proposition 26.34 Proca constraints are second class

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1273 · p. 913

Rests on

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 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 ¶
  - 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 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
  - depends_on definition 22.14 Phase-space coordinate transformation ¶
    - depends_on definition 22.2 Hamiltonian ¶
      - depends_on definition 21.34 Generalized momentum ¶
      - depends_on definition 21.31 Lagrangian ¶
depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
- depends_on proposition 26.10 The four outcomes ¶ ↺
- 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 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 proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
      - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - 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 proposition 26.10 The four outcomes ¶ ↺
    - proves proof ch:09-constrained-hamiltonian-dynamics@proof-5 ¶
  - proves proof ch:09-constrained-hamiltonian-dynamics@proof-6 ¶
- proves proof ch:09-constrained-hamiltonian-dynamics@proof-13 ¶
depends_on theorem 26.17 Counting the physical degrees of freedom ¶ ↺
proves proof ch:09-constrained-hamiltonian-dynamics@proof-14 ¶

Supports

depends_on proposition 113.3 Degrees of freedom before and after ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`cites`	→	Sur la théorie ondulatoire des électrons positifs et négatifs	derived	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1275
`depends_on`	→	First and second class	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1299
`depends_on`	→	Constraint structure of the free Maxwell field	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1299
`depends_on`	→	Counting the physical degrees of freedom	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1299
`depends_on`	←	Degrees of freedom before and after	declared	parts/11-qft-standard-model/15-exp-higgs-discovery.tex:161
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-14	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1302