theorem 25.22 Faddeev–Jackiw equations and brackets

open in the book · parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:847 · p. 882

Rests on

depends_on definition 25.21 First-order action in general coordinates ¶
- depends_on definition 13.103 Exterior derivative ¶
  - 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 definition 13.102 Wedge product ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on equation 13.239 eq:mfd-dual-basis ¶
- depends_on equation 25.20 eq:pq-phase-space-action ¶
depends_on lemma 16.18 Fundamental lemma ¶
- depends_on definition 16.15 Variation; the first variation ¶
  - depends_on definition 16.11 Admissible class; functional ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on equation 7.177 eq:ana-functional-derivative ¶
- depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
- proves proof ch:14-calculus-of-variations@proof-8 ¶
depends_on theorem 25.3 The bracket is a Lie bracket ¶
- depends_on equation 22.41 eq:ham-poisson-bracket ¶
- depends_on equation 22.56 eq:ham-symplectic-matrix ¶
- proves proof ch:08-poisson-quantum-bridge@proof-1 ¶
proves proof ch:08-poisson-quantum-bridge@proof-9 ¶

Supports

depends_on example 25.24 A charged particle in a strong magnetic field ¶

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`	→	First-order action in general coordinates	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:863
`depends_on`	→	Fundamental lemma	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:863
`depends_on`	→	The bracket is a Lie bracket	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:863
`depends_on`	←	A charged particle in a strong magnetic field	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:964
`proves`	←	ch:08-poisson-quantum-bridge@proof-9	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:866