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Preprint typeset in JHEP style - HYPER VERSION

Classical Field Theory

Gleb Arutyunov

ay a Institute for Theoretical Physics and Spinoza Institute, Utrecht University, 3508 TD Utrecht, The Netherlands Abstract:The aim of the course is to introduce the basic methods of classical eld theory and to apply them in a variety of physical models ranging from clas- sical electrodynamics to macroscopic theory of ferromagnetism. In particular, the course will cover the Lorentz-covariant formulation of Maxwell's electromagnetic the- ory, advanced radiation problems, elements of soliton theory. The students will get acquainted with the Lagrangian and Hamiltonian description of innite-dimensional dynamical systems, the concept of global and local symmetries, conservation laws. A special attention will be paid to mastering the basic computation tools which include the Green function method, residue theory, Laplace transform, elements of group theory, orthogonal polynomials and special functions.

Last Update 8.05.2011

Email: G.Arutyunov@phys.uu.nl

yCorrespondent fellow at Steklov Mathematical Institute, Moscow.

1. Classical Fields: General Principles2

1.1Lagrangian and Hamiltonian formalisms3

1.2Noether's theorem in classical mechanics9

1.3Lagrangians for continuous systems11

1.4Noether's theorem in eld theory15

1.5Hamiltonian formalism in eld theory20

2. Electrostatics21

2.1Laws of electrostatics21

2.2Laplace and Poisson equations26

2.3The Green theorems27

2.4Method of Green's functions29

2.5Electrostatic problems with spherical symmetry31

2.6Multipole expansion for scalar potential38

3. Magnetostatics41

3.1Laws of magnetostatics41

3.2Magnetic (dipole) moment42

3.3Gyromagnetic ratio. Magnetic moment of electron.44

4. Relativistic Mechanics46

4.1Newton's relativity principle46

4.2Einstein's relativity principle46

4.3Dening Lorentz transformations48

4.4Lorentz group and its connected components50

4.5Structure of Lorentz transformations53

4.6Addition of velocities57

4.7Lie algebra of the Lorentz group57

4.8Relativistic particle60

5. Classical Electrodynamics62

5.1Relativistic particle in electromagnetic eld62

5.2Lorentz transformations of the electromagnetic eld65

5.3Momentum and energy of a particle in a static gauge67

5.4Maxwell's equations and gauge invariance67

5.5Fields produced by moving charges69

5.6Electromagnetic waves72

{ 1 {

5.7Hamiltonian formulation of electrodynamics76

5.8Solving Maxwell's equations with sources79

5.9Causality principle84

6. Radiation85

6.1Fields of a uniformly moving charge86

6.2Fields of an arbitrary moving charge87

6.3Dipole Radiation91

6.4Applicability of Classical Electrodynamics100

6.5Darvin's Lagrangian101

7. Advanced magnetic phenomena105

7.1Exchange interactions106

7.2One-dimensional Heisenberg model of ferromagnetism108

8. Non-linear phenomena in media118

8.1Solitons120

9. Appendices124

9.1Appendix 1: Trigonometric formulae124

9.2Appendix 2: Tensors124

9.3Appendix 3: Functional derivative126

9.4Appendix 4: Introduction to Lie groups and Lie algebras127

10. Problem Set141

10.1Problems to section 1141

10.2Problems to section 2146

10.3Problems to section 3148

10.4Problems to section 4149

10.5Problems to section 5152

10.6Problems to section 6155

10.7Problems to section 7156

11. Recommended literature157

1. Classical Fields: General Principles

Classical eld theory is a very vast subject which traditionally includes the Maxwell theory of electromagnetism describing electromagnetic properties of matter and the Einstein theory of General Relativity. The main scope of classical eld theory is { 2 { to construct the mathematical description of dynamical systems with an innite number of degrees of freedom. As such, this discipline also naturally incorporates the classics aspects of uid dynamics. The basic mathematical tools involved are partial dierential equations with given initial and boundary conditions, theory of special functions, elements of group and representation theory.

1.1 Lagrangian and Hamiltonian formalisms

We start with recalling the two ways the physical systems are described in classical mechanics. The rst description is known as the Lagrangian formalism which is equivalent to the \principle of least action

1" (Maupertuis's principle). Consider a

point particle which moves in an-dimensional space with coordinates (q1;:::;qn) and in the potentialU(q). The Newtons equations describing the corresponding motion (trajectory) are mqi=@U@q i:(1.1) These equations can be obtained by extremizing the following functional S=Z t2 t

1dtL(q;_q;t) =Z

t2 t

1dtm_q22

U(q) :(1.2) HereSis the functional on the space of particle trajectories: to any trajectory which satises given initialqi(t1) =qiinand nalqi(t2) =qifconditions it puts in correspondence a number. This functional is called theaction. The specic function Ldepending on particle coordinates and momenta is calledLagrangian. According to the principle of stationary action, the actual trajectories of a dynamical system (particle) are the ones which deliver the extremum ofS.

Compute the variation of the action

S=Z t2 t

1dthddt

(m_qi) +@U@q ii q i+ total derivative; where we have integrated by parts. The total derivative term vanishes provided the end points of a trajectory are kept xed under the variation. The quantityS vanishes for anyqiprovided eq.(1.1) is satised. Note that in our particular example, the Lagrangian coincides with the dierence of the kinetic and the potential energy

L=TUand it does not explicitly depend on time.

In general, we simply regardLas an arbitrary function ofq, _qand time. The equations of motion are obtained by extremizing the corresponding action Sq i=ddt @L@_qi @L@q i= 01 More accurately, the principle of stationary action. { 3 { and they are called theEuler-Lagrange equations. We assume thatLdoes not depend on higher derivatives q,...qand so on, which re ects the fact that the corresponding dynamical system is fully determined by specifying coordinates and velocities. In- deed, for a system withndegrees of freedom there arenEuler-Lagrange equations of the second order. Thus, an arbitrary solution will depend on 2nintegration con- stants, which are determined by specifying, e.g. the initial coordinates and velocities.

SupposeLdoes not explicitly depend2ont, then

dLdt =@L@_qiqi+@L@q i_qi:

Substituting here

@L@q ifrom the Euler-Lagrange equations, we get dLdt =@L@_qiqi+ddt @L@_qi _qi=ddt @L@_qi_qi

Therefore, we nd that

ddt @L@_qi_qiL = 0 (1.3) as the consequence of the equations of motion. Thus, the quantity

H=@L@_qi_qiL;(1.4)

is conserved under the time evolution of our dynamical system. For our particular example,

H=m_q2L=m_q22

+U(q) =T+UE : Thus,His nothing else but the energy of our system; energy is conserved due to equations of motion.Dynamical quantities which are conserved during the time evolution of a dynamical system are called conservation laws or integrals of motion. Energy is our rst non-trivial example of a conservation law. Introduce a quantity called the (canonical) momentum p i=@L@_qi; p= (p1;:::;pn): For a point particlepi=m_qi. Suppose thatU= 0. Then _pi=ddt @L@_qi = 0 by the Euler-Lagrange equations. Thus, in the absence of the external potential, the momentumpis an integral of motion. This is our second example of a conservation law.2

This is homogenuity of time.

{ 4 { Now we remind the second description of dynamical systems which exploits the notion of the Hamiltonian. The conserved energy of a system expressed via canonical coordinates and momenta is called theHamiltonian

HH(p;q) =12mp2+U(q):

Let us again verify by direct calculation that it does not depend on time, dHdt =1m pi_pi+ _qi@U@q i=1m m2_qiqi+ _qi@U@q i= 0 due to the Newton equations of motion. Having the Hamiltonian, the Newton equations can be rewritten in the form _qj=@H@p j;_pj=@H@q j: These are the fundamental Hamiltonian equations of motion. Their importance lies in the fact that they are valid for arbitrary dependence ofHH(p;q) on the dynamical variablespandq.

In the general setting the Hamiltonian equations are obtained as follows. We take the full dierential of the

Lagrangian

dL=@L@q idqi+@L@_qid_qi= _pidqi+pid_qi= _pidqi+d(pi_qi)_qidpi;

where we have used the denition of the canonical momentum and the Euler-Lagrange equations. From here we nd

d(pi_qiL|{z}

H) = _qidpi_pidqi:

From the dierential equality the Hamiltonian equations follow. Transformation

H(p;q) =pi_qiL(q;_q)j_qi!pi

is the Legendre transform. The last two equations can be rewritten in terms of the single equation. Introduce two 2n-dimensional vectors x=p q ;rH= @H@p j@H@q j! and 2n2nmatrixJ: J=0? ?0 Then the Hamiltonian equations can be written in the form _x=J rH ;orJ_x=rH : { 5 { In this form the Hamiltonian equations were written for the rst time by Lagrange in 1808. A pointx= (x1;:::;x2n) denes a state of a system in classical mechanics. The set of all these points form aphase spaceP=fxgof the system which in the present case is just the 2n-dimensional Euclidean space with the metric (x;y) =P2n i=1xiyi. To get more familiar with the concept of a phase space, consider a one-dimensional example: the harmonic oscillator. The potential isU(q) =q22 . The Hamiltonian H=p22 +q22 , where we choosem= 1. The Hamiltonian equations of motion are given byordinarydierential equations: _q=p;_p=q=)q=q : Solving these equations with given initial conditions (p0;q0) representing a point in the phase space

3, we obtain a phase space curve

pp(t;p0;q0); qq(t;p0;q0): Through every phase space point there is one and only one phase space curve (unique- ness theorem for ordinary dierential equations). The tangent vector to the phase space curve is calledthe phase velocity vector or the Hamiltonian vector eld. By construction, it is determined by the Hamiltonian equations. The phase curve can consist of only one point. Such a point is called anequilibrium position. The Hamil- tonian vector eld at an equilibrium position vanishes. The law of conservation of energy allows one to nd the phase curves easily. On each phase curve the value of the total energyE=His constant. Therefore, each phase curve lies entirely in one energy level setH(p;q) =h. For harmonic oscillator p

2+q2= 2h

and the phase space curves are concentric circles and the origin. The matrixJserves to dene the so-calledPoisson bracketson the spaceF(P) of dierentiable functions onP: fF;Gg(x) = (JrF;rG) =Jij@iF@jG=nX j=1 @F@p j@G@q j@F@q j@G@p j The Poisson bracket satises the following conditions fF;Gg=fG;Fg; fF;fG;Hgg+fG;fH;Fgg+fH;fF;Ggg= 03

The two-dimensional plane in the present case.

{ 6 { for arbitrary functionsF;G;H. Thus, the Poisson bracket introduces onF(P) the structure of an innite- dimensional Lie algebra. The bracket also satises the Leibnitz rule fF;GHg=fF;GgH+GfF;Hg and, therefore, it is completely determined by its values on the basis elementsxi: fxj;xkg=Jjk which can be written as follows fqi;qjg= 0;fpi;pjg= 0;fpi;qjg=j i: The Hamiltonian equations can be now rephrased in the form _xj=fH;xjg ,_x=fH;xg=XH: It follows from Jacobi identity that the Poisson bracket of two integrals of motion is again an integral of motion. The Leibnitz rule implies that a product of two integrals of motion is also an integral of motion. The algebra of integrals of motion represents an important characteristic of a Hamiltonian system and it is closely related to the existence of a symmetry group. In the case under consideration the matrixJis non-degenerate so that there exists the inverse J 1=J which denes a skew-symmetric bilinear form!on phase space !(x;y) = (x;J1y): In the coordinates we consider it can be written in the form !=X jdp j^dqj:

This form is closed, i.e.d!= 0.

A non-degenerate closed two-form is called symplectic and a manifold endowed with such a form is called a symplectic manifold.Thus, the phase space we consider is the symplectic manifold. Imagine we make a change of variablesyj=fj(xk). Then _yj=@yj@x k|{z} A j k_xk=Aj kJkmrxmH=Aj kJkm@yp@x mrypeH { 7 { or in the matrix form _y=AJAt ryeH : The new equations foryare Hamiltonian with the new Hamiltonian iseH(y) =

H(f1(y)) =H(x) if and only if

AJA t=J : Hence, this construction motivates the following denition. Transformations of the phase space which satisfy the condition AJA t=J are called canonical 4.

Canonical transformations

5do not change the symplectic form!:

!(Ax;Ay) =(Ax;JAy) =(x;AtJAy) =(x;Jy) =!(x;y): In the case we considered the phase space was Euclidean:P=R2n. This is not always so. The generic situation is that the phase space is a manifold. Considera- tion of systems with general phase spaces is very important for understanding the structure of the Hamiltonian dynamics.

Short summary

A Hamiltonian system is characterized by a triple (P;f;g;H): a phase spaceP, a Poisson structuref;gand by a Hamiltonian functionH. The vector eldXHis called theHamiltonian vector eldcorresponding to the HamiltonianH. For any functionF=F(p;q) on phase space, the evolution equations take the form dFdt =fH;Fg=XHF : Again we conclude from here that the HamiltonianHis a time-conserved quantity dHdt =fH;Hg= 0: Thus, the motion of the system takes place on the subvariety of phase space dened byH=Econstant.4 In the case whenAdoes not depend onx, the set of all such matrices form a Lie group known as the real symplectic group Sp(2n;R) . The term \symplectic group" was introduced by Herman Weyl. The geometry of the phase space which is invariant under the action of the symplectic group is calledsymplectic geometry.

5Notice thatAJAt=Jimplies thatAtJA=J. Indeed, multiplying byJboth sides of the rst

equality from the right, we getAJAtJ=J2=?, which further impliesAtJ=J1A1=JA1. Finally, multiplying both sides of the last expression from the right byA, we obtain the desired formula. { 8 {

1.2 Noether's theorem in classical mechanics

Noether's theorem is one of the most fundamental and general statements concern- ing the behavior of dynamical systems. It relates symmetries of a theory with its conservation laws. It is clear that equations of motion are unchanged if we add to a Lagrangian a total time derivative of a function which depends on the coordinates and time only:

L!L+ddt

G(q;t). Indeed, the change of the action under the variation will be

S!S0=S+Z

t2 t

1dtddt

G(q;t) =S+@G@q

iqijt=t2t=t1: Since in deriving the equations of motion the variation is assumed to vanish at the initial and nal moments of time, we see thatS0=Sand the equations of motion are unchanged. Let now an innitezimal transformationq!q+qbe such that the variation of the Lagrangian takes the form (without usage of equations of motion!)6of a total time derivative of some functionF:

L=dFdt

Transformationqis called a symmetry of the action. Now we are ready to discuss Noether's theorem. Suppose thatq0=q+qis a symmetry of the action. Then

L=@L@q

iqi+@L@_qi_qi=@L@q iqi+@L@_qiddt qi=dFdt

By the Euler-Lagrange equations, we get

L=ddt @L@_qi q i+@L@_qiddt qi=dFdt

This gives

L=ddt @L@_qiqi =dFdt As the result, we nd the quantity which is conserved in time dJdt ddt @L@_qiqiF = 0:

This quantity

J=@L@_qiqiF=piqiF

is called Noether's current. Now we consider some important applications.6 As we have already seen, a variation of the Lagrangiancomputed on the equations of motionis always a total derivative! { 9 { Momentum conservation. Momentum conservation is related to the freedom of arbitrary choosing the origin of the coordinate system. Consider a Lagrangian L=m2 _q2i:

Consider a displacement

0i=qi+ai)qi=ai;

_q0i= _qi)_qi= 0: Obviously, under this transformation the Lagrangian remains invariant and we can takeF= 0 orF= any constant. Thus,

J=piqi=piai;

Sinceaiarbitrary, all the componentspiare conserved.

Angular momentum conservation. Consider again

L=m2 _q2i and make a transformation q

0i=qi+ijqj)qi=ijqj:

Then,

L=m_qiij_qj:

Thus, ifijis anti-symmetric, the variation of the Lagrangian vanishes. Again, we can takeF= 0 orF= any constant and obtain

J=piqi=piijqj;

Sinceijis arbitrary, we nd the conservation of angular momentum compo- nents J j i=piqjpjqi: Particle in a constant gravitational eld. The Lagrangian L=m2 _z2mgz :

Shiftz!z+a,i.e.z=a. We getL=mga=ddt

(mgat). Thus, the quantity

J=m_zzF=m_za+mgat

is conserved. This is a conservation law of the initial velocity _z+gt= const. { 10 { Conservation of energy.Energy conservation is related to the freedom of arbitrary choosing the origin of time (you can perform you experiment today or after a several years but the result will be the same provided you use the same initial conditions).quotesdbs_dbs17.pdfusesText_23

[PDF] [PDF] Classical Field Theory - scienceuunl project csg

Preprint typeset in JHEP style - HYPER VERSION

Classical Field Theory

Gleb Arutyunov

Last Update 8.05.2011

Email: G.Arutyunov@phys.uu.nl

Contents

1. Classical Fields: General Principles2

1.1Lagrangian and Hamiltonian formalisms3

1.2Noether's theorem in classical mechanics9

1.3Lagrangians for continuous systems11

1.4Noether's theorem in eld theory15

1.5Hamiltonian formalism in eld theory20

2. Electrostatics21

2.1Laws of electrostatics21

2.2Laplace and Poisson equations26

2.3The Green theorems27

2.4Method of Green's functions29

2.5Electrostatic problems with spherical symmetry31

2.6Multipole expansion for scalar potential38

3. Magnetostatics41

3.1Laws of magnetostatics41

3.2Magnetic (dipole) moment42

3.3Gyromagnetic ratio. Magnetic moment of electron.44

4. Relativistic Mechanics46

4.1Newton's relativity principle46

4.2Einstein's relativity principle46

4.3Dening Lorentz transformations48

4.4Lorentz group and its connected components50

4.5Structure of Lorentz transformations53

4.6Addition of velocities57

4.7Lie algebra of the Lorentz group57

4.8Relativistic particle60

5. Classical Electrodynamics62

5.1Relativistic particle in electromagnetic eld62

5.2Lorentz transformations of the electromagnetic eld65

5.3Momentum and energy of a particle in a static gauge67

5.4Maxwell's equations and gauge invariance67

5.5Fields produced by moving charges69

5.6Electromagnetic waves72

5.7Hamiltonian formulation of electrodynamics76

5.8Solving Maxwell's equations with sources79

5.9Causality principle84

6. Radiation85

6.1Fields of a uniformly moving charge86

6.2Fields of an arbitrary moving charge87

6.3Dipole Radiation91

6.4Applicability of Classical Electrodynamics100

6.5Darvin's Lagrangian101

7. Advanced magnetic phenomena105

7.1Exchange interactions106

7.2One-dimensional Heisenberg model of ferromagnetism108

8. Non-linear phenomena in media118

8.1Solitons120

9. Appendices124

9.1Appendix 1: Trigonometric formulae124

9.2Appendix 2: Tensors124

9.3Appendix 3: Functional derivative126

9.4Appendix 4: Introduction to Lie groups and Lie algebras127

10. Problem Set141

10.1Problems to section 1141

10.2Problems to section 2146

10.3Problems to section 3148

10.4Problems to section 4149

10.5Problems to section 5152

10.6Problems to section 6155

10.7Problems to section 7156

11. Recommended literature157

1. Classical Fields: General Principles

1.1 Lagrangian and Hamiltonian formalisms

1" (Maupertuis's principle). Consider a

1dtL(q;_q;t) =Z

1dtm_q22

Compute the variation of the action

1dthddt

L=TUand it does not explicitly depend on time.

SupposeLdoes not explicitly depend2ont, then

Substituting here

Therefore, we nd that

H=@L@_qi_qiL;(1.4)