Generalized Potential and Mathematical Principles of Nonlinear Analysis
GGeneralized Potential and Mathematical Principles of Nonlinear Analysis
Peng Yue ∗ University of Electronic Science and technology of China, Chengdu, China
In the past hundred years, chaos has always been a mystery to human beings, including thebutterfly effect discovered in 1963 and the dissipative structure theory which won the chemistryNobel Prize in 1977. So far, there is no quantitative mathematical-physical method to solve andanalyze these problems. In this paper, the idea of using field theory to study nonlinear systems isput forward, and the concept of generalized potential is established mathematically. The physicalessence of generalized potential promoting the development of nonlinear field is extended and thespatiotemporal evolution law of generalized potential is clarified. Then the spatiotemporal evolutionlaw of conservative system and pure dissipative system is clarified. Acceleration field, conservativevector field and dissipation vector field are established to evaluate the degree of conservation anddissipation of physical field. Finally, the development route of new field research and the preconditionof promoting engineering application in the future are discussed.
For a long time, the researches on nonlinear problemshave attracted much attention. The reason for this en-thusiasm for research is that, on the one hand, due tothe continuous in-depth study of turbulence, bucklingand other mechanical theories, the problems found haveobvious nonlinear characteristics [1], which greatly re-stricts related theoretical breakthroughs and engineeringinnovation. On the other hand, the well-known butter-fly effect, dissipation mechanism and other factors alsoshow the influence of nonlinear factors in the researchesof medical statistics [2], economics [3], sociology [4] andother disciplines, which also makes the research on non-linear problems attract the attention of researchers inother subjects. Therefore, it is necessary to develop in-novative mathematical analysis methods to describe non-linear phenomena.At present, in addition to relying on phenomenologicalstatistics to obtain the basic laws of nonlinear dynam-ics [1][2][3][4], international researches on nonlinear prob-lems are mainly based on two major ideas. One is to relyon the continuous development of mathematical theory,through geometric methods or the solution of nonlinearpartial differential equations [5] to obtain the basic lawsof system nonlinearity. The other is to add the necessaryphysical conditions to linearize the nonlinear problem,and develop and improve the approximate characteristicsof nonlinearity through the study of linear problems [6].However, one obstacle is that the development of mathe-matical theory depends on sufficient inspiration and con-tinuous updating of philosophical methodology. Anotherobstacle is that the physical simplification method of lin-earization will lead to the loss of key information of non-linear problems. For example, the current common initialvalue sensitive problems cannot be reflected by linearizedmodels. Based on the above two problems, the nonlineartheory of classical mechanics is already in a bottleneckperiod, and it is urgent to develop new analysis methodsto deal with the current problems. ∗ Emial: [email protected]
Usually, any nonlinear system can be expressed by thefollowing general expressions:˙ x i = f i ( x j ) , f i ∈ C ( R n ) (1)For example, Lorentz system can be expressed as:˙ x = σ ( x − x )˙ x = rx − x − x x ˙ x = x x − bx (2)At present, the common method for studying abovethe system is Lagrange method. The above-mentionedsystem is taken as the motion equation of the particlesystem, and the stability of the system can be judged bystudying the singularity of the system [7]. This methodis widely used in related projects. However, this methodhas great limitations in revealing the nonlinear dissipa-tion characteristics of the system at the system level, forexample, it is very difficult to judge the occurrence ofbutterfly effect and reveal the characteristics of dissipa-tive structure in chaos systems. In the field of mechanics,the common analysis methods include Lagrange methodand Euler method. In this paper, Euler method is used todescribe the nonlinear system. For example, for Lorentzsystem (2), “phase flow” can be expressed as FIG. 1 toFIG. 3. FIG. 1. Phase flow diagram on XOY plane for Lorentz system
In addition, it can be seen that the streamline diagramof phase flow fits well with the traditional phase diagramof nonlinear system. a r X i v : . [ phy s i c s . c l a ss - ph ] D ec FIG. 2. Phase flow diagram on XOZ plane for Lorentz systemFIG. 3. Comparison between streamline diagram (blue) ofphase flow and traditional phase diagram (red) for Lorentzsystem
Streamline, as one of the physical quantities of fluidmechanics, refers to the curve which is tangent to thevelocity vector at every point in the flowfield. It is a curvecomposed of different fluid particles at the same timeand reveals the spatial distribution law of velocity, whichcoincides with the relationship between momentum andposition in phase diagram. It inspires a new method tosolve PDEs and will be discussed in the future becauseit is not related to the content discussed in this paper.From FIG. 1 to FIG. 3 provide a good foundation forthe introduction of nonlinear analysis into field theory.In the history of conservative field research, firstly, La-grange [8] described the gravitational field with gravita-tional potential. The gravity at any point is equal to thenegative gradient of the gravitational potential at thatpoint. Then, Laplace [9] gives the gravitational potentialequation in rectangular coordinates. Laplace [9] assumesthat the Laplace equation holds when the attractive par-ticle is in the body. Finally, Poisson [10] fixed the error.Poisson [10] pointed out that if the point is in the interiorof the attractor, the Laplace equation should be changedto Poisson equation in practical physics research. There-fore, we can understand that potential is the reason topromote the development of conservative field [11]. As weall know, nonlinear field includes conservative field anddissipative field. According to the expansion of the con-servative field researches, we believe that the nonlinearfield also has a “push potential” to promote its develop- ment. In order to study this “push potential”, here is theconcept of generalized potential.
Definition 1.
For Ω ⊂ R n , M = (Ω , S ) , existencetensor field σ ij ∈ T M . if conjugate gradient of vector ϕ i ∈ T M satisfy condition σ ij = ϕ i ∂ j , the vector ϕ i isgeneralized potential of tensor field σ ij . In order to explain why conjugate gradient is used inthe definition, lemma 1 need to be studied.
Lemma 1.
The curl of vector’s gradient is zero, and thecurl of vector’s conjugate gradient is not zero.Proof.
Due to arbitrary vector ϕ i ∈ T M , M = (Ω , S ),the curl of vector s’ gradient is e k ∂ k × e i e j ∂ i ϕ j = (cid:0) e k × e i (cid:1) e j ϕ j,ki = (cid:15) ki •• m e m e j ϕ j,ki For partial derivatives, the order of derivation is trans-formed, and its value does not change. (cid:15) ki •• m e m e j ϕ j,ki = (cid:15) ki •• m e m e j ϕ j,ik By converting the index again, we can obtain (cid:15) ki •• m e m e j ϕ j,ik = (cid:15) ik •• m e m e j ϕ j,ik = − (cid:15) ki •• m e m e j ϕ j,ki = 0It is proved that the curl of vector’s gradient is zero.Then, assuming that the curl of the conjugate gradientof the vector is zero and always holds e k ∂ k × e i e j ϕ i ∂ j ≡ − e k ∂ k × e i e j ϕ i ∂ j We can get the following formula e k ∂ k × e i e j ϕ i ∂ j = (cid:0) e k × e i (cid:1) e j ϕ i,kj = (cid:15) ki ••• m e m e j ϕ i,kj According to the rule of index conversion, we can ob-tain − (cid:15) ki ••• m e m e j ϕ i,kj = (cid:15) • ikm •• e m e j ϕ i,kj = (cid:15) • imk •• e k e j ϕ i,mj Combine the above two formulas, we can get a funnyconclusion k ≡ m According to the definition of tensor index, it is obviousthat the curl of vector’s conjugate gradient is not zero.Thus it can be seen that the conjugate gradient is dif-ferent from the gradient. The gradient of the vector hasthe expression problem of rotating field. However, theconjugate gradient represents the second-order tensor,which can be homeomorphic mapping with the originaltensor. Therefore, any tensor can be expressed as conju-gate gradient form of vector.In order to establish the relationship between conju-gate gradient and gradient, lemma 2 should be studied.
Lemma 2.
The divergence of the conjugate gradient ofan arbitrary vector is equal to the gradient of the diver-gence of this vector.Proof.
Due to arbitrary vector ϕ i ∈ T M , M = (Ω , S ),the divergence of the conjugate gradient of the vector is e k ∂ k · e i e j ϕ i ∂ j = e j δ ki ϕ i,jk = e j ϕ i,ji and the gradient of the divergence of vector ϕ i is e j ∂ j (cid:0) e k ∂ k · e i ϕ i (cid:1) = e j δ ki ϕ i,kj = e j ϕ i,ij Because the order of derivation of partial derivatives canbe interchanged arbitrarily, it can be obtained that e j ϕ i,ji = e j ϕ i,ij It can be concluded that the divergence of the conju-gate gradient of an arbitrary vector is equal to the gra-dient of the divergence of this vector.In the field of classical mechanics, Newton’s secondlaw is usually considered when describing the relationshipbetween the change of motion state and the force actingon it. So we can get the most important theorem of thispaper.
Theorem 1.
For stress tensor field σ ij ∈ T M , M =(Ω , S ) . The generalized potential of the stress field is ϕ i ∈ T M , then the evolution law of the stress field satis-fies ϕ • ,i • i, • j = ρ du j dt (3) where ρ is the density of nonlinear system, t is developingtime of the nonlinear system and u j is the velocity vectorof nonlinear system.Proof. Generally, an object is subjected to two kinds offorce, surface force F planej and mass force F massj . Hy-pothetically, if the force caused by mass is equivalentlydistributed to the surface, then it can be mathematicallyequivalent to sum surface force F j . F j = F planej + F massj According to Gauss theorem, above force can be de-scribed by stress state, n i is the surface normal vector. F j = (cid:90) ∂τ n i σ ij d ( ∂τ ) = (cid:90) τ σ •• ,iij, • dτ so the force per unit volume can be expressed as f j = dF j dτ = σ •• ,iij, • Assuming that the generalized potential of the stressfield is ϕ i ∈ T M , according to lemmas 1 and 2, it is easyto obtain the following expression σ •• ,iij, • = ϕ • , • ii,j • = ϕ • ,i • i, • j Newton’s second law with the force per unit volume inMechanics can be expressed as: f j = ρ du j dt Combining all of the above formulas, we can obtain thefinal result ϕ • ,i • i, • j = ρ du j dt In this way, the theory has been proved.This theorem reveals the spatiotemporal evolution lawof nonlinear systems in classical mechanics. According tothe famous formula in field theory. It can be simply un-derstood that the sum of Laplacian operator and doublecurl operator is the operator on the left of formula (3) asshown in following ϕ • ,i • i, • j = ϕ • , • ij,i • + (cid:15) • nmj •• (cid:15) • kim •• ϕ i,nk In the previous introduction, it has been explained thatthe physical field represented by the Laplacian operatoris a conservative field. Therefore, theorem 2 can be ob-tained and the proof is omitted.
Theorem 2.
In the framework of classical mechanics,nature exists a generalized potential ϕ i in the nonlin-ear system. The generalized potential is the fundamentalreason that drives the development of nonlinear systems.The spatial evolution law of the generalized potential is ϕ • , • ij,i • + (cid:15) • nmj •• (cid:15) • kim •• ϕ i,nk = ρ du j dt (4) where ρ is the density of nonlinear system, t is developingtime of the nonlinear system and u j is the velocity vectorof nonlinear system. In addition, we can get two important corollaries,which reveal conditions for the occurrence of conserva-tive system and pure dissipative system.
Inference 1.
The conservative field satisfies the follow-ing relations, the Laplacian operator of the generalizedpotential is a constant value, and the acceleration drivesthe generalized potential to be spatially distributed in theform of double curl operator. L j = ϕ • , • ij,i • , L j ∈ T M (5) At this time, the space-time evolution law can be trans-formed into (cid:15) • nmj •• (cid:15) • kim •• ϕ i,nk = ρ du j dt − L j (6) where ρ is the density of nonlinear system, t is developingtime of the nonlinear system and u j is the velocity vectorof nonlinear system. Inference 2.
The pure-dissipation field satisfies the fol-lowing relations, the double curl operator of the gener-alized potential is a constant value, and the accelerationdrives the generalized potential to be spatially distributedin the form of Laplacian. D j = (cid:15) • nmj •• (cid:15) • kim •• ϕ i,nk , D j ∈ T M (7) At this time, the space-time evolution law can be trans-formed into ϕ • , • ij,i • = ρ du j dt − D j (8) where ρ is the density of nonlinear system, t is developingtime of the nonlinear system and u j is the velocity vectorof nonlinear system. Formula (5) reveals the basic evolution law of generalchaotic system, and formula (6) and formula (7) are twolimits of chaotic motion. Therefore, the following phys-ical quantities can be defined to evaluate the degree ofconservation and dissipation in any chaotic system.
Definition 2.
For Ω ⊂ R n , M = (Ω , S ) , existence ten-sor field σ ij ∈ T M . L is the Laplace operator and D isthe double curl operator. L j ∈ T M is the conservativevector field and D j ∈ T M is the dissipation vector fieldfor the original tensor field σ ij . The development speedof complex system is u j ∈ T M and the generalized poten-tial is ϕ j ∈ T M , then these nonlinear development law ofcomplex system is as follows: ρ du j dt = L j + D j , L j = L ( ϕ j ) , D j = D ( ϕ j ) (9) where ρ is the density of complex system, t is developingtime of the complex system Finally, an example is given to illustrate the influenceof generalized potential on nonlinear system. If the dis-tribution of generalized potential in three-dimensionalspace is in the following form ϕ = σ ( x − x ) ϕ = rx − x − x x ϕ = x x − bx (10)It is noticed that above equations look like Lorentz sys-tem, but they are the spatial distribution of the general-ized potential, not the spatial distribution of the originalfield, so it is not Lorentz system. Let’s name it the new-1nonlinear system. Its generalized potential space distri-bution is shown in FIG. 1 to FIG. 3. The distribution ofdissipation vector is shown in FIG. 4 to FIG. 7.It can be seen from FIG. 4, FIG. 5 and FIG. 6 thatthe small disturbance at the center point will bring vi-bration of the distant space acceleration, which revealsthe famous ”Butterfly Effect” [12]. For the new-1 non-linear system, the maximal amplifying coefficient is be-tween 115 and 155. In fact, due to the superposition of FIG. 4. Dissipation (green) and conservative (red) vector fielddistribution and acceleration field (blue) on XOY plane fornonlinear system New-1FIG. 5. Dissipation (green) and conservative (red) vector fielddistribution and acceleration field (blue) on XOZ plane fornonlinear system New-1 vectors, the small disturbances in different places will beintegrated to balance. Therefore, if we want to calculatethe disturbance amplifying coefficient in space, we needto consider all the small disturbance values and stackthem. The cancellation of the nonlinear interaction canbe demonstrated in the experiment of turbulent inter-action changing into laminar flow published in 2019 [1].In addition, it can be seen from the streamline diagramFIG. 7 that the dissipative structure of nonlinear dis-turbance is reciprocating, and it can be seen that systemwill develop from one equilibrium state to another. Here,the dissipative structure theory [13] proposed by Ilya Pri-gogine is verified mathematically and physically. So wecan get the following general law in physics.
Theorem 3.
All nonlinear processes spontaneously un-dergo a conservative and dissipative reciprocating process.The generalized potential is distributed in the space in theform of divergence gradient operator, where the spatialdistribution of the generalized potential on Laplace oper-ator represents the conservative behavior of the nonlinearsystem, and the spatial distribution on the double curloperator of the generalized potential represents the dissi-pative behavior of the nonlinear system, until the kineticenergy of the nonlinear system become zero and chaosstops.
FIG. 6. Dissipation (green) and conservative (red) vector fielddistribution and acceleration field (blue) on YOZ plane fornonlinear system New-1FIG. 7. Streamlines of dissipation (green) and conservative(red) vector field distribution and acceleration field (blue) fornonlinear system New-1
Nonlinearity is the essential law of nature, so it is notonly a physical law, but also a new natural philosophcalmethodology. Due to the regular generalized potentialdistribution (Fig.1 and Fig.2) will lead to random andirregular distribution of kinematic physical quantities inspace (Fig.4, Fig.5 and Fig.6) , the double curl opera-tor of complex system will lead to the failure of Lapla-cian determinism, Therefore, generalized potential andits principle can not only be applied to fluid mechanics,solid mechanics, electrodynamics, even economics, polit-ical science, history and other disciplines, but also con-tribute to development of natural philosophy principle.In summary, mathematically speaking, the fundamen-tal cause of nonlinear phenomena is the double curl op-erator of the generalized potential. Due to the influenceof different initial value and different boundary condi-tions, the spatial distribution of generalized potential isdifferent, and the development law of nonlinear systemhas also undergone great changes. Its application scenar-ios could be different physical scenarios of analytical me-chanics, fluid mechanics, solid mechanics and electrody-namics research. Therefore, the generalized potential caneffectively unify classical mechanics. In addition, gener-alized potential can be used as an effective method tosolve nonlinear problems in engineering. This is an openand challenging new field. Determining generalized po-tentials with different boundary conditions and differentinitial value will be the focus of future research in thisfield. [1] J. Kuchnen, B. Song, D. Scarselli, and et.al. Destabiliz-ing turbulence in pipe flow.
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