theorem 26.17 Counting the physical degrees of freedom

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:479 · p. 904

Rests on

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 definition 26.9 Consistency conditions ¶
      - 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 5.15 Basis ¶
    - depends_on definition 22.14 Phase-space coordinate transformation ¶
      - depends_on definition 22.2 Hamiltonian ¶
- 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 4.33 Vector space ¶
  - 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 ¶
      - depends_on definition 26.3 Singular Lagrangian ¶
      - depends_on remark 26.4 The constant-rank theorem is owed by Part II ¶
      - proves proof ch:09-constrained-hamiltonian-dynamics@proof-1 ¶
    - 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 ¶

Supports

depends_on corollary A.642 The constraints are first class ¶
depends_on definition 26.23 Gauge conditions ¶
- depends_on proposition 26.24 The Faddeev–Popov determinant ¶
  - depends_on remark 26.26 What the factor is doing ¶
depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
- 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 ¶
depends_on proposition 26.34 Proca constraints are second class ¶ ↺
depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
- depends_on notation A.636 Densities and smearing ¶
  - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
    - depends_on lemma A.638 Naturality ¶
      - depends_on proposition A.639 The first two brackets ¶
- depends_on proposition 26.43 The hypersurface-deformation algebra ¶

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`	→	Second-class constraints come in pairs	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:490
`depends_on`	→	The Frobenius theorem is owed by Part II	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:490
`depends_on`	→	First-class constraints generate gauge transformations	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:490
`depends_on`	←	The constraints are first class	declared	appendices/A-long-proofs.tex:31367
`depends_on`	←	Gauge conditions	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:792
`depends_on`	←	Constraint structure of the free Maxwell field	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`	←	Reparametrization invariance and the mass-shell constraint	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1071
`depends_on`	←	The Hamiltonian structure of general relativity	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1649
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-6	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:493