definition 26.8 Total Hamiltonian

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:235 · p. 901

ground object -- no derivation owed

Rests on

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 ¶
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 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 ¶
- proves proof ch:09-constrained-hamiltonian-dynamics@proof-2 ¶

Supports

depends_on definition 26.16 Extended Hamiltonian ¶
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 notation A.646 Gradings and brackets ¶
      - depends_on definition A.647 Koszul–Tate differential ¶
      - depends_on lemma A.652 Nilpotency, order by order ¶
    - depends_on theorem 26.51 Existence and uniqueness of the BRST charge ¶
  - depends_on proposition 26.48 Consistency of the state conditions ¶
depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
- depends_on definition 26.16 Extended Hamiltonian ¶ ↺
- depends_on postulate 26.47 Dirac quantization ¶ ↺
- 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 26.24 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 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`	→	Poisson bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:246
`depends_on`	→	The canonical Hamiltonian is defined on the constraint surface	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:246
`depends_on`	←	Extended Hamiltonian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:476
`depends_on`	←	Properties of the Dirac bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:628
`depends_on`	←	First-class constraints generate gauge transformations	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:411