definition 24.43 Momentum map

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1453 · p. 865

ground object -- no derivation owed

Rests on

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 13.102 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 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 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 13.83 Covector ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.98 $k$-form ¶ ↺
- depends_on definition 24.8 Symplectic manifold ¶ ↺

Supports

depends_on definition A.561 Coadjoint action and equivariance ¶
- 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 example 24.44 Linear and angular momentum ¶
- depends_on example A.572 Rotational reduction of the central-force problem ¶ ↺
depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
- 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 proposition A.566 The isotropy group acts, and the quotient is smooth ¶ ↺
- depends_on theorem A.563 Marsden–Weinstein reduction ¶ ↺
- 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 ¶ ↺
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`	→	Lie group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1464
`depends_on`	→	Hamiltonian vector field	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1464
`depends_on`	→	Symplectomorphism	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1464
`depends_on`	←	Coadjoint action and equivariance	declared	appendices/A-long-proofs.tex:27083
`depends_on`	←	Linear and angular momentum	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1500
`depends_on`	←	The level set of the momentum map is the symplectic orthogonal of the orbit	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1568
`depends_on`	←	Noether, symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1514