proposition 26.32 Constraint structure of the free Maxwell field

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1152 · p. 912

Rests on

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 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 ¶
proves proof ch:09-constrained-hamiltonian-dynamics@proof-13 ¶

Supports

depends_on proposition 26.34 Proca constraints are second class ¶
- depends_on proposition 113.3 Degrees of freedom before and after ¶
depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The four outcomes	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1174
`depends_on`	→	Primary constraints	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1174
`depends_on`	→	Counting the physical degrees of freedom	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1174
`depends_on`	←	Proca constraints are second class	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1299
`depends_on`	←	Constraint structure of Yang–Mills theory	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1438
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-13	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1177