theorem 22.22 Poincaré

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:724 · p. 813

Rests on

depends_on definition 7.125 Multiple integral ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on definition 7.39 Darboux sums and the definite integral ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
depends_on lemma 22.20 The symplectic condition ¶
- depends_on definition 5.135 The symplectic group ¶
  - depends_on definition 5.113 Non-degenerate form ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
  - depends_on equation 5.138 eq:lin-bilinear-expansion ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on definition 5.113 Non-degenerate form ¶ ↺
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-54 ¶
- depends_on equation 22.18 eq:ham-generating ¶
- depends_on proposition 7.105 Clairaut–Schwarz ¶
  - depends_on definition 7.20 Continuity 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 equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-64 ¶
- proves proof ch:05-hamiltonian-mechanics@proof-7 ¶
proves proof ch:05-hamiltonian-mechanics@proof-9 ¶

Supports

depends_on proposition 32.40 A Hamiltonian section preserves area ¶
depends_on remark 22.23 What the invariant does not depend on ¶

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`	→	Multiple integral	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:738
`depends_on`	→	The symplectic condition	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:738
`depends_on`	←	A Hamiltonian section preserves area	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1042
`depends_on`	←	What the invariant does not depend on	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:786
`proves`	←	ch:05-hamiltonian-mechanics@proof-9	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:741