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Classical Field Theory Preprint typeset in JHEP style - HYPER VERSION Classical Field Theory Gleb Arutyunova∗y 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 field 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 infinite-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: [email protected] yCorrespondent fellow at Steklov Mathematical Institute, Moscow. Contents 1. Classical Fields: General Principles 2 1.1 Lagrangian and Hamiltonian formalisms 3 1.2 Noether's theorem in classical mechanics 9 1.3 Lagrangians for continuous systems 11 1.4 Noether's theorem in field theory 15 1.5 Hamiltonian formalism in field theory 20 2. Electrostatics 21 2.1 Laws of electrostatics 21 2.2 Laplace and Poisson equations 26 2.3 The Green theorems 27 2.4 Method of Green's functions 29 2.5 Electrostatic problems with spherical symmetry 31 2.6 Multipole expansion for scalar potential 38 3. Magnetostatics 41 3.1 Laws of magnetostatics 41 3.2 Magnetic (dipole) moment 42 3.3 Gyromagnetic ratio. Magnetic moment of electron. 44 4. Relativistic Mechanics 46 4.1 Newton's relativity principle 46 4.2 Einstein's relativity principle 46 4.3 Defining Lorentz transformations 48 4.4 Lorentz group and its connected components 50 4.5 Structure of Lorentz transformations 53 4.6 Addition of velocities 57 4.7 Lie algebra of the Lorentz group 57 4.8 Relativistic particle 60 5. Classical Electrodynamics 62 5.1 Relativistic particle in electromagnetic field 62 5.2 Lorentz transformations of the electromagnetic field 65 5.3 Momentum and energy of a particle in a static gauge 67 5.4 Maxwell's equations and gauge invariance 67 5.5 Fields produced by moving charges 69 5.6 Electromagnetic waves 72 { 1 { 5.7 Hamiltonian formulation of electrodynamics 76 5.8 Solving Maxwell's equations with sources 79 5.9 Causality principle 84 6. Radiation 85 6.1 Fields of a uniformly moving charge 86 6.2 Fields of an arbitrary moving charge 87 6.3 Dipole Radiation 91 6.4 Applicability of Classical Electrodynamics 100 6.5 Darvin's Lagrangian 101 7. Advanced magnetic phenomena 105 7.1 Exchange interactions 106 7.2 One-dimensional Heisenberg model of ferromagnetism 108 8. Non-linear phenomena in media 118 8.1 Solitons 120 9. Appendices 124 9.1 Appendix 1: Trigonometric formulae 124 9.2 Appendix 2: Tensors 124 9.3 Appendix 3: Functional derivative 126 9.4 Appendix 4: Introduction to Lie groups and Lie algebras 127 10. Problem Set 141 10.1 Problems to section 1 141 10.2 Problems to section 2 146 10.3 Problems to section 3 148 10.4 Problems to section 4 149 10.5 Problems to section 5 152 10.6 Problems to section 6 155 10.7 Problems to section 7 156 11. Recommended literature 157 1. Classical Fields: General Principles Classical field 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 field theory is { 2 { to construct the mathematical description of dynamical systems with an infinite number of degrees of freedom. As such, this discipline also naturally incorporates the classics aspects of fluid dynamics. The basic mathematical tools involved are partial differential 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 first description is known as the Lagrangian formalism which is equivalent to the \principle of least action1" (Maupertuis's principle). Consider a point particle which moves in a n-dimensional space with coordinates (q1; : : : ; qn) and in the potential U(q). The Newtons equations describing the corresponding motion (trajectory) are @U mq¨i = : (1.1) −@qi These equations can be obtained by extremizing the following functional Z t2 Z t2 mq_2 S = dt L(q; q;_ t) = dt U(q) : (1.2) t1 t1 2 − Here S is the functional on the space of particle trajectories: to any trajectory i i i i which satisfies given initial q (t1) = qin and final q (t2) = qf conditions it puts in correspondence a number. This functional is called the action. The specific function L depending on particle coordinates and momenta is called Lagrangian. According to the principle of stationary action, the actual trajectories of a dynamical system (particle) are the ones which deliver the extremum of S. Compute the variation of the action Z t2 h d i @U i i δS = dt (mq_ ) + i δq + total derivative ; − t1 dt @q where we have integrated by parts. The total derivative term vanishes provided the end points of a trajectory are kept fixed under the variation. The quantity δS vanishes for any δqi provided eq.(1.1) is satisfied. Note that in our particular example, the Lagrangian coincides with the difference of the kinetic and the potential energy L = T U and it does not explicitly depend on time. − In general, we simply regard L as an arbitrary function of q,_q and time. The equations of motion are obtained by extremizing the corresponding action δS d @L @L = = 0 δqi dt @q_i − @qi 1More accurately, the principle of stationary action. { 3 { and they are called the Euler-Lagrange equations. We assume that L does not depend ... on higher derivativesq ¨, q and so on, which reflects the fact that the corresponding dynamical system is fully determined by specifying coordinates and velocities. In- deed, for a system with n degrees of freedom there are n Euler-Lagrange equations of the second order. Thus, an arbitrary solution will depend on 2n integration con- stants, which are determined by specifying, e.g. the initial coordinates and velocities. Suppose L does not explicitly depend2 on t, then dL @L @L = q¨i + q_i : dt @q_i @qi @L Substituting here @qi from the Euler-Lagrange equations, we get dL @L d @L d @L = q¨i + q_i = q_i : dt @q_i dt @q_i dt @q_i Therefore, we find that d @L q_i L = 0 (1.3) dt @q_i − as the consequence of the equations of motion. Thus, the quantity @L H = q_i L; (1.4) @q_i − is conserved under the time evolution of our dynamical system. For our particular example, mq_2 H = mq_2 L = + U(q) = T + U E: − 2 ≡ Thus, H is 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 first non-trivial example of a conservation law. Introduce a quantity called the (canonical) momentum @L p = ; p = (p ; : : : ; p ) : i @q_i 1 n i For a point particle pi = mq_ . Suppose that U = 0. Then d @L p_ = = 0 i dt @q_i by the Euler-Lagrange equations. Thus, in the absence of the external potential, the momentum p is an integral of motion. This is our second example of a conservation law. 2This 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 the Hamiltonian 1 H H(p; q) = p2 + U(q) : ≡ 2m Let us again verify by direct calculation that it does not depend on time, dH 1 @U 1 @U = p p_ +q _i = m2q_ q¨ +q _i = 0 dt m i i @qi m i i @qi due to the Newton equations of motion. Having the Hamiltonian, the Newton equations can be rewritten in the form j @H @H q_ = ; p_j = j : @pj −@q These are the fundamental Hamiltonian equations of motion. Their importance lies in the fact that they are valid for arbitrary dependence of H H(p; q) on the ≡ dynamical variables p and q. In the general setting the Hamiltonian equations are obtained as follows. We take the full differential of the Lagrangian @L i @L i i i i i i dL = dq + dq_ =p _idq + pidq_ =p _idq + d(piq_ ) − q_ dpi ; @qi @q_i where we have used the definition of the canonical momentum and the Euler-Lagrange equations. From here we find i i i d(piq_ − L) =q _ dpi − p_idq : | {z } H From the differential equality the Hamiltonian equations follow. Transformation i H(p; q) = p q_ − L(q; q_)j i i q_ !pi is the Legendre transform. The last two equations can be rewritten in terms of the single equation. Introduce two 2n-dimensional vectors @H ! p @pj x = ; H = @H q r @qj and 2n 2n matrix J: × 0 1 J = − : 1 0 Then the Hamiltonian equations can be written in the form x_ = J H; or J x_ = H: · r · −∇ { 5 { In this form the Hamiltonian equations were written for the first time by Lagrange in 1808.
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