theorem 24.45 Noether, symplectic form

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1503 · p. 866

Rests on

depends_on definition 24.43 Momentum map ¶
- depends_on definition 14.2 Lie group ¶
- 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 13.98 $k$-form ¶ ↺
    - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - depends_on equation 22.4 eq:ham-pdot ¶
  - depends_on equation 22.3 eq:ham-qdot ¶
- depends_on definition 24.17 Symplectomorphism ¶
  - depends_on definition A.70 Pullback ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
  - depends_on definition 24.8 Symplectic manifold ¶ ↺
depends_on equation 24.5 eq:sym-hamiltonian-field ¶
depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
- depends_on definition 22.29 Poisson bracket ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
  - 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 equation 22.46 eq:ham-jacobi ¶
- proves proof ch:07-symplectic-geometry@proof-4 ¶
proves proof ch:07-symplectic-geometry@proof-14 ¶

Supports

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`	→	Momentum map	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1514
`depends_on`	→	eq:sym-hamiltonian-field	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1514
`depends_on`	→	Poisson bracket from the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1514
`depends_on`	←	Invariant Hamiltonians descend with their flows	declared	appendices/A-long-proofs.tex:27413
`proves`	←	ch:07-symplectic-geometry@proof-14	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1518