lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1550 · p. 866

Rests on

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 13.83 Covector ¶
      - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on equation 13.239 eq:mfd-dual-basis ¶
  - 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 5.37 Linear transformation ¶
    - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
    - 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 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 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 13.98 $k$-form ¶ ↺
  - depends_on definition 24.8 Symplectic manifold ¶ ↺
depends_on theorem 13.59 Regular value theorem ¶
- depends_on definition 13.52 Differential; pushforward ¶
  - depends_on definition 13.49 Smooth map between manifolds ¶
    - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶ ↺
      - depends_on definition 13.43 Coordinate system ¶ ↺
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
  - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
    - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 13.58 Hypersurface ¶
  - depends_on definition 13.54 Embedded submanifold ¶
    - depends_on definition 13.53 Immersion, submersion, embedding ¶
      - depends_on definition 13.52 Differential; pushforward ¶ ↺
      - depends_on definition 6.7 Homeomorphism ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 13.49 Smooth map between manifolds ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-10 ¶
proves proof ch:07-symplectic-geometry@proof-15 ¶

Supports

depends_on proposition A.565 Freeness makes every value regular ¶
- depends_on lemma A.567 The differential of the momentum map along the orbit ¶
  - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
- depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
  - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶ ↺
depends_on theorem A.563 Marsden–Weinstein reduction ¶
- depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
- depends_on example A.572 Rotational reduction of the central-force problem ¶
depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺
depends_on theorem 24.47 Marsden–Weinstein reduction ¶
- 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 ¶ ↺

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`	→	Symplectic manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1568
`depends_on`	→	Momentum map	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1568
`depends_on`	→	Regular value theorem	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1568
`depends_on`	←	Freeness makes every value regular	declared	appendices/A-long-proofs.tex:27181
`depends_on`	←	Marsden–Weinstein reduction	declared	appendices/A-long-proofs.tex:27134
`depends_on`	←	The radical is the isotropy orbit	declared	appendices/A-long-proofs.tex:27291
`depends_on`	←	Marsden–Weinstein reduction	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1613
`proves`	←	ch:07-symplectic-geometry@proof-15	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1572