lemma 22.20 The symplectic condition

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:588 · p. 811

Rests on

depends_on definition 5.135 The symplectic group ¶
- depends_on definition 5.113 Non-degenerate form ¶
  - depends_on definition 5.110 Bilinear map ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
- depends_on equation 5.138 eq:lin-bilinear-expansion ¶
- 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 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 equation 22.18 eq:ham-generating ¶
depends_on proposition 7.105 Clairaut–Schwarz ¶
- depends_on definition 7.20 Continuity at a point ¶
  - depends_on definition 7.16 Limit ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-64 ¶
proves proof ch:05-hamiltonian-mechanics@proof-7 ¶

Supports

depends_on corollary 22.24 Invariance of the phase-space volume ¶
- depends_on remark 22.25 This is Liouville's theorem ¶
depends_on proposition 22.27 Properties of the Lagrange bracket ¶
- depends_on remark 22.28 Where the content of the fundamental brackets lies ¶
depends_on proposition 22.30 Properties of the Poisson bracket ¶
- depends_on remark 22.31 The bracket is a Lie algebra structure ¶
  - depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶
depends_on remark 22.33 What the duality is good for ¶
depends_on remark 22.28 Where the content of the fundamental brackets lies ¶ ↺
depends_on theorem 22.22 Poincaré ¶
- depends_on proposition 32.40 A Hamiltonian section preserves area ¶
- depends_on remark 22.23 What the invariant does not depend on ¶

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`	→	The symplectic group	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:597
`depends_on`	→	eq:ham-generating	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:597
`depends_on`	→	Clairaut–Schwarz	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:597
`depends_on`	←	Invariance of the phase-space volume	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:800
`depends_on`	←	Properties of the Lagrange bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:891
`depends_on`	←	Properties of the Poisson bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1003
`depends_on`	←	What the duality is good for	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1167
`depends_on`	←	Where the content of the fundamental brackets lies	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:950
`depends_on`	←	Poincaré	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:738
`proves`	←	ch:05-hamiltonian-mechanics@proof-7	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:600