proposition 25.12 The algebra of generators closes

open in the book · parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:424 · p. 877

Rests on

depends_on definition 25.11 Generators of the Galilei transformations ¶
- depends_on definition 22.29 Poisson bracket ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
  - 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 25.6 Infinitesimal canonical transformation ¶
  - depends_on definition 22.29 Poisson bracket ¶ ↺
  - depends_on definition 24.17 Symplectomorphism ¶
    - depends_on definition A.70 Pullback ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.98 $k$-form ¶
    - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on definition 13.105 Closed form ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 24.2 Symplectic vector space ¶
depends_on equation 22.44 eq:ham-poisson-fundamental ¶
depends_on theorem 22.38 Poisson's theorem ¶
- depends_on equation 22.46 eq:ham-jacobi ¶
- depends_on theorem 22.34 Evolution of a phase-space function ¶
  - depends_on equation 22.4 eq:ham-pdot ¶
  - depends_on equation 22.41 eq:ham-poisson-bracket ¶
  - depends_on equation 22.3 eq:ham-qdot ¶
  - proves proof ch:05-hamiltonian-mechanics@proof-14 ¶
- proves proof ch:05-hamiltonian-mechanics@proof-17 ¶
proves proof ch:08-poisson-quantum-bridge@proof-6 ¶

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type	direction	node	provenance	where
`depends_on`	→	Generators of the Galilei transformations	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:446
`depends_on`	→	eq:ham-poisson-fundamental	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:446
`depends_on`	→	Poisson's theorem	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:446
`proves`	←	ch:08-poisson-quantum-bridge@proof-6	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:450