definition 26.12 First and second class

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:338 · p. 903

ground object -- no derivation owed

Rests on

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 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
- 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 ¶

Supports

depends_on proposition 26.13 The first-class functions close ¶
- depends_on proposition 26.43 The hypersurface-deformation algebra ¶
- depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
depends_on proposition 26.19 Second-class constraints come in pairs ¶
- depends_on definition 26.20 Dirac bracket ¶
  - 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 example 26.22 Holonomic constraints ¶
  - depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
  - depends_on theorem 26.21 Properties of the Dirac bracket ¶
    - depends_on postulate 26.47 Dirac quantization ¶
      - depends_on definition 26.50 The BRST charge ¶
      - depends_on proposition 26.48 Consistency of the state conditions ¶
- depends_on proposition 26.24 The Faddeev–Popov determinant ¶ ↺
- depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
  - depends_on proposition A.619 The Dirac bracket is a projected Hamiltonian flow ¶
  - depends_on remark 26.26 What the factor is doing ¶ ↺
- depends_on theorem 26.17 Counting the physical degrees of freedom ¶
  - depends_on corollary A.642 The constraints are first class ¶
  - depends_on definition 26.23 Gauge conditions ¶ ↺
  - 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 proposition 26.43 The hypersurface-deformation algebra ¶ ↺
- depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
depends_on proposition 26.34 Proca constraints are second class ¶ ↺

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 final constraint set	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:350
`depends_on`	→	Poisson bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:350
`depends_on`	←	The first-class functions close	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:357
`depends_on`	←	Second-class constraints come in pairs	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:563
`depends_on`	←	Proca constraints are second class	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1299