proposition 26.29 Reparametrization invariance and the mass-shell constraint

Rests on

• depends_on proposition 26.5 Primary constraints ¶
  • depends_on definition 26.3 Singular Lagrangian ¶
    • depends_on equation 22.2 eq:ham-hessian ¶
    • depends_on proposition 22.3 Invertibility of the Legendre map ¶
      • depends_on corollary A.292 Inverse function theorem ¶
        • depends_on proposition 7.104 Chain rule in several variables ¶
        • depends_on theorem A.286 Implicit function theorem ¶
        • proves proof app:A-long-proofs@proof-186 ¶
      • depends_on equation 21.43 eq:lag-generalized-momentum ¶
      • proves proof ch:05-hamiltonian-mechanics@proof-1 ¶
  • depends_on remark 26.4 The constant-rank theorem is owed by Part II ¶
  • proves proof ch:09-constrained-hamiltonian-dynamics@proof-1 ¶
• depends_on theorem 26.17 Counting the physical degrees of freedom ¶
  • depends_on proposition 26.19 Second-class constraints come in pairs ¶
    • depends_on definition 26.12 First and second class ¶
      • depends_on definition 26.11 The final constraint set ¶
        • depends_on proposition 26.10 The four outcomes ¶
        • depends_on remark 26.7 The regularity assumption, stated once ¶
      • depends_on definition 22.29 Poisson bracket ¶
        • depends_on definition 7.97 Partial derivative; gradient ¶
        • depends_on definition 22.14 Phase-space coordinate transformation ¶
    • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
      • depends_on definition 5.110 Bilinear map ¶
        • depends_on definition 5.37 Linear transformation ¶
      • depends_on definition 5.113 Non-degenerate form ¶
        • depends_on definition 5.110 Bilinear map ¶ ↺
      • depends_on definition 5.111 Symmetric and antisymmetric forms ¶
        • depends_on definition 5.110 Bilinear map ¶ ↺
        • depends_on equation 5.137 eq:lin-bilinear-matrix ¶
      • depends_on equation 5.139 eq:lin-bilinear-congruence ¶
      • proves proof ch:03-linear-algebra-representations@proof-54 ¶
    • proves proof ch:09-constrained-hamiltonian-dynamics@proof-7 ¶
  • depends_on remark 26.18 The Frobenius theorem is owed by Part II ¶
  • depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
    • depends_on definition 26.8 Total Hamiltonian ¶
      • depends_on definition 22.29 Poisson bracket ¶ ↺
      • depends_on proposition 26.6 The canonical Hamiltonian is defined on the constraint surface ¶
        • depends_on equation 26.4 eq:cd-canonical-hamiltonian ¶
        • depends_on proposition 26.5 Primary constraints ¶ ↺
        • proves proof ch:09-constrained-hamiltonian-dynamics@proof-2 ¶
    • depends_on proposition 26.10 The four outcomes ¶ ↺
    • proves proof ch:09-constrained-hamiltonian-dynamics@proof-5 ¶
  • proves proof ch:09-constrained-hamiltonian-dynamics@proof-6 ¶
• depends_on theorem 40.5 Energy–momentum relation ¶
  • depends_on definition 40.4 Four-momentum ¶
    • depends_on definition 40.2 Four-velocity ¶
      • depends_on definition 38.12 Proper time ¶
        • depends_on definition 38.6 Minkowski metric and interval ¶
        • depends_on proposition 38.5 The invariant interval ¶
      • depends_on definition 40.1 Four-vector ¶
        • depends_on definition 38.6 Minkowski metric and interval ¶ ↺
        • depends_on equation 38.14 eq:lor-invariance-condition ¶
  • depends_on proposition 40.3 Normalization of the four-velocity ¶
    • depends_on definition 38.6 Minkowski metric and interval ¶ ↺
    • depends_on equation 40.2 eq:rdyn-fourvelocity ¶
    • proves proof ch:04-relativistic-dynamics@proof-1 ¶
  • proves proof ch:04-relativistic-dynamics@proof-2 ¶
• proves proof ch:09-constrained-hamiltonian-dynamics@proof-12 ¶

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