Application of Differential Invariant Method for Solving the Electromagnetic Fields

FU JingliXIANG ChunCAO ShanGUO Yongxin

1 College of Mechanical and Automotive Engineering,Zhejiang University of Water Resources and Electric Power,Hangzhou 310018,China 2 Institute of Mathematical Physics,School of Science,Zhejiang Sci-Tech University,Hangzhou 310018,China 3 Department of Physics,School of Physics,Liaoning University,Shenyang 110036,China

Abstract:We study the first integral and the solution of electromagnetic field by Lie symmetry technique and the differential invariant method. The definition and properties of differential invariants are introduced and the infinitesimal generators of Lie symmetries and the differential invariants of electromagnetic field are obtained. The first integral and the solution of electromagnetic field are given by the Lie symmetry technique and the differential invariants method. A typical example is presented to illustrate the application of our theoretical results.

Key words:differential invariant;first integral;Lie symmetry;electromagnetic field

Introduction

The symmetry theory has been widely used in mathematics,physics,mechanics,and dynamics.In early years,Lie group theory has been a popular and useful method in the study of ordinary differential equations (ODEs) and partial differential equations (PDEs)[1-6].Many researches[7-12]have showed that one can use the differential invariant method to reduce the order or to solve the differential equation with Lie group transformation.Afterward many researchers[13-16]studied the symmetries and conservation laws of nonlinear mathematical physics equations.In other words,given ODEs or PDEs with known Lie symmetry technique,one can obtain the corresponding differential invariants,and then get the equations after reduction of order.Furthermore,the solutions of the equations can be obtained.

As early as a few decades ago,Lie symmetry technique was widely introduced to solve the problems of constraint system[17-35].The basic idea of Lie symmetry technique is to keep the equation of motion invariants under the infinitesimal transformations.We can obtain generational functions and normal functions from determining equations and structural equations.Then,the conserved quantities can be obtained from Lie theorem with the infinitesimal transformation and structural equations.Nowadays,many good and useful results have been obtained about the electromagnetic field.However,the application of Lie symmetry and the differential invariant method to the electromagnetic field to obtain the corresponding first integral has not been studied.

In recent years,the differential invariant method plays a central role in a wide variety of problems arising in geometry,differential equations,mathematical physics,and applications.And in the paper by Zhang and Mei[36],they made an introduction of applying differential invariant method to nonholonomic systems.In this paper,we will apply this method to electromagnetic field to obtain first integral,and further solve it.

1 Differential Invariant

Now,let us consider a one-parameter and multi-degree of freedom Lie group transformation

x*=X(x;ε),

(1)

y*=Y(x)=(y1(x),y2(x),…,yn(x)),

(2)

Now consider the application of Lie groups of point transformation to the study of a second or higher-order ODE

y(n)=f(x,y,y′,…,y(n-1)),

(3)

Assume that Eq.(3) admits a one-parameter Lie group of point transformations

x*=X(x,y;ε)=x+εξ(x,y)+O(ε2),

(4)

y*=Y(x,y;ε)=y+εη(x,y)+O(ε2),

(5)

with infinitesimal generator

(6)

Thenth-order Eq.(3) represented by the surface

F(x,y,y1,…,yn-1)=yn-f(x,y,y1,…,yn-1)=0,

(7)

wherey′=y1,y″=y2,…,y(n)=yn.The surface Eq.(7) is an invariant if and only of it admits the groups (4) and (5),i.e.,

X(n)F=0,whenF=0,

whereX(n)is thenth extension of the infinitesimal generatorX,

n=1,2,…,l.

(8)

TheoremThe extended infinitesimalsη(k)satisfy the recursion relation

η(k)(x,y,y1,…,yk)=

Dη(k-1)(x,y,y1,…,yk-1)-ykDξ,

k=1,2,…

(9)

whereη(0)=η(x,y).In particular,

(10)

Hence,it follows thatF(x,y,y1,…,yn) is some function of the group’s invariants

u(x,y),v1(x,y,y1),…,vn(x,y,y1,…,yn)，

(11)

(12)

Equation (7) becomes

G(u,v1,v2,…,vn)=0,

(13)

In terms of differential invariants,solving thenth-order Eq.(3) reduces to solving the (n-1)th-order one

(14)

wherev(x,y,y1)=v1(x,y,y1),if

v=Φ (u;C1,C2,…,Cn-1)

(15)

is the general solution of Eq.(14),whereC1,C2,…,Cn-1are arbitrary constants,then the general solution of thenth-order Eq.(3) is found by solving ODE

v(x,y,y′)=Φ(u(x,y);C1,C2,…,Cn-1),

(16)

which reduces to a quadrature since it admits the groups (4) and (5).

2 Lie Symmetry of Electromagnetic Field

(17)

uμuμ=1,

(18)

(19)

Now,defining the general momentum

(20)

so the equation of motion can be expressed in the form

(21)

We assume that the system has one-parameter Lie point symmetry.Thus,the infinitesimal transformations with respect to time and generalized coordinates can be introduced as

(22)

whereεis a parameter,andξandηare infinitesimals.The group is determined by its infinitesimal generator

(23)

The generator of the first extended group is given by

(24)

Its second extension has the form

(25)

Thus,the basic idea of Lie symmetry is to keep Eq.(21) invariant under the infinitesimal transformations Eq.(22) that must satisfy the determining equation

(26)

Then,we have

(27)

Obviously,we can get the infinitesimal transformations from Eq.(27).

3 Differential Invariants and First Integral of Electromagnetic Field

Now,we discuss an electromagnetic field with differential invariants.We have worked out the infinitesimal transformation (ξ(t,xμ),ημ(t,xμ))from Eq.(27).Then,using Eqs.(12) and (23),we can obtain the invariant of the first extension (u,v).If (u,v) is the first integral of the electromagnetic field,then the solution of the electromagnetic field can be obtained.

In this part,an example about electromagnetic field is considered.The new conserved quantities have been constructed from the different differential invariants.

ExampleLet us consider the motion of charged particle in the external electromagnetic field,there is reaction between particle and field.Let the charge of particle ise,four-dimensional velocity isuμ=dxμ/dt,the four-dimensional potential of electromagnetic field isAμ,and the Lagrange function of this system is

L=-m0-eAμ(x)uμ,

(28)

and satisfies constraints (18) and (19).Using Eq.(20),the general momentums of the external electromagnetic field are written as

(29)

(30)

(31)

where

Fμv=?μAv-?vAμ.

(32)

The infinitesimal transformation can be introduced as

(33)

and the determining equation of the system under the infinitesimal transformation is

(34)

(35)

From Eq.(35) we can obtain the infinitesimal generators as

ξ1=1,η1=0,

(36)

ξ2=0,η2=1,

(37)

ξ3=1,η3=1,

(38)

ξ4=t,η4=-xμ.

(39)

According to the method of differential invariants,the corresponding characteristic equations as

(40)

(41)

(42)

(43)

It is obvious that invariants of the first extension of Eqs.(40)-(43) are given by

u1(t,xμ)=C1,v1(t,xμ)=0,

(44)

u2(t,xμ)=C2,v2(t,xμ)=C3,

(45)

u3(t,xμ)=t-xμ,v3(t,xμ)=C4,

(46)

(47)

whereC1,C2,C3,andC4are arbitrary constants.The differential invariants do not satisfy the condition of Eqs.(44)- (46),the corresponding differential invariant of Eq.(47) is

(48)

Hence,the relation betweenu4andv4is

(49)

(50)

Substituting Eqs.(49) and (50) into Eq.(47),we can have

(51)

(52)

whereCis an arbitrary constant.Obviously,in this example,Eqs.(44)-(47) are the first integral of the electromagnetic field.

4 Results

5 Conclusions

In this paper,we recommend a new method of differential invariants for obtaining the first integral of electromagnetic field.Using this method,we have obtained the Lie symmetries,differential invariants and first integral of electromagnetic field,and the solutions of electromagnetic field are also obtained.We can use constructing differential invariant method for seeking the first integrals of non conservative mechanical systems,nonholonomic mechanical systems,mechanic-electrical coupling systemsetc.