1. 3. Classical Physics in Galilean and Minkowski Spacetimes 3.1. The Action Principle in Galilean Spacetime 3.2. Symmetries and Conservation Laws 3.3. The Hamiltonian 3.4. Poisson Brackets and Translation Operators 3.5. The Action Principle in Minkowski Spacetime 3.6. Classical Electrodynamics 3.7. Geometry in Classical Physics

2. Purpose: Set out classical mechanics in a form suitable for generalization to general relativity & quantum mechanics. Two basic ideas: 1. Action (contains all dynamical information of the system ). 2. Symmetry ~ conservation law. Comment: Not all systems can be described by an action, e.g. systems with dissipation. However, all known fundamental (microscopic) interaction can.

3. 3.1. The Action Principle in Galilean Spacetime Euler-Lagrange Equations: Assumptions: 1. State of a microscopic system is uniquely specified by (Equations of motion are 2 nd order in time) 2. Least action principle. The action S of a system is defined as= Lagrangian. Least action principle: Actual evolution of the system minimizes S, i.e., δS = 0.

4. Euler-Lagrange equations Generalized momentum p i conjugate to q i is defined as For L = T  V, the Euler-Lagrange equations become Newton’s equations: or

5. Spacetime Symmetries Invariance under Time Translations: Galilean time has its own metric dτ 2 = g(t) dt 2 wheret = time coordinate, g(t) = metric tensor, and  = linear measure of the time manifold, i.e., d  = "length" of time between 2 points at t and t + dt. t = aτ+b → g(τ) = 1/a 2 Any t with g(t) = const is called a proper time coordinate. Newton's 1 st law can be used to check if t is a proper time. By definition, the Lagrangian of a free particle is the same for all proper time t, i.e., L 0 is invariant under proper time translations. ↔t = aτ+b

6. Invariance under Spatial Translations: Space in Galilean spacetime is Euclidean. → in Cartesian coordinates, g = I. g is invariant under any spatial translation → Euclidean space is homogenous. In an inertial frame, L 0 is invariant under spatial translations: ↔x is an inertial frame Invariance under Rotations: The Euclidean metric tensor is invariant under any spatial rotations. → Euclidean space is isotropic. In an inertial frame, L 0 must be invariant under any rotations: where R is a rotation operator and

7. Invariance under Galilean Transformations: Coordinates of S and S', moving with relative velocity v, are related by x = x  v tt = t If S is an inertial system in which Newton's 1 st law holds, so is S'. →Newtonian physics is invariant under Galilean transformations. It obeys Galilean relativity. For a free particle with Lagrangianwhere the Euler-Lagrange equations are where

8. Under the Galilean transformation The Euler-Lagrange equation can remain unchanged iff → Lagrangian of free particle is determined by spacetime symmetries.

9. Lagrangian L 0 = T is determined solely by the symmetries of the Galilean spacetime. L of a system of non-interacting particles must be a sum of T i of all particles. If the particles interact via a potential 1. Invariance under spatial translation requires, then 2. Invariance under Galilean transformation requires 3. Invariance under rotation requires V to depend only on rotational scalars such as etc →where is not of this form since V is produced by external sources.

10. 3.2. Symmetries and Conservation Laws Conservation of Energy: Invariance under time translation →or → → where  energy of the system. Invariance of time translation →conservation of energy ( for any isolated system )

11. Noether's Theorem Consider transformation or with L is invariant under the transformation if i.e., (Noether's theorem)→

12. Examples Let x j → x j + εa. If L is invariant, the Noether's theorem gives Conservation of Total Linear Momentum for a system of particles : i.e., the total momentum P = Σ j p j is conserved if L is invariant spatial translations. Conservation of Total Angular Momentum: See Exercise 3.1.

13. Miscellaneous Interpretations A symmetry transformation can be interpreted in 2 ways. Active point of view Portion of the system being studied is switched from x to x + a. Passive point of view Origin of the coordinate system is switched by –a. Finite Transformations Finite transformations = (path-dependent) integral of infinitesimal transformation. Conservation under infinitesimal transformations → conservation over finite version of same transformations.

14. 3.3. The Hamiltonian Lagrangian formulism: state variables = Hamiltonian formulism : state variables = (q, p ) ( Legendre transformation ) Note: H is a function defined for every value of (q, p ). But E is defined only on the actual trajectories. → Hamilton's equations

15. 3.4. Poisson Brackets and Translation Operators Lagrangian and Hamiltonian formulisms allow theories beyond Newtonian mechanics. E.g.,statistical mechanics,quantum mechanics. Poisson bracket (on trajectory) → or Liouville operator

16. (q,p) is restricted to the actual trajectories (q(t),p(t)) by means of the density function: → → Liouville equation →

17. Translation Operators →  exp(i t H ) is a temporal translation operator, and H is a generator of time translation.  exp(i a · P ) is a spatial translation operator, and P is a generator of spatial translation.

18. 3.5. The Action Principle in Minkowski Spacetime A Minkowski spacetime is defined as a spacetime manifold in which a class of inertial Cartesian coordinate systems exists such that Distance between 2 points in the manifold is called a proper time interval, . For infinitesimal separations, Ifthen time is defined as while spatial position is given by the 3-D vector x = x i, i = 1,2,3. A coordinate system is called inertial if it can be obtained from an inertial Cartesian coordinate system by a transformation that leaves the form of g unchanged.

19. Isometries Isometry = Symmetry transformation that leaves the form of g unchanged. → or Matrix form: where Λ is an isometry ↔ g and g' have the same functional form. Isometries convert one inertial system into another.

20. Examples of Isometries Translations:  Lorentz Transformation: Proper Improper with Rotation about the x 1 axis by an angle  : Lorentz group

21. Poincare Transformation: Poincare groupwith Boost along x 1 by v : Proper Improper

22. Tensors Let f(x) be a scalar function of the Cartesian coordinates x μ. Under a Poincare transformation → An object V  that transforms like dx  is called a contravariant 4-vector. An object V  that transforms like   f is called a covariant 4-vector. Objects with m upper and n lower indices and transform like are called mixed 4-tensors of rank m+n. Caution: 4-tensors are not necessarily true tensors since only  with constant elements are considered. A contraction of all tensor indices is a Lorentz scalar invariant under all Lorentz transformations. E.g., η μν U μ V ν

23. Lagrangian of a Free Particle Motion of particle = path x μ (τ) in Minkowski spacetime (  = proper time ).  is a scalar → is a 4-vector called the 4-velocity. Principle of relativity → EOMs have same form in every inertial frame. i.e., they are invariant under all isometric (Poincare) transformations. → Action is a Lorentz scalar and translationally invariant. For a free particle: →

24. On the actual path of the particle, X = ½ c 2 → or A simple choice is for v << c.

25. Energy-Momentum 4-Vectors The canonical momentum conjugate to x μ are called an energy- momentum 4-vector or 4-momentum. In Cartesian coordinates, so that for a free particle, The 1-form p μ is conserved due to translational invariance. Vector version: where

26. The velocity of a particle with coordinatesis →

27. Lagrangian of Many Free Particles For a system of N non-interacting particles Equation of continuity: (number of total particles is conserved ) → Prove it! Equation of continuity:

28. The associated flow of quantity A carried by the particles is defined as Setting A to be the electric charge, we obtain the electromagnetic current. Setting A to be the 4-momentum, we get the (energy-momentum) stress tensor T is symmetric ( T μν = T νμ ) and conserved (  ν T μν = 0 ). A perfect fluid is a fluid that has a rest frame in which its density is spatially uniform and the average velocity of its particles is zero. Its stress tensor is (see exercise 3.4), where  is the energy density and p the pressure.

29. 3.6. Classical Electrodynamics Maxwell's Equations in Heaviside-Lorentz units: Gauss' law: No monopoles: Faraday's law: Ampere's law: Note: some GUTs (see chapter 12) suggest the existence of magnetic monopoles. The homogeneous eqs are satisfied automatically by setting Introducing the 4-vector we define the anti-symmetric field strength 4-tensor as with

30. In Cartesian coordinates, we have → The inhomogeneous Maxwell eqs can be combined as where is the 4-current density. Caution: index conventions not strictly observed here.

31. Field Lagrangian The Maxwell's equations can be derived from the least action principle. A μ (x) is a 4-vector dynamic variable with label x. Since d 4 x is a Lorentz scalar, so are S and L. becomes 

32. Interaction with a Charged Particle For a single particle of charge q where

33. Equations of Motion The dynamic variables are x μ (τ) and A μ (x). The x in A μ (x) has 2 interpretations. 1. x is just a label when A μ (x) is treated as a dynamic variable, as in L and L int. 2. x is a dynamic variable when A μ (x) is a potential acting on a particle, as in L int. Analogous interpretation also applies to j e μ (x). In the variation of each dynamic variable, all others should be taken as fixed. Thus, for the particle degrees of freedom x μ (τ), we set δA μ = 0 so that δ L = 0

34. Using we have → is a 3-D Euclidean vector.

35. Rate of work and Lorentz force

36. Gauge Transformation Under a gauge transformation is invariant: Gauge invariance of F can be traced to its antisymmetry. "~" relates L s that give the same EOMs, e.g., L s that differ by a divergence.  L be gauge invariant →  conserved current Symmetry of gauge invariance → Conservation of electric charges.

37. 3.7. Geometry in Classical Physics 3.7.1.More on Tensors 3.7.2.Differential Forms, Dual Tensors and Maxwell's Equations 3.7.3.Configuration Space and Its Relatives 3.7.4.The Symplectic Geometry of Phase Space

38. 3.7.1.More on Tensors A rank ( m n ) tensor T is a multi-linear function that maps m 1-forms and n vectors to a number in K. A vector is a linear function that maps a 1-form to a number. A 1-form is a linear function that maps a vector to a number. In a coordinate system { x a }, whereare vectors and1-forms. whereare basis vectors andbasis 1-forms.

39. Vectors and 1-Forms Let = gradient to scalar field f.= tangent to curve C(λ). → Given a coordinate system {x a }, the natural bases for vectors and 1-forms are andrespectively. These so-called coordinate bases satisfy

40. Under a coordinate transformation, →

41. Direct Product The direct product of a ( m n ) tensor S with a ( m n ) tensor T is a ( m + m n + n ) tensor S  T such that →

42. 3.7.2. Differential Forms, Dual Tensors and Maxwell's Equations A p-form is a totally antisymmetric tensor of rank ( 0 p ): → independent p-forms in an n-D manifold.  only one n-form, which must be proportional to the Levi- Civita tensor density if abc… 

43. Wedge Product Between 1-Forms A totally antisymmetrized version of the direct product is called the wedge product . The wedge product between two 1-forms is defined as → For a 2-form The set of C n 2 independent 2-forms is a basis for the 2-forms in an n-D manifold.

44. Wedge Product Between Arbitrary Forms Wedge (exterior) products are associative: p­-form The wedge product of p-form ω and q-form σ is therefore where P is a permutation of the p+q labels of the vectors.

45. Cross Product 3-D vectors with Cartesian coordinates: The rank 3 Levi-Civita symbol ε abc is defined only for Cartesian coordinates. Raising index using the Euclidean metric tensor g ab = δ ab gives → This works only because 2-forms in 3-D transform like 1-forms.

46. Dual Tensors Consider the set of basis p-forms in an d-D manifold There are onlyindependent p-forms in B. Similarly, p-vectors can be constructed with the help of the wedge product. → dim  p V of p-vectors = dim  d-p *V of (d–p)-forms   bijection between  p V and  d-p *V ( they are dual to each other ). is dual to is the volume d-form with components Thus

47. Exterior Derivatives The exterior derivative is a derivation that increases the degree of a form by 1. Furthermore, we require, for forms β and γ of arbitrary degrees, that 1. Linearity: 2. (Generalized) Leibniz rule: 3. Nilpotence: A derivation that satisfies property (2) is called an antiderivation. Reminder: the Leibniz rule for a derivation d is Combining (1) and (2) gives for any constants a and b.

48. p-forms Taking a scalar function f as a 0-form, we have a 1-form Consistency check: Note that is a 1-form while the ordinary derivative dx a is a 1-vector. Also, most authors (Lawrie is one) use the symbol d to denote For a 1-form

49. For a p-form If ω is itself the exterior derivative of another (p  1)-form Thus, the nilpotent assumption is consistent.

50. E3E3 Dual of the volume n-form:

52. is the volume n-form.   Similarly

53. Poincare Lemma A p-form ω is said to be exact if there exists a (p  1)-form σ such that A p-form ω is said to be closed if  an exact form is also closed, but not vice versa. Poincare lemma: Any closed form in a n-D manifold is exact in any region that is homeomorphic to the open unit ball S n  1 [see §4.19 of Schutz for proof]. R n is homeomorphic to S n  1  a closed form is exact in any region describable by a single coordinate patch.  Every closed form is exact locally but not necessarily so globally. The study of this is called the cohomology theory.

54. Maxwell Equations Faraday 2-form Reminder: E and B are not the spatial parts of some vectors in Minkowski space.  Either E j or E j can be used to denote the jth component of E.

55.  Homogeneous Maxwell equations:

56. Non-homogenous Maxwell equations:

57. 3.7.3. Configuration Space and Its Relatives Newtonian dynamics of N particles with Galilean relativity: Positions of all particles at time t = point in the configuration space Q. Q = manifold with 3N generalized coordinates { q i } as coordinates. State of a system = point in a 6N-D manifold T Q. T Q = tangent bundle with Q as base & tangent space T at a point in Q as fibre. [See §2.11, Schutz, for a definition of a fibre bundle]. A point in T Q has coordinates ( { q i }, { v i } ), where { v i } are coordinates in the fibre at point { q i } in Q. The actual velocities of the particles are denoted as Evolution of system is represented a curve in T Q. A vector field over Q is called a cross section of the bundle.

58. Lagrangian L({q i }, {v i } ) is a scalar field on T Q. Let g i j (q) = metric tensor field on Q. Generalized momentum conjugate to q i is p i is a 1-form obtained by lowering the indices of v i. The set of all p i at a point P in Q forms a vector space called the cotangent space T P *. The fibre bundle formed using T* as typical fibre is called a cotangent bundle T* Q.

59. 3.7.4. The Symplectic Geometry of Phase Space Symplectic Manifold: The fibre bundle T*Q is called a phase space in Hamiltonian dynamics. The state of the system is given by a point in T*Q with coordinates Symplectic form: If Ω is used as volume form for dual operations, the bundle T* Q is called a symplectic manifold. A vector field V is written as The underbars serve as a reminder that and are components of a vector. By definition:

60. Canonical 1-form Ω is exact:whereis called the canonical 1-form. History of system is represented by a curve with tangent vector d/dt. Components of the velocity vectors of the particles =  = H + L ~ Legendre transform between H & L. ~ Transform between T Q and T* Q.

61. Symplectic 2-form Ω is a 2-form → * associates vectors and 1-forms For a vector V: If Ω is non-degenerate, i.e. the 2-vector exists with we can associate a 1-form uniquely with a vector: exists iff For the phase space described by coordinates where I is the 3N  3N unit matrix and A degenerate symplectic 2-form implies mismatch between the numbers of independent coordinates and momenta. Example: Electromagnetism where the gauge degrees of freedom are “unphysical”. Hamiltonian description of such systems require special techniques.

62. Hamiltonian Dynamics associated with a scalar fieldThe Hamiltonian vector field is defined by In components : → Hamilton’s equations:

63. Symmetries The study of (continuous) symmetries is best conducted using mathematical techniques devised by S. Lee, namely, Lie derivatives, Lie group, and Lie algebra. An introduction of these can be found in Schutz. [see Ex 5.6, Schutz] First, the Hamiltonian vector fields form a Lie algebra: Hence,  By definition, the Lie derivative of a vector field along another vector field is If, W is said to be Lie-dragged by V and is a coordinate basis for vectors.

64. Consider V H, whose integral curves are possible trajectories of the system. → → → A is conserved ↓ → integral curves of V A, or simply A, is “Lie-dragged” (the same) along the trajectories of the system. ↓ → H is invariant along the integral curves of V A, i.e., H has a symmetry.