proposition A.556 The Jacobian of the actions is the lattice matrix

open in the book · appendices/A-long-proofs.tex:26825 · p. 3061

Rests on

depends_on definition A.554 Actions on the torus ¶
- depends_on equation 23.49 eq:hj-action-variable ¶
- depends_on equation 24.3 eq:sym-canonical-forms ¶
- depends_on proposition A.552 The component is a torus ¶
  - depends_on lemma A.548 The joint flow is a translation action on the level set ¶
    - depends_on lemma A.546 The Hamiltonian fields of the integrals ¶
      - depends_on equation A.844 eq:app-hj-liouville-arnold-involution ¶
      - depends_on equation 24.8 eq:sym-bracket-homomorphism ¶
      - depends_on equation 24.5 eq:sym-hamiltonian-field ¶
      - proves proof app:A-long-proofs@proof-330 ¶
    - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on equation 13.267 eq:mfd-lie-def ¶
      - depends_on equation 13.273 eq:mfd-lie-vector ¶
      - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-31 ¶
    - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶ ↺
    - proves proof app:A-long-proofs@proof-332 ¶
  - depends_on lemma A.549 The action is transitive ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on theorem A.286 Implicit function theorem ¶
      - proves proof app:A-long-proofs@proof-186 ¶
    - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
    - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on lemma A.546 The Hamiltonian fields of the integrals ¶ ↺
      - depends_on theorem 13.133 Frobenius ¶
      - depends_on theorem 13.59 Regular value theorem ¶
      - proves proof app:A-long-proofs@proof-331 ¶
    - proves proof app:A-long-proofs@proof-333 ¶
  - depends_on theorem A.550 Quoted: discrete subgroups of a real vector space ¶
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - proves proof app:A-long-proofs@proof-334 ¶
depends_on proposition A.552 The component is a torus ¶ ↺
depends_on proposition 22.18 The mixed generating functions ¶
- depends_on definition 22.15 Canonical transformation ¶
  - 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 ¶
- depends_on equation 22.18 eq:ham-generating ¶
- proves proof ch:05-hamiltonian-mechanics@proof-6 ¶
proves proof app:A-long-proofs@proof-337 ¶

Supports

depends_on example A.557 The check that fixes the convention ¶

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`	→	Actions on the torus	declared	appendices/A-long-proofs.tex:26838
`depends_on`	→	The component is a torus	declared	appendices/A-long-proofs.tex:26838
`depends_on`	→	The mixed generating functions	declared	appendices/A-long-proofs.tex:26838
`depends_on`	←	The check that fixes the convention	declared	appendices/A-long-proofs.tex:26896
`proves`	←	app:A-long-proofs@proof-337	declared	appendices/A-long-proofs.tex:26842