definition 22.29 Poisson bracket

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:955 · p. 816

ground object -- no derivation owed

Rests on

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 3.43 Map ¶
  - 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 4.33 Vector space ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - 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 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 notation A.646 Gradings and brackets ¶
      - depends_on theorem 26.51 Existence and uniqueness of the BRST charge ¶
    - depends_on proposition 26.48 Consistency of the state conditions ¶
depends_on definition 26.12 First and second class ¶
- 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 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 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 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 ¶ ↺
depends_on definition 26.8 Total Hamiltonian ¶
- depends_on definition 26.16 Extended Hamiltonian ¶
- depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
- 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 definition 25.11 Generators of the Galilei transformations ¶
- depends_on proposition 25.12 The algebra of generators closes ¶
depends_on definition 25.6 Infinitesimal canonical transformation ¶
- depends_on definition 25.11 Generators of the Galilei transformations ¶ ↺
depends_on definition 25.2 Poisson algebra ¶
depends_on definition 24.36 Poisson manifold ¶
- depends_on definition 25.2 Poisson algebra ¶ ↺
- depends_on definition 24.37 Casimir function ¶
  - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
    - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
    - depends_on remark 24.42 The intermediate axis ¶
  - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
  - depends_on theorem 24.38 Symplectic foliation, quoted ¶
- depends_on definition 24.40 Lie–Poisson bracket ¶
  - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶ ↺
  - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶ ↺
- depends_on theorem 24.38 Symplectic foliation, quoted ¶ ↺
depends_on notation 26.28 Field systems ¶
- depends_on definition 26.40 The $3+1$ decomposition ¶
  - depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
    - depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
      - depends_on lemma A.630 Codazzi equation ¶
      - depends_on lemma A.631 The normal–normal Ricci contraction ¶
  - depends_on proposition 26.41 The ADM form of the Einstein–Hilbert Lagrangian ¶
    - depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶ ↺
- depends_on notation A.636 Densities and smearing ¶ ↺
- depends_on notation 26.36 Gauge fields and their SI dimensions ¶
  - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶ ↺
depends_on notation 26.1 Weak and strong equality ¶
- depends_on notation 26.28 Field systems ¶ ↺
depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
- depends_on definition 24.37 Casimir function ¶ ↺
- depends_on theorem 24.45 Noether, symplectic form ¶
  - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶

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`	→	Partial derivative; gradient	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:962
`depends_on`	→	Phase-space coordinate transformation	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:962
`depends_on`	←	Dirac bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:608
`depends_on`	←	First and second class	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:350
`depends_on`	←	Total Hamiltonian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:246
`depends_on`	←	Generators of the Galilei transformations	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:421
`depends_on`	←	Infinitesimal canonical transformation	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:262
`depends_on`	←	Poisson algebra	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:100
`depends_on`	←	Poisson manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1246
`depends_on`	←	Field systems	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1049
`depends_on`	←	Weak and strong equality	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:85
`depends_on`	←	Poisson bracket from the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:442