Mathematical Methods of Classical Physics

1 Vicente Cort es, Alexander S. HauptMathematical Methods ofClassical Physics25th May 2017 [math-ph] 24 May 2017 AcknowledgementsWe are very grateful to David Lindemann for careful proof-reading of earlier versionsof the manuscript of this book and for numerous constructive comments which helpedimprove the presentation of this book. We thank Thomas Leistner for drawing ourattention to the reference [6] and Thomas Mohaupt for useful remarks concerningChapter ..12 Lagrangian Mechanics.. Lagrangian mechanical systems and their equations of motion .. Integrals of motion .. Motion in a radial potential .. in Newton s gravitational potential.

2 153 Hamiltonian Mechanics.. Symplectic geometry and Hamiltonian systems .. Relation between Lagrangian and Hamiltonian systems .. formulation for the Lagrangian systems ofExample .. Legendre transform .. Linearization and stability .. 284 Hamilton-Jacobi theory.. 375 Classical field theory.. The Lagrangian, the action and the Euler-Lagrange equations .. Automorphisms and conservation laws .. Why are conservation laws called conservation laws? .. Examples of field theories .. models .. Yang-Mills theory .. Einstein-Hilbert Lagrangian .. The energy-momentum tensor .. 72 AExercises.

3 79 References.. 91viiviiiContentsIndex.. 93 List of SymbolsAconnection one-form, 66AY MG(E)subset of Yang-MillsG-connections, 66 AutG(E)gauge group of the vector bundleEwithG-reductionFG(E),65 Bmagnetic field, 62 Diff+(M)subgroup of orientation preserving diffeomorphisms ofM,50 DivPtotal divergence, 48d covariant exterior derivative, 62dvolvolume element onS, 46E(total) energy, 5 Eacomponent of Euler-Lagrange operator, 48 Ekinkinetic energy, 5 Epotpotential energy, 5 Eelectric field, 62F(E)frame bundle ofE, 61F curvature of a connection , 61 fLegendre transform of a smooth functionf, 27gmetric onM(Riemannian or pseudo-Riemannian), 6 HHamiltonian, 21 Isom+(N,h)

4 Subgroup of orientation preserving isometries of the metrichonN, 60J(Noether) current, 57J0charge density, 57 Jetk(S,T)jet bundle overS, 46jk(f)smooth section of the jet bundle Jetk(S,T), 46 Jflux density, 57 Kfield (in this book,K=RorC), 61 Llength of the angular momentum vectorL, 14 LLagrangian (function), 5L2linearized Lagrangian, 30ixxList of SymbolsLangular momentum vector, 13 Msmooth manifold (configuration space), 5mmass of a point particle, 5ndimension ofM, 8pr(k)Zk-th prolongation ofZ, 51pmomentum vector, 12Q(Noether) charge, 57(q1, .. ,qn, q1, .. , qn)induced/canonical local coordinates onT Massociated withlocal coordinates(x1, .. ,xn)onM, 7(q1, .. ,qn,p1.)

5 ,pn)Darboux coordinates, 20 RicRicci curvature tensor, 69 Saction, 6scalscalar curvature of(M,g), 69S[f]action functional of a Classical field theory, 45 Ssource manifold (possibly with boundary), 45ttime, 5 Ttarget manifold, 45T Mtangent bundle ofM, 5T component of the energy-momentum tensor, 73 Vpotential, 5 Veffeffective potential, 14(x1, .. ,xn)local coordinates on an open subsetU M, 7 XfHamiltonian vector field associated with a smooth functionf, 20X(M)set of all smooth vector fields onM, 11 Xververtical lift ofX, 11 Euler-Lagrange one-form, 8 icomponent of the Euler-Lagrange one-form in some localcoordinate system, 8 i jKronecker delta. Its value is defined to be 1 if the indices areequal, and 0 otherwise, 33 smooth curve inM, 6 gravitational coupling constant, 69 Liouville form, 20 cosmological constant, 71 Hessian matrix of the potentialV, 32 canonical symplectic form, 20 0frequency of a small oscillation, 32 canonical projection fromT MtoM, 7 (f)tension off, 59 0period of a small oscillation, 32 eigenvector of , 34 , Euclidean scalar product onRn, 5 Hodge star operator, 62 Laplace operator, 59 List of Symbolsxi ( )time derivate, f(t) =f (t), f(t) =f (t), 2 covariant derivative or connection, 9 We use an adapted version ofEinstein s summation conven-tionthroughout the book.

6 Upper and lower indices appearingwith the same symbol within a term are to be summed write the symbol to indicate whenever this conventionis employed. Owing to the aforementioned convention, thesummation indices can be (and are usually) omitted belowthe symbol , 7( ) matrix transposition, 33 Chapter 1 IntroductionAbstractWe define the framework of Classical Physics as considered in the presentbook. We briefly point out its place in the history of Physics and its relation tomodern Physics . As the prime example of a theory of Classical Physics we introduceNewtonian mechanics and discuss its limitations. This leads to and motivates thestudy of different formulations of Classical mechanics, such as Lagrangian andHamiltonian mechanics, which are the subjects of later chapters.

7 Finally, we explainwhy in this book, we take a Mathematical perspective on central topics of Physics refers to the collection of physical theories that do not use quantumtheory and often predate modern quantum Physics . They can be traced back to Newton(17th century) and in some sense even further all the way to Aristotle, Archimedes,and other Greek philosophers of antiquity (starting in the 4th century BC). However,this does by far not mean that theories of Classical Physics are exclusively a subjectof the past. They continue to play important roles in modern Physics , for example inthe study of macroscopic systems, such as fluids and planetary motions, where theeffects of the quantum behavior of the microscopic constituents are mechanics is arguably the first mathematically rigorous and self-contained theory of Classical Physics .

8 In its traditional formulation, Newton s theorycomprises three physical laws known as Newton s laws of motion, describing therelationship between a body (usually assumed to be a point particle of constant massm>0) and forces acting upon it. They also quantify the resulting motion of the bodyin response to those forces and can be summarised, in an inertial reference frame, law:A body at rest will stay at rest and a body in uniform motion will stay in motionat constant velocity, unless acted upon by a net IntroductionSecond law:The forceFacting on a body is equal to the mass of the body times the accelerationaof the body:F=m a.( )Third law:For every action force there is a corresponding reaction force which is equal inmagnitude and opposite in direction.

9 This is often abbreviated asactio= a point particle moving inR3, its acceleration at timetis given by thesecond time derivate of its position vector, denoted bya(t) = x. Throughout theentire book, we work in theC -category, unless stated otherwise, that is, manifoldsand maps between them are usually assumed to be smooth. The forceFis generallyallowed to depend on positionx, velocity x, and timet, that isF=F(x, x,t). Eq. ( )then turns into x=1mF(x, x,t).( )FormandF(x, x,t)given, this is a set of second-order ordinary differential equationsknown asNewton s equations of motion. Note thatF=0if and only if the motiont x(t)is linear and therefore Newton s first law is a special case of the secondlaw.

10 Also, the third law is a consequence of the second law in combination withconservation of momentum, which ultimately follows from translational invariance(see Corollary ). Despite this redundancy in Newton s laws, all three highlightdifferent aspects of important concepts of modern Physics , namely the notion ofinertial frame in Einstein s theory of relativity and the relation between differentconcepts of mass (inertial versus gravitational) in gravitational theories (see, forexample, [5]).We remark that eq. ( ) is also sometimes written asF= p, wherep=m xis themomentumof the particle. This allows for the consideration of bodies withnon-constant mass, such as rockets consuming their fuel, where however the secondlaw needs to be applied to the total system including the lost (or gained) order to solve eq.