proposition 22.30 Properties of the Poisson bracket

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:979 · p. 816

Rests on

depends_on equation 22.42 eq:ham-poisson-matrix ¶
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 ¶
depends_on proposition 7.105 Clairaut–Schwarz ¶ ↺
proves proof ch:05-hamiltonian-mechanics@proof-12 ¶

Supports

depends_on remark 22.31 The bracket is a Lie algebra structure ¶
- depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶

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`	→	eq:ham-poisson-matrix	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1003
`depends_on`	→	The symplectic condition	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1003
`depends_on`	→	Clairaut–Schwarz	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1003
`depends_on`	←	The bracket is a Lie algebra structure	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1096
`proves`	←	ch:05-hamiltonian-mechanics@proof-12	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1007