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  1. 5. Applications in Physics • Thermodynamics • Hamiltonian mechanics • Electromagnetism • Dynamics of a perfect fluid • Cosmology

  2. A. Thermodynamics 5.1 Simple systems 5.2 Maxwell and other mathematical identities 5.3 Composite thermodynamic systems: Caratheodory's theorem

  3. 5.1. Simple Systems 1-component fluid: (1st law) 1-forms on 2-D manifold with coordinates (V,U).  (δQ is not exact ) δQ is a 1-form in 2-space → its ideal is closed (see Ex 4.31(b)) Frobenius' theorem (§4.26) → 2nd law: holds for all systems in thermodynamic equilibrium.

  4. 5.2. Maxwell and Other Mathematical Identities → Switching to (S,V) gives → (Maxwell identity) Switching to (T,V) gives → → Switching to (T,V) gives →

  5. 5.3. Composite Thermodynamic Systems: Caratheodory's Theorem 1-forms on 2N-D manifold with coordinates (Vi,Ui). Frobenius’ theorem:  (See Ex 4.30 ) δQ integrable → equilibrium submanifolds defined by S = constant. → Equilibrium states on different submanifolds cannot be bridged adiabatically. Question: is the converse true? i.e., Not every state reachable adiabatically → S exisits ? c.f. 2nd law: Heat can’t be transfer from cold to hot regions in a closed system without doing work. Caratheodory: 2nd law → S exisits.

  6. Proof: If δQ is not integrable, then  V, W , s.t. in the neighborhod of P, but i.e., the subspace KPof vector fields that annul δQ do not form a hypersurface. e.g., δQ is not integrable → Any states near P can be reached adiabatically. Not all states near P can be reached adiabatically → δQ is integrable

  7. B. Hamiltonian Mechanics • Hamiltonian Vector Fields • Canonical Transformation • Map between Vectors & 1-forms Provided by ω • Poisson Bracket • Many Particle System: Symplectic Forms • Linear Dynamical Systems: Symplectic Inner Product & Conserved Quantities • Fibre Bundle Structure of the Hamiltonian Equations See Frankel

  8. 5.4. Hamiltonian Vector Fields Lagrangian: Momenta: Hamiltonian: Euler-Lagrange eq: Hamilton’s eqs: Let M be the manifold with coordinates ( qi). Then the set ( qi, qi,t ) is the tangent bundle T(M). The set ( qi, pi ) can be taken as the cotangent bundle T*(M), or a symplectic manifold with the Poincare (symplectic) 2-form

  9. Definition: Symplectic Forms: A 2-form on M2n is symplectic ( M is then a symplectic manifold ) if 1. 2. ω is non-degenerate, i.e., iXω is non-singular. Definition: Interior Product: by s.t. if α is a 0-form if α is a 1-form if α is a p-form Properties: Components:

  10. If U is tangent to system trajectory Eq(4.67): → U is called a Hamiltonian vector field. (Conservative system)

  11. 5.5. Canonical Transformation A coordinate transformation is canonical if This can be achieved through a generating function F. E.g., given F = F(q,Q ) s.t. & we have

  12. 5.6. Map between Vectors & 1-forms Provided by ω Define or Note difference in order of indices with eq(5.27) See Ex 5.3 However, ω is not a metric since Ex 5.5: A hamiltonian vector field corresponds to an exact 1-form, i.e.

  13. 5.7. Poisson Bracket then Let → → Note sign difference with eq(5.31), which can be traced to eq(5.27)

  14. 5.8. Many Particle System: Symplectic Forms Symplectic = German for plaiting together Symplectic Form Phase space = Symplectic manifold Ex 5.6 8

  15. 5.9. Linear Dynamical Systems: Symplectic Inner Product & Conserved Quantities Linear system: Linearity: if & are solutions, so is  Solution sub-manifold is also a vector space. A manifold that is also a vector space is a flat manifold (M is isomorphic to Rn )

  16. Treating elements of phase space as vectors, we set The (anti-symmetric) symplectic inner product is defined as The symplectic inner product is time-independent if Y(i) are solution curves. Proof: Reminder:

  17. For time independent T i jand Vi j , we have → → i.e., if is a solution, so is Define the canonical energy by ECis conserved on solution curves. If Y is a solution, then

  18. Let U be a vector field on configuration space s.t. c.f. Ex 5.8 If is a solution, so is The canonical U-momentum is defined by PUis conserved on solution curves. For the Klein-Gordon eq. 4-current density = jμ is conserved: Setting ( DoF =  ) we have

  19. 5.10. Fibre Bundle Structure of the Hamiltonian Equations qidefines a configuration space manifold M. Evolution of system is a curve qi(t) on M. Lagrangian L( qi, qi, t ) is a function on the tangent bundle T(M). is a 1-form field on M, i.e., phase space is a cotangent bundle T*(M) Proof: Consider a new set of coordinates and new momenta Since qi, t and Q j , t are elements of the (tangent) fibre of T(M), they transform like contravariant vectors, i.e., → ( pi & Pj  are 1-forms )

  20. Phase space { qi, pi } is a cotangent bundle T*(M). H is a function on T*(M). The symplectic form is coordinate free. Proof: → →  →  since where QED Reminder: System with constraints leads to non-trivial bundles.

  21. C. Electromagnetism • Rewriting Maxwell’s Equations Using Differential Forms • Charge & Topology • The Vector Potential • Plane Waves: A Simple Example

  22. 5.11. Rewriting Maxwell’s Equations Using Differential Forms Maxwell’s equations in vacuum with sources, Gaussian units with c = 1: Faraday 2-form: →  

  23. i,j,k cyclic.  corresponds to the homogeneous eqs.

  24.  Inhomogeneous eqs: Metric volume form = →

  25. i,j,k cyclic.

  26.  Inhomogeneous eqs are given by Magnetic monopole:

  27. Alternative Approach See §7.2, Frankel §4.6, Flanders

  28. → where

  29. → Ex 5.14  Inhomogeneous eqs are given by

  30. 12. Charge & Topology Charge = Topology 1. Wheeler: Wormhole (handle) → Pair of  charges. • Objections: • Origin of wormhole unknown. • Linkage of distant pair of charge unacceptable. 2. Sorkin: Wormhole creating pair of nearby charges of same sign.

  31. 5.13. The Vector Potential ← F is invariant under a gauge transformation: A cannot be defined in region with magnetic monopole. Ex 5.16

  32. 5.14. Plane Waves: A Simple Example Let → Static fields ignored →

  33. D. Dynamics of a Perfect Fluid • Role of Lie Derivatives • The Comoving Time-Derivative • Equation of Motion • Conservation of Vorticity

  34. 5.15. Role of Lie Derivatives • Perfect fluid: • No viscosity. • No heat conduction (adiabatic). • Quantities conserved in any fluid element( local conservation laws ) : • Mass. • Entropy. • Vorticity. Conservation laws are more transparent within the framework of Lie derivatives.

  35. 5.16. The Comoving Time-Derivative Equation of continuity: where τ is the volume 3-form: = time-derivative operator in a frame travelling with the fluid element. Proof : Let ( x, y, z, t ) be the coordinates of a fluid particle in the Galilean space-time. The tangent U to the “world-line” of the fluid particle is ( parameter of world-line = t ) The time-derivative operator in a frame travelling with the fluid element is LU.

  36. i = x, y, z.  W if W is purely spatial, i.e., W t = 0  if W is purely spatial This holds if W is replaced by any purely spatial (n0) tensor. Reminder: The Galilean space-time is a fibre bundle with t as base. Ex 5.19

  37. 5.17. Equation of Motion Adiabatic flow: specific entropy S conserved → Euler’s equation of motion ( see Landau & Lifshitz, “Fluid Mechanics”, §2 ) : ( p = pressure, Φ = gravitational potential ) In Cartesian coordinates: • Equation valid only in Cartesian coordinates because : • Index mismatch (allowable only in orthonormal bases). • j V i is a tensor only for transformations with coordinate independent Λi’j .

  38. is not a tensor equation in general coordinates. Usual remedy is to introduce a covariant derivative (see Chap 6). An alternative approach via Lie derivative is as follows. Index mismatch can be resolved using V i = Vi for Cartesian coordinates : Non-tensorial transformation behavior is resolved using ( d involves only spatial derivatives ) →

  39. 5.18. Conservation of Vorticity ( d involves only spatial derivatives ) Since d both sides → Case I : p = p(ρ ) → ( Helmholtz circulation theorem )

  40. Case II : p = p(ρ, S ) Since  → dS  both sides of gives ( Ertel’s Theorem ) Since any two 3-forms are proportional in our 3-D space, we can write α = some scalar function, τ = volume 3-form →

  41. Ex 5.21 : Ex.5.22

  42. E. Cosmology • The Cosmological Principle • Lie Algebra of Maximal Symmetry • The Metric of a Spherically Symmetric 3-Space • Construction of the Killing Vectors • Open, Closed, & Flat Universes

  43. 5.19. The Cosmological Principle General relativity → Cosmology Assuming universe to be homogeneous & isotropic in the large scale, D.G. → only 3 cosmology models (different initial metrics) are possible: Flat, Open, Closed. This result can be derived without using general relativity or Riemannian geometry. • Mass distribution of the universe: • Small scale [ 1015 m (nuclear) ~ 1017 m (interstellar) ] : lumpy. • Star cluster = Galaxy : lumpy • Cluster of galaxies ( 101 – 103 galaxies ) : lumpy • Cluster of galaxy clusters = Supercluster : lumpy • Beyond superclusters : homogeneous & isotropic

  44. Since the universe is evolving, the “observed” homogeneity is an interpolation to the “present time”. Spacetime is thus treated as a foliation with leaves of constant time hypersurfaces. A hypersurface is space-like is g is positive-definite on all vectors tangent to it. Definition of homogeneity Let G be the isometry Lie group of manifold S with metric tensor field g. The Lie algebra G of G is that of the Killing vector fields of g. Elements of G are diffeomorphisms of S onto itself. The action of G on S is transitive if  P, Q S,  gG s.t. g(P) = Q. A manifoldS is homogeneous if its isometry group acts transitively on it, i.e., the geometry is the same everywhere on S. Elements of G which leaves a point P on S fixed form a subgroup HP of G. HP is called the isotropy group of P.

  45. The isotropy group HP of P maps any curve through P to another curve through it. • HP : TP→ TP (c.f. adjoint representation of a Lie group) A manifold S is isotropic about P if its HP = SO(m). If S is isotropic about all P, it is isotropic. A cosmology model M is a homogeneous cosmology if it has a foliation of homogeneous space-like hypersurfaces. Similarly for isotropic cosmology. The universe is observed to be homogeneous on the large scale about us. Cosomological principle: likewise for all observers in the universe. Ex 5.23

  46. 5.20. Lie Algebra of Maximal Symmetry Let S be a 3-D manifold & ξ a Killing vector field on it, i.e., → where = Christoffel symbol

  47. is symmetric in i & j → ½ n(n+1) eqs for n variables ξj . • ξ is over-determined for n > 1. → A general g may have no Killing vector fields. Task: Find criteria for g to have the maximal set of Killing vectors. (1) k eq. gives (2) i → j → k →i : (3) (1)+(3)(2) : where

  48. For a given g, if ξi and ξi , j at point P are known, then all higher derivatives of ξ at P are known. → ξ is known in any neighborhood of P where ξ is analytic. Given ξi , the symmetric part of ξi , j is given by Hence, a Killing vector field on S is determined given some appropriate values at a single point P S. Number of independent choices of ηi is n. That of Ai j is ½ n(n1). → Maximal number of Killing vector fields is ½ n(n+1). In which case, M is maximally symmetric.