definition 22.14 Phase-space coordinate transformation

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:369 · p. 809

ground object -- no derivation owed

Rests on

depends_on definition 22.2 Hamiltonian ¶
- depends_on definition 21.34 Generalized momentum ¶
- depends_on definition 21.31 Lagrangian ¶

Supports

depends_on definition 22.15 Canonical transformation ¶
- depends_on proposition 22.17 Transformation conditions ¶
  - depends_on remark 22.21 The count ¶
- depends_on proposition 22.18 The mixed generating functions ¶
  - depends_on definition 23.2 Hamilton's principal function and the Hamilton–Jacobi equation ¶
    - depends_on remark 23.12 Two functions, two roles ¶
  - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
    - depends_on example A.557 The check that fixes the convention ¶
  - depends_on remark 22.19 Which form exists ¶
  - depends_on theorem 23.30 Motion in action–angle variables ¶
    - depends_on example 23.24 Harmonic oscillator ¶
    - depends_on theorem 23.34 Adiabatic invariance of the action ¶
      - depends_on example 23.35 The slowly stiffened oscillator ¶
      - depends_on example 23.27 Charged particle in a uniform magnetic field ¶
      - depends_on remark 23.36 How good the invariance really is ¶
      - depends_on remark 23.37 Why this matters observationally ¶
    - depends_on theorem 23.31 Liouville–Arnold ¶
      - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      - depends_on remark 23.33 Integrability is exceptional ¶
      - depends_on remark 23.32 What that proof imports ¶
      - depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
      - depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
- depends_on remark 22.16 Why the class is larger than the point transformations ¶
depends_on definition 22.26 Lagrange bracket ¶
depends_on definition 22.29 Poisson bracket ¶
- 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 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 26.29 Reparametrization invariance and the mass-shell constraint ¶
      - depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
    - 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 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 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 lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
      - depends_on lemma A.638 Naturality ¶
  - 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`	→	Hamiltonian	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:378
`depends_on`	←	Canonical transformation	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:393
`depends_on`	←	Lagrange bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:857
`depends_on`	←	Poisson bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:962