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).