remark 22.31 The bracket is a Lie algebra structure

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1082 · p. 817

remark -- no derivation owed by its kind

Rests on

depends_on definition 5.128 Derivation ¶
- depends_on definition 5.121 Algebra ¶
  - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
    - depends_on definition 4.2 Binary operation ¶ ↺
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on definition 5.127 Lie algebra ¶
- depends_on definition 5.121 Algebra ¶ ↺
depends_on proposition 22.30 Properties of the Poisson bracket ¶
- 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 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 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 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 theorem 7.34 Rolle ¶
      - 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 22.39 Poisson's theorem does not manufacture new constants ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Derivation	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1096
`depends_on`	→	Lie algebra	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1096
`depends_on`	→	Properties of the Poisson bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1096
`depends_on`	←	Poisson's theorem does not manufacture new constants	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1305