proposition 24.15 Poisson bracket from the symplectic form

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:429 · p. 853

Rests on

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 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
- 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 24.14 Hamiltonian vector field ¶
- depends_on definition 24.8 Symplectic manifold ¶
  - depends_on definition 13.105 Closed form ¶
    - depends_on definition 13.103 Exterior derivative ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.102 Wedge product ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
  - depends_on definition 13.98 $k$-form ¶ ↺
  - depends_on definition 24.2 Symplectic vector space ¶
    - 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 22.4 eq:ham-pdot ¶
- depends_on equation 22.3 eq:ham-qdot ¶
depends_on equation 22.46 eq:ham-jacobi ¶
proves proof ch:07-symplectic-geometry@proof-4 ¶

Supports

depends_on definition 24.37 Casimir function ¶
- depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
  - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
    - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
  - depends_on remark 24.42 The intermediate axis ¶
- depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
- depends_on theorem 24.38 Symplectic foliation, quoted ¶
depends_on theorem 24.45 Noether, symplectic form ¶
- depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶

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`	→	Poisson bracket	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:442
`depends_on`	→	Hamiltonian vector field	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:442
`depends_on`	→	eq:ham-jacobi	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:442
`depends_on`	←	Casimir function	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1254
`depends_on`	←	Noether, symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1514
`proves`	←	ch:07-symplectic-geometry@proof-4	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:446