proposition 26.48 Consistency of the state conditions

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1846 · p. 920

Rests on

depends_on equation 26.16 eq:cd-first-class-algebra ¶
depends_on postulate 26.47 Dirac quantization ¶
- depends_on postulate 25.27 Dirac's correspondence rule ¶
  - depends_on equation 22.41 eq:ham-poisson-bracket ¶
  - depends_on theorem 25.3 The bracket is a Lie bracket ¶
    - depends_on equation 22.41 eq:ham-poisson-bracket ¶ ↺
    - depends_on equation 22.56 eq:ham-symplectic-matrix ¶
    - proves proof ch:08-poisson-quantum-bridge@proof-1 ¶
- depends_on theorem 26.21 Properties of the Dirac bracket ¶
  - depends_on definition 26.20 Dirac bracket ¶
    - 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 26.19 Second-class constraints come in pairs ¶
      - depends_on definition 26.12 First and second class ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
      - proves proof ch:09-constrained-hamiltonian-dynamics@proof-7 ¶
  - 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 26.19 Second-class constraints come in pairs ¶ ↺
  - proves proof ch:09-constrained-hamiltonian-dynamics@proof-8 ¶
  - proves proof ch:09-constrained-hamiltonian-dynamics@prooflink-1 ¶
- depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
  - depends_on definition 26.8 Total Hamiltonian ¶ ↺
  - depends_on proposition 26.10 The four outcomes ¶
    - depends_on definition 26.9 Consistency conditions ¶
      - depends_on equation 26.7 eq:cd-evolution ¶
      - depends_on equation 26.2 eq:cd-primary ¶
    - depends_on remark 26.7 The regularity assumption, stated once ¶
    - proves proof ch:09-constrained-hamiltonian-dynamics@proof-3 ¶
  - proves proof ch:09-constrained-hamiltonian-dynamics@proof-5 ¶
proves proof ch:09-constrained-hamiltonian-dynamics@proof-17 ¶

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type	direction	node	provenance	where
`depends_on`	→	eq:cd-first-class-algebra	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1857
`depends_on`	→	Dirac quantization	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1857
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-17	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1860