definition 22.15 Canonical transformation

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:381 · p. 809

ground object -- no derivation owed

Rests on

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 definition 21.45 Action ¶
- depends_on definition 16.96 Action; Hamilton's principle ¶
  - depends_on definition 21.31 Lagrangian ¶ ↺
  - depends_on equation 16.11 eq:calcvar-basic-functional ¶
- depends_on equation 21.37 eq:lag-lagrangian ¶
depends_on theorem 22.6 Hamilton's equations ¶
- depends_on definition 22.2 Hamiltonian ¶ ↺
- depends_on equation 21.39 eq:lag-el-conservative ¶
- depends_on equation 21.43 eq:lag-generalized-momentum ¶
- proves proof ch:05-hamiltonian-mechanics@proof-2 ¶

Supports

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 24.57 The half-integer, and the honest status of the construction ¶
    - 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 phenomenon 32.58 Regular and chaotic motion coexist ¶
      - depends_on phenomenon 32.59 The solar system is chaotic ¶
      - depends_on remark 23.33 Integrability is exceptional ¶ ↺
    - depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
depends_on remark 22.16 Why the class is larger than the point transformations ¶

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`	→	Phase-space coordinate transformation	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:393
`depends_on`	→	Action	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:393
`depends_on`	→	Hamilton's equations	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:393
`depends_on`	←	Transformation conditions	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:423
`depends_on`	←	The mixed generating functions	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:489
`depends_on`	←	Why the class is larger than the point transformations	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:407