proposition 26.25 The Liouville measure of a second-class surface

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:855 · p. 909

Rests on

depends_on definition 24.20 Liouville volume ¶
- depends_on definition 13.102 Wedge product ¶
  - depends_on definition 13.98 $k$-form ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on equation 13.239 eq:mfd-dual-basis ¶
- 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 theorem 24.12 Darboux ¶
  - depends_on definition 24.8 Symplectic manifold ¶ ↺
  - depends_on theorem A.68 Darboux ¶
    - depends_on definition A.67 Symplectic manifold ¶
      - depends_on definition 13.105 Closed form ¶ ↺
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on equation 5.144 eq:lin-symplectic-normal-form ¶
    - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
    - proves proof app:A-long-proofs@proof-57 ¶
  - proves proof ch:07-symplectic-geometry@prooflink-1 ¶
depends_on proposition 26.19 Second-class constraints come in pairs ¶
- depends_on definition 26.12 First and second class ¶
  - depends_on definition 26.11 The final constraint set ¶
    - depends_on proposition 26.10 The four outcomes ¶
      - depends_on definition 26.9 Consistency conditions ¶
      - depends_on remark 26.7 The regularity assumption, stated once ¶
      - proves proof ch:09-constrained-hamiltonian-dynamics@proof-3 ¶
    - depends_on remark 26.7 The regularity assumption, stated once ¶ ↺
  - 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 5.15 Basis ¶
    - depends_on definition 22.14 Phase-space coordinate transformation ¶
      - depends_on definition 22.2 Hamiltonian ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
- proves proof ch:09-constrained-hamiltonian-dynamics@proof-7 ¶
depends_on proposition 24.3 Even dimension and the canonical basis ¶
- depends_on definition 24.2 Symplectic vector space ¶ ↺
- depends_on equation 22.56 eq:ham-symplectic-matrix ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
- proves proof ch:07-symplectic-geometry@proof-1 ¶
proves proof ch:09-constrained-hamiltonian-dynamics@proof-11 ¶

Supports

depends_on proposition A.619 The Dirac bracket is a projected Hamiltonian flow ¶
depends_on remark 26.26 What the factor is doing ¶

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`	→	Liouville volume	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:876
`depends_on`	→	Second-class constraints come in pairs	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:876
`depends_on`	→	Even dimension and the canonical basis	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:876
`depends_on`	←	The Dirac bracket is a projected Hamiltonian flow	declared	appendices/A-long-proofs.tex:29876
`depends_on`	←	What the factor is doing	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1004
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-11	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:880