equation 5.138 eq:lin-bilinear-expansion

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• depends_on definition 5.135 The symplectic group ¶
  • depends_on definition 22.40 Symplectic matrix ¶
  • depends_on definition 24.4 Symplectic group ¶
    • depends_on proposition 24.5 Properties of the symplectic group ¶
      • depends_on remark 24.6 Connectedness, and what it rests on ¶
      • depends_on remark 24.7 Contrast with the orthogonal group ¶
    • depends_on proposition 24.30 Linear non-squeezing ¶
      • depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
        • depends_on proposition 24.34 Capacities exist if and only if non-squeezing holds ¶
        • depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
  • depends_on lemma 22.20 The symplectic condition ¶
    • 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 ¶
  • depends_on proposition 5.139 Dimension of the symplectic group ¶
    • depends_on proposition 24.5 Properties of the symplectic group ¶ ↺
    • depends_on remark 22.21 The count ¶
  • depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶
  • depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
    • depends_on corollary 22.24 Invariance of the phase-space volume ¶ ↺
    • depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
    • depends_on proposition 24.5 Properties of the symplectic group ¶ ↺