moving quasi
orthonormal frame field.
♠ Let α be an arclength parametrized particle and B be a magnetic field in the ordinary space. We call the particle α as a
quasi-velocity magnetic particle if the quasi tangent field of the particle meets
the subsequent Lorentz force equation:
∇tqtq=ϕ(tq)=B×tq.
♠ Lorentz force ϕ of the magnetic field B with the quasi-velocity magnetic particle is given in the quasi frame by
the subsequent equations.
ϕ(tq)=ϰ1nq+ϰ2bq,ϕ(nq)=-ϰ1tq+ϖbq,ϕ(bq)=-ϰ2tq-ϖnq,
where ϖ=ϕ(nq)⋅bq is a sufficiently smooth function and ϰ1,ϰ2 are quasi-curvatures. Also, magnetic field B is given by
B=ϖtq-ϰ2nq+ϰ1bq.
Let α(s,ω,t) be the motion of regular particles in space. The flow of α can easily be
displayed seeing as
∂α∂t=β1tq+β2nq+β3bq,
where β1,β2,β3 are tangential, normal and binormal potentials of particle.
♠ Time derivatives of the quasi frame are given by
∇ttq=∂β2∂s +ϰ1β1-ϰ3β3nq+∂β3∂s+ϰ2β1+ϰ3β2bq,∇tnq=-∂β2∂s\+ϰ1β1-ϰ3β3tq+χbq,∇tbq=-∂β3∂s+ϰ2β1+ϰ3β2tq-χnq,
where χ=∇tnq⋅bq.
♠ Differentation of Lorentz forces are obtained by
∇sϕ(tq)=-(ϰ12+ϰ22)tq+(ϰ1'-ϰ2ϰ3)nq+(ϰ2'+ϰ1ϰ3)bq,∇sϕ(nq)=-(ϰ1'+ϰ2ϖ)tq-(ϰ12+ϰ3ϖ)nq+(ϖ'-ϰ1ϰ2)bq,∇sϕ(bq)=(ϖϰ1-ϰ2')tq-(ϖ'+ϰ2ϰ1)nq-(ϰ22+ϖϰ3)bq.
♠ Flows of Lorentz forces of the quasi frame are given by
∇tϕ(tq)=-(ϰ1∂β2∂s +ϰ1β1-ϰ3β3+ϰ2∂β3∂s+ϰ2β1+ϰ3β2)tq+∂ϰ1∂t-ϰ2χnq+∂ϰ2∂t+ϰ1χbq,∇tϕ(nq)=-∂ϰ1∂t+ϖ∂β3∂s+ϰ2β1+ϰ3β2tq-ϰ1∂β2∂s +ϰ1β1-ϰ3β3+ϖχnq+∂ϖ∂t-ϰ1∂β3∂s+ϰ2β1+ϰ3β2bq,
∇tϕ(bq)=ϖ∂β2∂s+ϰ1β1-ϰ3β3-∂ϰ2∂ttq-∂ϖ∂t+ϰ2∂β2∂s +ϰ1β1-ϰ3β3nq-ϰ2∂β3∂s+ϰ2β1+ϰ3β2+ϖχbq.
4.Quasi magnetic Lorentz flux surfaces
The magnetic flux equation theory or the theory of flux systems has had an enormous
impact flow of the magnetic flux between two distinct points on a given surface
causes the exact same angular momentum for the charged particle. As opposed to the
traditional spectral analysis approach, which is used to built magnetic flux
surfaces at small scales we rather choose to focus on the geometric methods in order
to avoid the flaws of the aforementioned method, i.e., the nondeterministic
polynomial of exponential complexity.
Magnetic ϕ(tq),ϕ(nq),ϕ(bq) flux surfaces afford typical characterizations of connectivity,
dynamics, and the spatial structure of magnetic fields, magnetic field connectivity
in the interstellar medium and in the interplanetary and dynamo problems, in
laboratory plasmas, and magnetic reconnection. Here, the different kinds of magnetic
flux surfaces are computed through the modifications of the evolution equations,
which define the motion of magnetically driven particles and corresponding gradient
flows. We mainly consider the orthonormal curvilinear coordinates and derive the
solution families of the equations of the Lorentz force associated with magnetic
flux surfaces of the distinct kinds. As will be seen in the next section all three
kinds of flux surfaces formed by quasi-tangential, quasi-normal, quasi-binormal
vectors have different configurations and particular behaviors.
Case 1. Magnetic ϕ(tq) flux
♠ The magnetic ϕ(tq) flux mFϕ(tq) is given by
mFϕ(tq)=∫F([{∂ϰ1∂s-ϰ2ϰ3}{∂ϰ2∂t+ϰ1χ}-{∂ϰ2∂s+ϰ1ϰ3}{∂ϰ1∂t-ϰ2χ}]ϖ-ϰ2[ϰ12+ϰ22∂ϰ2∂t+ϰ1χ-∂ϰ2∂s+ϰ1ϰ3]ϰ1{∂β2∂s+ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}])+[{∂ϰ1∂s-ϰ2ϰ3}]ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}]-{ϰ12+ϰ22}{∂ϰ1∂t-ϰ2χ}]ϰ1)dπ.
Magnetic ϕ(tq) flux mFϕ(tq) is given by
mFϕ(tq)=∫FB⋅∇sϕ(tq)×∇tϕ(tq)dπ.
With short calculations, we obtain that
∇sϕ(tq)×∇tϕ(tq)=([∂ϰ1∂s-ϰ2ϰ3][∂ϰ2∂t+ϰ1χ]-[∂ϰ2∂s+ϰ1ϰ3][∂ϰ1∂t-ϰ2χ])tq+([ϰ12+ϰ22][∂ϰ2∂t+ϰ1χ]-[∂ϰ2∂s+ϰ1ϰ3][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}])nq+([∂ϰ1∂s-ϰ2ϰ3][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}]-[ϰ12+ϰ22][∂ϰ1∂t-ϰ2χ])bq.
Magnetic flux density of ϕ(tq) is given by
mLϕ(tq)=([∂ϰ1∂s-ϰ2ϰ3][∂ϰ2∂t+ϰ1χ]-[∂ϰ2∂s+ϰ1ϰ3][∂ϰ1∂t-ϰ2χ])ϖ-ϰ2([ϰ12+ϰ22][∂ϰ2∂t+ϰ1χ]-[∂ϰ2∂s+ϰ1ϰ3][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}])+([∂ϰ1∂s-ϰ2ϰ3][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}]-[ϰ12+ϰ22][∂ϰ1∂t-ϰ2χ])ϰ1.
Moreover, ϕ(tq) flux is obtained in the following way
mFϕ(tq)=∫F([{∂ϰ1∂s-ϰ2ϰ3}{∂ϰ2∂t+ϰ1χ}-{∂ϰ2∂s+ϰ1ϰ3}{∂ϰ1∂t-ϰ2χ}]ϖ-ϰ2[{ϰ12+ϰ22}{∂ϰ2∂t+ϰ1χ}-{∂ϰ2∂s+ϰ1ϰ3}{ϰ1∂β2∂s +ϰ1β1-ϰ3β3+ϰ2∂β3∂s+ϰ2β1+ϰ3β2}]+[∂ϰ1∂s-ϰ2ϰ3{ϰ1∂β2∂s +ϰ1β1-ϰ3β3+ϰ2∂β3∂s+ϰ2β1+ϰ3β2}-{ϰ12+ϰ22}∂ϰ1∂t-ϰ2χ]ϰ1)dπ.
From the ferromagnetic model, the flux density is given by
mLϕ(tq)ferro=B⋅∇sϕ(tq)×ϕ(tq)×∇s2ϕ(tq).
Similarly, we can obtain that
∇sϕ(tq)×ϕ(tq)×∇s2ϕ(tq)=([ϰ1'-ϰ2ϰ3]ϰ1[∂∂s{ϰ12+ϰ22}+ϰ1{∂ϰ1∂s-ϰ2ϰ3}+ϰ2{∂ϰ2∂s+ϰ1ϰ3}]+[ϰ2'+ϰ1ϰ3]ϰ2[∂∂s{ϰ12+ϰ22}+ϰ1{∂ϰ1∂s-ϰ2ϰ3}+ϰ2{∂ϰ2∂s+ϰ1ϰ3}])tq+(ϰ1[∂∂sϰ12+ϰ22+ϰ1∂ϰ1∂s-ϰ2ϰ3+ϰ2∂ϰ2∂s+ϰ1ϰ3][ϰ12+ϰ22]+[ϰ2'+ϰ1ϰ3][ϰ1{∂∂s⌈∂ϰ2∂s+ϰ1ϰ3⌉-[ϰ12+ϰ22]ϰ2+ϰ3{∂ϰ1∂s-ϰ2ϰ3}]-ϰ2[∂∂s{∂ϰ1∂s-ϰ2ϰ3}+ϰ3{∂ϰ2∂s+ϰ1ϰ3}+{ϰ12+ϰ22}ϰ1])nq+([ϰ12+ϰ22]ϰ2[∂∂sϰ12+ϰ22+ϰ1∂ϰ1∂s-ϰ2ϰ3+ϰ2∂ϰ2∂s+ϰ1ϰ3]-[ϰ1'-ϰ2ϰ3][ϰ1{∂∂s⌈∂ϰ2∂s+ϰ1ϰ3⌉-⌈ϰ12+ϰ22⌉ϰ2+ϰ3⌈∂ϰ1∂s-ϰ2ϰ3⌉}-ϰ2[∂∂s{∂ϰ1∂s-ϰ2ϰ3}+ϰ3{∂ϰ2∂s+ϰ1ϰ3}+{ϰ12+ϰ22}ϰ1}])bq.
Similarly, ferromagnetic magnetic ϕ(tq) flux is given by
mFϕ(tq)ferro=∫F(ϖ[{ϰ1'-ϰ2ϰ3}ϰ1{∂∂s(ϰ12+ϰ22)+ϰ1∂ϰ1∂s-ϰ2ϰ3+ϰ2∂ϰ2∂s+ϰ1ϰ3}+{ϰ2'+ϰ1ϰ3}ϰ2{∂∂s(ϰ12+ϰ22 )+ϰ1∂ϰ1∂s-ϰ2ϰ3+ϰ2∂ϰ2∂s+ϰ1ϰ3}]-ϰ2[ϰ1{∂∂s ( ϰ12+ϰ22)+ϰ1∂ϰ1∂s-ϰ2ϰ3+ϰ2∂ϰ2∂s+ϰ1ϰ3}{ϰ12+ϰ22}+{ϰ2'+ϰ1ϰ3}×{ϰ1∂∂s∂ϰ2∂s+ϰ1ϰ3-[ϰ12+ϰ22]ϰ2+ϰ3∂ϰ1∂s-ϰ2ϰ3-ϰ2(∂∂s∂ϰ1∂s-ϰ2ϰ3+ϰ3∂ϰ2∂s+ϰ1ϰ3+[ϰ12+ϰ22]ϰ1)}]+ϰ1[{ϰ12+ϰ22}ϰ2{∂∂sϰ12+ϰ22+ϰ1∂ϰ1∂s-ϰ2ϰ3+ϰ2∂ϰ2∂s+ϰ1ϰ3}-{ϰ1'-ϰ2ϰ3}{ϰ1\pari∂∂s∂ϰ2∂sϰ1ϰ3-[ϰ12+ϰ22]ϰ2+ϰ3∂ϰ1∂s-ϰ2ϰ3-ϰ2∂∂s∂ϰ1∂s-ϰ2ϰ3+ϰ3∂ϰ2∂s+ϰ1ϰ3+ [ϰ12+ϰ22]ϰ1}])dπ.
The magnetic ϕ(tq) flux surface condition is given by
([∂ϰ1∂s-ϰ2ϰ3][∂ϰ2∂t+ϰ1χ]-[∂ϰ2∂s+ϰ1ϰ3][∂ϰ1∂t-ϰ2χ])ϖ-ϰ2((ϰ12+ϰ22)[∂ϰ2∂t+ϰ1χ]-[∂ϰ2∂s+ϰ1ϰ3][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}])+([∂ϰ1∂s-ϰ2ϰ3][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}]-(ϰ12+ϰ22)[∂ϰ1∂t-ϰ2χ])ϰ1=0.
The magnetic ϕ(tq) flux surface is given by the ferromagnetic condition
ϖ([ϰ1'-ϰ2ϰ3]ϰ1[∂∂s{ϰ12+ϰ22}+ϰ1{∂ϰ1∂s-ϰ2ϰ3}+ϰ2{∂ϰ2∂s+ϰ1ϰ3}]+[ϰ2'+ϰ1ϰ3]ϰ2\nt[∂∂s{ϰ12+ϰ22}+ϰ1{∂ϰ1∂s-ϰ2ϰ3}+ϰ2{∂ϰ2∂s+ϰ1ϰ3}])-ϰ2(ϰ1[∂∂s{ϰ12+ϰ22}+ϰ1{∂ϰ1∂s-ϰ2ϰ3}+ϰ2{∂ϰ2∂s+ϰ1ϰ3}][ϰ12+ϰ22]+[ϰ2'+ϰ1ϰ3][ϰ1{∂∂s∂ϰ2∂s+ϰ1ϰ3-(ϰ12+ϰ22)ϰ2+ϰ3∂ϰ1∂s-ϰ2ϰ3}-ϰ2{∂∂s∂ϰ1∂s-ϰ2ϰ3+ϰ3∂ϰ2∂s+ϰ1ϰ3+(ϰ12+ϰ22)ϰ1}])+ϰ1([ϰ12+ϰ22]ϰ2[∂∂s{ϰ12+ϰ22}+ϰ1{∂ϰ1∂s-ϰ2ϰ3}+ϰ2{∂ϰ2∂s+ϰ1ϰ3}]-[ϰ1'-ϰ2ϰ3][ϰ1{∂∂s∂ϰ2∂s+ϰ1ϰ3-(ϰ12+ϰ22)ϰ2+ϰ3∂ϰ1∂s-ϰ2ϰ3}-ϰ2{∂∂s∂ϰ1∂s-ϰ2ϰ3+ϰ3∂ϰ2∂s+ϰ1ϰ3+(ϰ12+ϰ22)ϰ1}])=0.
The magnetic flux density and flow lines in the axis is represented by quadrupole
magnets under the action of the Lorentz force ϕ(tq). The magnetic flux density on time of flight is fed to the particle
tracing algorithm by a logical expression. Here, the intricacy of the magnetic flux
surface is demonstrated in a static and uniform magnetic surface by using the
analytical and numerical results. To obtain the visualization of the evolved systems
of the magnetic ϕ(tq) flux we use the basic numerical algorithms to solve the above equations
at Matlab and Comsol software. This technique presents a new approach to formulating
the relationship between local frames of reference and flux coordinates in Fig. 1.
Case 2. Magnetic ϕ(nq) flux
The magnetic ϕ(nq) flux mFϕ(nq) is given by
mFϕ(nq)=∫F([{∂ϖ∂s-ϰ1ϰ2}{ϰ1∂β2∂s+ϰ1β1-ϰ3β3+ϖχ}-{ϰ12+ϰ3ϖ}{∂ϖ∂t-ϰ1∂β3∂s+ϰ2β1+ϰ3β2}]ϖ-ϰ2[{∂ϰ1∂s+ϰ2ϖ}{∂ϖ∂t-ϰ1∂β3∂s+ϰ2β1+ϰ3β2}-{∂ϖ∂s-ϰ1ϰ2}{∂ϰ1∂t+ϖ∂β3∂s+ϰ2β1+ϰ3β2}]+ϰ1[{∂ϰ1∂s+ϰ2ϖ}{ϰ1∂β2∂s\ +ϰ1β1-ϰ3β3+ϖχ}-{ϰ12+ϰ3ϖ}{∂ϰ1∂t+ϖ∂β3∂s+ϰ2β1+ϰ3β2}])dπ.
Firstly, we compute that
∇s2ϕ(nq)=(ϰ1[ϰ12+ϰ3ϖ]-∂∂s[∂ϰ1∂s+ϰ2ϖ]-ϰ2[∂ϖ∂s-ϰ1ϰ2])tq-(∂∂s[ϰ12+ϰ3ϖ]+[∂ϰ1∂s+ϰ2ϖ]ϰ1+ϰ3[∂ϖ∂s-ϰ1ϰ2])nq+(∂∂s[∂ϖ∂s-ϰ1ϰ2]-[∂ϰ1∂s+ϰ2ϖ]ϰ2-[ϰ12+ϰ3ϖ]ϰ3)bq.
It is also true that
ϕ(nq)×∇s2ϕ(nq)=ϖ(∂∂s[ϰ12+ϰ3ϖ]+[∂ϰ1∂s+ϰ2ϖ]ϰ1+ϰ3[∂ϖ∂s-ϰ1ϰ2])tq+(ϰ1[∂∂s{∂ϖ∂s-ϰ1ϰ2}-{∂ϰ1∂s+ϰ2ϖ}ϰ2-{ϰ12+ϰ3ϖ}ϰ3]+ϖ[ϰ1{ϰ12+ϰ3ϖ}-∂∂s{∂ϰ1∂s+ϰ2ϖ}-ϰ2{∂ϖ∂s-ϰ1ϰ2}])nq+ϰ1(∂∂s[ϰ12+ϰ3ϖ]+[∂ϰ1∂s+ϰ2ϖ]ϰ1+ϰ3[∂ϖ∂s-ϰ1ϰ2])bq.
Thus, we can easily obtain that
∇tϕ(nq)=-(∂ϰ1∂t+ϖ[∂β3∂s+ϰ2β1+ϰ3β2])tq-(ϰ1[∂β2∂s +ϰ1β1-ϰ3β3]+ϖχ)nq+(∂ϖ∂t-ϰ1[∂β3∂s+ϰ2β1+ϰ3β2])bq.
As a result, we reach the following identities
mLϕ(nq)=([∂ϖ∂s-ϰ1ϰ2][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϖχ]-(ϰ12+ϰ3ϖ)[∂ϖ∂t-ϰ1{∂β3∂s+ϰ2β1+ϰ3β2}])ϖ-ϰ2([∂ϰ1∂s+ϰ2ϖ][∂ϖ∂t-ϰ1{∂β3∂s+ϰ2β1+ϰ3β2}]-[∂ϖ∂s-ϰ1ϰ2][∂ϰ1∂t+ϖ{∂β3∂s+ϰ2β1+ϰ3β2}])+ϰ1([∂ϰ1∂s+ϰ2ϖ]×[ϰ1{∂β2∂s\ +ϰ1β1-ϰ3β3}+ϖχ]-(ϰ12+ϰ3ϖ)[∂ϰ1∂t+ϖ{∂β3∂s+ϰ2β1+ϰ3β2}]).
By ferromagnetic spin for ϕ(nq), we obtain
mLϕ(nq)ferro=B⋅∇sϕ(nq)×ϕ(nq)×∇s2ϕ(nq).
Therefore, under the assumptions of the Heisenberg ferromagnetic model, it can be
computed that
∇sϕ(nq)×ϕ(nq)×∇tq2ϕ(nq)=-([ϰ12+ϰ3ϖ]ϰ1[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}]+[ϖ'-ϰ1ϰ2][ϰ1{∂∂s∂ϖ∂s-ϰ1ϰ2-∂ϰ1∂s+ϰ2ϖϰ2-(ϰ12+ϰ3ϖ)ϰ3}+ϖ{ϰ1(ϰ12+ϰ3ϖ)-∂∂s∂ϰ1∂s+ϰ2ϖ-ϰ2∂ϖ∂s-ϰ1ϰ2}])tq+(ϰ1[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}][ϰ1'+ϰ2ϖ]+[ϖ'-ϰ1ϰ2]×ϖ[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}])nq+(ϖ[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}][ϰ12+ϰ3ϖ]-[ϰ1'+ϰ2ϖ][ϰ1{∂∂s∂ϖ∂s-ϰ1ϰ2-∂ϰ1∂s+ϰ2ϖϰ2-(ϰ12+ϰ3ϖ)ϰ3}+ϖ{ϰ1(ϰ12+ϰ3ϖ)-∂∂s∂ϰ1∂s+ϰ2ϖ-ϰ2∂ϖ∂s-ϰ1ϰ2}])bq.
Similarly, ferromagnetic magnetic ϕ(nq) flux is given by
mFϕ(nq)ferro=∫F(-ϖ[{ϰ12+ϰ3ϖ}ϰ1{∂∂s(ϰ12+ϰ3ϖ) +∂ϰ1∂s+ϰ2ϖϰ1+ϰ3∂ϖ∂s-ϰ1ϰ2}+[ϖ'-ϰ1ϰ2][ϰ1{∂∂s∂ϖ∂s-ϰ1ϰ2-∂ϰ1∂s+ϰ2ϖϰ2-(ϰ12+ϰ3ϖ)ϰ3}+ϖ{ϰ1(ϰ12+ϰ3ϖ)-∂∂s∂ϰ1∂s+ϰ2ϖ-ϰ2∂ϖ∂s-ϰ1ϰ2}])-ϰ2(ϰ1[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}][ϰ1'+ϰ2ϖ]+[ϖ'-ϰ1ϰ2]ϖ[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}])+ϰ1(ϖ[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}][ϰ12+ϰ3ϖ]-[ϰ1'+ϰ2ϖ][ϰ1{∂∂s∂ϖ∂s-ϰ1ϰ2-∂ϰ1∂s+ϰ2ϖϰ2-(ϰ12+ϰ3)ϰ3}+ϖ{ϰ1(ϰ12+ϰ3ϖ)-∂∂s∂ϰ1∂s+ϰ2ϖ-ϰ2∂ϖ∂s-ϰ1ϰ2}])dπ.
The condition of magnetic ϕ(nq) flux surface is given by
([∂ϖ∂s-ϰ1ϰ2][ϰ1{∂β2∂s +ϰ1β1-ϰ3β3}+ϖχ]-[ϰ12+ϰ3ϖ][∂ϖ∂t-ϰ1{∂β3∂s+ϰ2β1+ϰ3β2}])ϖ-ϰ2([∂ϰ1∂s+ϰ2ϖ][∂ϖ∂t-ϰ1{∂β3∂s+ϰ2β1+ϰ3β2}]-[∂ϖ∂s-ϰ1ϰ2][∂ϰ1∂t+ϖ{∂β3∂s+ϰ2β1+ϰ3β2}])+ϰ1([∂ϰ1∂s+ϰ2ϖ][ϰ1{∂β2∂s+ϰ1β1-ϰ3β3}+ϖχ]-[ϰ12+ϰ3ϖ][∂ϰ1∂t+ϖ{∂β3∂s+ϰ2β1+ϰ3β2}])=0.
The magnetic ϕ(nq) flux surface is given by the following ferromagnetic condition
-ϖ([ϰ12+ϰ3ϖ]ϰ1[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}]+[ϖ'-ϰ1ϰ2][ϰ1{∂∂s∂ϖ∂s-ϰ1ϰ2-∂ϰ1∂s+ϰ2ϖϰ2-(ϰ12+ϰ3ϖ)ϰ3}+ϖ{ϰ1(ϰ12+ϰ3ϖ)-∂∂s∂ϰ1∂s+ϰ2ϖ-ϰ2∂ϖ∂s-ϰ1ϰ2}])-ϰ2(ϰ1[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}][ϰ1'+ϰ2ϖ]+[ϖ'-ϰ1ϰ2]ϖ[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}])+ϰ1(ϖ[∂∂s{ϰ12+ϰ3ϖ}+{∂ϰ1∂s+ϰ2ϖ}ϰ1+ϰ3{∂ϖ∂s-ϰ1ϰ2}][ϰ12+ϰ3ϖ]-[ϰ1'+ϰ2ϖ][ϰ1{∂∂s∂ϖ∂s-ϰ1ϰ2-∂ϰ1∂s+ϰ2ϖϰ2-(ϰ12+ϰ3ϖ)ϰ3}+ϖ{ϰ1(ϰ12+ϰ3ϖ )-∂∂s∂ϰ1∂s+ϰ2ϖ-ϰ2∂ϖ∂s-ϰ1ϰ2}])=0.
We consider the similar method as in the first case to extract the following
demonstration. In Fig. 2, the magnetic flux
density of the particle is shown when it is assumed under the action of the Lorentz
force ϕ(nq).
Case 3. Magnetic ϕ(bq) flux
The magnetic ϕ(bq) flux mFϕ(bq) is given by
mFϕ(bq)=∫F(ϖ[{∂ϖ∂s+ϰ2ϰ1}{ϰ2∂β3∂s+ϰ2β1+ϰ3β2+ϖχ}-{ϰ22+ϖϰ3}{∂ϖ∂t+ϰ2∂β2∂s\ +ϰ1β1-ϰ3β3}]-ϰ2[{-∂ϰ2∂s+ϖϰ1}{ϰ2∂β3∂s+ϰ2β1+ϰ3β2+ϖχ}-{ϰ22+ϖϰ3}{ϖ∂β2∂s\ +ϰ1β1-ϰ3β3-∂ϰ2∂t}]+ϰ1[{∂ϖ∂s+ϰ2ϰ1}{ϖ ∂β2∂s\ +ϰ1β1-ϰ3β3-∂ϰ2∂t}-{∂ϖ∂t+ϰ2∂β2∂s\ +ϰ1β1-ϰ3β3}{-∂ϰ2∂s+ϖϰ1}])dπ.
With short calculations, we have
∇sϕ(bq)×∇tϕ(bq)=([∂ϖ∂s+ϰ2ϰ1][ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}+ϖχ]-[ϰ22+ϖϰ3]×[∂ϖ∂t+ϰ2{∂β2∂s +ϰ1β1-ϰ3β3}])tq+([-∂ϰ2∂s+ϖϰ1][ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}+ϖχ]-[ϰ22+ϖϰ3][ϖ{∂β2∂s+ϰ1β1-ϰ3β3}-∂ϰ2∂t])nq+([∂ϖ∂s+ϰ2ϰ1][ϖ{∂β2∂s+ϰ1β1-ϰ3β3}-∂ϰ2∂t]-[∂ϖ∂t+ϰ2{∂β2∂s\ +ϰ1β1-ϰ3β3}][-∂ϰ2∂s+ϖϰ1])bq.
Flux density of ϕ(bq) is given by
mLϕ(bq)=ϖ([∂ϖ∂s+ϰ2ϰ1][ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}+ϖχ]-[ϰ22+ϖϰ3][∂ϖ∂t+ϰ2{∂β2∂s +ϰ1β1-ϰ3β3}])-ϰ2([-∂ϰ2∂s+ϖϰ1][ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}+ϖχ]-[ϰ22+ϖϰ3][ϖ{∂β2∂s +ϰ1β1-ϰ3β3}-∂ϰ2∂t])+ϰ1([∂ϖ∂s+ϰ2ϰ1][ϖ{∂β2∂s +ϰ1β1-ϰ3β3}-∂ϰ2∂t]-[∂ϖ∂t+ϰ2{∂β2∂s +ϰ1β1-ϰ3β3}][-∂ϰ2∂s+ϖϰ1]).
Using the above equation in phase, we obtain that
mFϕ(bq)=∫F(ϖ[{∂ϖ∂s+ϰ2ϰ1}{ϰ2∂β3∂s+ϰ2β1+ϰ3β2+ϖχ}-{ϰ22+ϖϰ3}{∂ϖ∂t+ϰ2∂β2∂s+ϰ1β1-ϰ3β3}]-ϰ2[{-∂ϰ2∂s+ϖϰ1}{ϰ2∂β3∂s+ϰ2β1+ϰ3β2+ϖχ}-{ϰ22+ϖϰ3}{ϖ∂β2∂s\ +ϰ1β1-ϰ3β3-∂ϰ2∂t}]+ϰ1[{∂ϖ∂s+ϰ2ϰ1}{ϖ(∂β2∂s+ϰ1β1-ϰ3β3)-∂ϰ2∂t}-{∂ϖ∂t+ϰ2∂β2∂s\ +ϰ1β1-ϰ3β3}{-∂ϰ2∂s+ϖϰ1}])dπ.
Also, ferromagnetic model for ϕ(bq), we get that
mLϕ(bq)ferro=B⋅∇sϕ(bq)×ϕ(bq)×∇s2ϕ(bq).
By using the quasi frame, we have
∇sϕ(bq)×ϕ(bq)×∇tq2ϕ(bq)=(ϰ2[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}][ϰ22+ϖϰ3]-[ϖ'+ϰ2ϰ1][ϖ{∂∂s-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+ (ϰ22+ϖϰ3 )ϰ2}ϰ2{-∂ϰ2∂s+ϖϰ1ϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1+(ϰ22+ϖϰ3)ϰ3}])tq-([ϖϰ1-ϰ2'][ϖ{∂∂s-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+(ϰ22+ϖϰ3 )ϰ2}-ϰ2{-∂ϰ2∂s+ϖϰ1ϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1}+{ϰ22+ϖϰ3}ϰ3])+(ϰ22+ϖϰ3)ϖ(ϰ2[-∂ϰ2∂s+ϖϰ1]-[∂ϖ∂s+ϰ2ϰ1]ϰ3-∂∂s[ϰ22+ϖϰ3])nq+([ϖϰ1-ϰ2']ϰ2[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}]-[ϖ'+ϰ2ϰ1]ϖ[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}])bq.
Also, we find that
mLϕ(bq)ferro=ϖ(ϰ2[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}][ϰ22+ϖϰ3]-[ϖ'+ϰ2ϰ1][ϖ{∂∂s-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+(ϰ22+ϖϰ3)ϰ2}-ϰ2{-∂ϰ2∂s+ϖϰ1ϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1+(ϰ22+ϖϰ3)ϰ3}])+ϰ1([ϖϰ1-ϰ2']ϰ2×[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}]-[ϖ'+ϰ2ϰ1]ϖ[ϰ2×{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}])+ϰ2([ϖϰ1-ϰ2'][ϖ{∂∂s×-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+(ϰ22+ϖϰ3)ϰ2}-ϰ2{-∂ϰ2∂s+ϖϰ1ϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1+(ϰ22+ϖϰ3)ϰ3}]+[ϰ22+ϖϰ3]ϖ[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}]).
Thus, we immediately obtain that
mFϕ(bq)ferro=∫F(ϖ[ϰ2{ϰ2-∂ϰ2∂s+ϖϰ1-∂ϖ∂s+ϰ2ϰ1ϰ3-∂∂s(ϰ22+ϖϰ3)}{ϰ22+ϖϰ3}-{ϖ'+ϰ2ϰ1}{ϖ(∂∂s-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+ϰ22+ϖϰ3ϰ2)-ϰ2(-∂ϰ2∂s+ϖϰ1\bracrϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1+ϰ22+ϖϰ3ϰ3)}]+ϰ1[{ϖϰ1-ϰ2'}ϰ2×{ϰ2-∂ϰ2∂s+ϖϰ1-∂ϖ∂s+ϰ2ϰ1ϰ3-∂∂s(ϰ22+ϖϰ3)}-{ϖ'+ϰ2ϰ1}ϖ{ϰ2-∂ϰ2∂s+ϖϰ1-∂ϖ∂s+ϰ2ϰ1ϰ3-∂∂s(ϰ22+ϖϰ3)}]+ϰ2[{ϖϰ1-ϰ2'}{ϖ(∂∂s-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+ϰ22+ϖϰ3ϰ2)-ϰ2-∂ϰ2∂s+ϖϰ1ϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1+ϰ22+ϖϰ3ϰ3}+{ϰ22+ϖϰ3}ϖ{ϰ2×-∂ϰ2∂s+ϖϰ1-∂ϖ∂s+ϰ2ϰ1ϰ3-∂∂s(ϰ22+ϖϰ3)}])dπ.
The magnetic ϕ(bq) flux surface condition is given by
ϖ([∂ϖ∂s+ϰ2ϰ1][ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}+ϖχ]-[ϰ22+ϖϰ3][∂ϖ∂t+ϰ2{∂β2∂s +ϰ1β1-ϰ3β3}])-ϰ2([-∂ϰ2∂s+ϖϰ1][ϰ2{∂β3∂s+ϰ2β1+ϰ3β2}+ϖχ]-[ϰ22+ϖϰ3][ϖ{∂β2∂s +ϰ1β1-ϰ3β3}-∂ϰ2∂t])+ϰ1([∂ϖ∂s+ϰ2ϰ1][ϖ{∂β2∂s +ϰ1β1-ϰ3β3}-∂ϰ2∂t]-[∂ϖ∂t+ϰ2{∂β2∂s +ϰ1β1-ϰ3β3}][-∂ϰ2∂s+ϖϰ1])=0.
The magnetic ϕ(bq) flux surface is given by the ferromagnetic condition
ϖ(ϰ2[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}][ϰ22+ϖϰ3]-[ϖ'+ϰ2ϰ1][ϖ{∂∂s-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+(ϰ22+ϖϰ3)ϰ2}-ϰ2{-∂ϰ2∂s+ϖϰ1ϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1+(ϰ22+ϖϰ3)ϰ3}])+ϰ1([ϖϰ1-ϰ2']ϰ2×[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}]-[ϖ'+ϰ2ϰ1]ϖ[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}])+ϰ2([ϖϰ1-ϰ2'][ϖ{∂∂s-∂ϰ2∂s+ϖϰ1+∂ϖ∂s+ϰ2ϰ1ϰ1+(ϰ22+ϖϰ3) ϰ2}-ϰ2{-∂ϰ2∂s+ϖϰ1ϰ1-∂∂s∂ϖ∂s+ϰ2ϰ1+(ϰ22+ϖϰ3)ϰ3}]+[ϰ22+ϖϰ3]ϖ[ϰ2{-∂ϰ2∂s+ϖϰ1}-{∂ϖ∂s+ϰ2ϰ1}ϰ3-∂∂s{ϰ22+ϖϰ3}])=0.
We consider the similar method as in the first and second case to extract the
following demonstration. In Fig. 3, the
magnetic flux density of the particle is shown when it is assumed under the action
of the Lorentz force ϕ(bq).
5.Conclusion
Flows perform an essential part in geometric design and style, applied physics, and
structural motion. In ths paper, we have investigated a different approach by
considering the directional flows of the velocity magnetic particles and the quasi
frame. We have provided several different characterizations of curvatures regarding
plenty of differential equations in space. These characterizations may further be
used to investigate the directional flows of magnetic particles. Another purpose of
future studies will be to explore the unified formulations of the systems composed
of arbitrary dyons, magnetic and electric charges of manifolds with magnetic flux
lines. The magnetohydrodynamic model of the magnetic surfaces reduced from the
magnetic flux surfaces of distinct types will also be the subject of another
project. This project eventually aims to investigate significant features of the
flux maximizing and minimizing flows.
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