lemma A.548 The joint flow is a translation action on the level set

open in the book · appendices/A-long-proofs.tex:26612 · p. 3058

Rests on

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 ¶
  - depends_on definition 13.124 Integral curve; complete vector field ¶
    - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶
  - depends_on theorem A.74 Flow of a time-dependent vector field ¶
    - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on definition 9.2 Linear equation; homogeneity ¶
      - depends_on remark 9.3 Normal form and first-order systems ¶
      - depends_on theorem 9.8 Picard–Lindelöf ¶
      - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
    - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on definition 7.53 $\ee$ ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof app:A-long-proofs@proof-52 ¶
    - depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
    - proves proof app:A-long-proofs@proof-54 ¶
  - depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-27 ¶
- 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 ¶

Supports

depends_on corollary A.553 Quasi-periodic motion ¶
depends_on lemma A.549 The action is transitive ¶
- depends_on proposition A.552 The component is a torus ¶
  - depends_on corollary A.553 Quasi-periodic motion ¶ ↺
  - depends_on definition A.554 Actions on the torus ¶
    - depends_on lemma A.555 The actions are well defined ¶
    - 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 proposition A.556 The Jacobian of the actions is the lattice matrix ¶ ↺
depends_on proposition A.552 The component is a torus ¶ ↺

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`	→	The Hamiltonian fields of the integrals	declared	appendices/A-long-proofs.tex:26624
`depends_on`	→	Commuting fields have commuting flows	declared	appendices/A-long-proofs.tex:26624
`depends_on`	→	Existence, uniqueness and smoothness of the flow	declared	appendices/A-long-proofs.tex:26624
`depends_on`	←	Quasi-periodic motion	declared	appendices/A-long-proofs.tex:26752
`depends_on`	←	The action is transitive	declared	appendices/A-long-proofs.tex:26652
`depends_on`	←	The component is a torus	declared	appendices/A-long-proofs.tex:26724
`proves`	←	app:A-long-proofs@proof-332	declared	appendices/A-long-proofs.tex:26628