equation 5.139 eq:lin-bilinear-congruence

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• depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  • 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 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 22.33 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 ¶ ↺
  • depends_on definition 24.2 Symplectic vector space ¶
    • depends_on definition 24.8 Symplectic manifold ¶
      • depends_on definition 24.9 The canonical form on a cotangent bundle ¶
        • depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
      • depends_on definition 24.14 Hamiltonian vector field ¶
        • depends_on definition 24.43 Momentum map ¶
        • depends_on definition 24.50 Prequantum operator ¶
        • depends_on proposition 25.4 Leibniz rule and derivations ¶
        • depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
        • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      • depends_on definition 24.20 Liouville volume ¶
        • depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
        • depends_on theorem 24.21 Liouville ¶
        • depends_on theorem 24.24 Poincaré recurrence ¶
      • depends_on definition 24.53 Real polarization and polarized sections ¶
        • depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶
      • depends_on definition 24.49 Prequantum datum ¶
        • depends_on definition 24.53 Real polarization and polarized sections ¶ ↺
        • depends_on definition 24.50 Prequantum operator ¶ ↺
        • depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      • depends_on definition 24.17 Symplectomorphism ¶
        • depends_on definition 25.6 Infinitesimal canonical transformation ¶
        • depends_on definition 24.28 Ball and cylinder ¶
        • depends_on definition 24.43 Momentum map ¶ ↺
        • depends_on theorem 24.18 Symplectic form of the transformation condition ¶
      • 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 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 theorem 24.12 Darboux ¶
        • depends_on definition 24.20 Liouville volume ¶ ↺
        • depends_on remark 24.13 What Darboux's theorem forbids ¶
      • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶ ↺
      • depends_on theorem 24.38 Symplectic foliation, quoted ¶
      • depends_on theorem 24.47 Marsden–Weinstein reduction ¶ ↺
    • depends_on lemma A.564 Dimension and double orthogonal ¶
      • depends_on proposition A.565 Freeness makes every value regular ¶ ↺
      • depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺
    • depends_on proposition 24.3 Even dimension and the canonical basis ¶
      • depends_on definition 24.4 Symplectic group ¶ ↺
      • depends_on proposition 26.25 The Liouville measure of a second-class surface ¶ ↺
  • depends_on proposition 26.19 Second-class constraints come in pairs ¶
    • depends_on definition 26.20 Dirac bracket ¶
      • depends_on definition 26.23 Gauge conditions ¶
        • depends_on proposition 26.24 The Faddeev–Popov determinant ¶
      • depends_on example 26.22 Holonomic constraints ¶
      • depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
      • depends_on theorem 26.21 Properties of the Dirac bracket ¶
        • depends_on postulate 26.47 Dirac quantization ¶
    • depends_on proposition 26.24 The Faddeev–Popov determinant ¶ ↺
    • depends_on proposition 26.25 The Liouville measure of a second-class surface ¶ ↺
    • depends_on theorem 26.17 Counting the physical degrees of freedom ¶
      • depends_on corollary A.642 The constraints are first class ¶
      • depends_on definition 26.23 Gauge conditions ¶ ↺
      • depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
        • depends_on proposition 26.34 Proca constraints are second class ¶
        • depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
      • depends_on proposition 26.34 Proca constraints are second class ¶ ↺
      • depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
      • depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
        • depends_on notation A.636 Densities and smearing ¶
        • depends_on proposition 26.43 The hypersurface-deformation algebra ¶
    • depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
  • depends_on proposition 24.3 Even dimension and the canonical basis ¶ ↺
  • depends_on theorem A.68 Darboux ¶
    • depends_on corollary A.69 No local invariants ¶
      • depends_on remark 24.13 What Darboux's theorem forbids ¶ ↺
    • depends_on example A.80 The phase space of one particle in space ¶
    • depends_on theorem 24.12 Darboux ¶ ↺
  • depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺
• depends_on theorem 5.116 Sylvester's law of inertia ¶
  • depends_on definition 5.117 Signature ¶
    • depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      • depends_on proposition 5.95 The focal polar equation of a conic ¶
  • depends_on definition 10.6 Type of a second-order operator ¶
    • depends_on definition 10.79 Parabolic boundary ¶
      • depends_on theorem 10.80 Parabolic maximum principle ¶
        • depends_on corollary 10.81 Uniqueness for the heat equation ¶
        • depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
    • depends_on example 10.9 ex:pde-three-types ¶
    • depends_on proposition 10.25 The backward heat problem is ill posed ¶
    • depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
    • depends_on theorem 10.8 The type is a coordinate invariant ¶
      • depends_on lemma A.90 Affine normalization ¶
        • depends_on lemma A.91 Smallness after rescaling ¶
      • depends_on proposition 10.13 Canonical form in two variables ¶
        • depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶ ↺
        • depends_on theorem 10.55 d'Alembert's formula ¶
      • depends_on proposition 10.11 What a characteristic surface is ¶
        • depends_on corollary 10.12 Characteristics of the three canonical operators ¶
        • depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
        • depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
  • depends_on proposition 30.30 Stability bounds ¶
    • depends_on phenomenon 30.43 Two bulk wave speeds ¶
      • depends_on phenomenon 30.47 The three seismic phases ¶
      • depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
    • depends_on phenomenon 30.32 Poisson contraction ¶
      • depends_on proposition 30.33 Conversion among the moduli ¶
        • depends_on definition 30.63 Flexural rigidity ¶
        • depends_on example 30.35 From single crystals to the isotropic aggregate ¶
        • depends_on proposition A.682 The plane-stress law ¶
        • depends_on proposition 30.44 The speed ratio is fixed by Poisson's ratio ¶
        • depends_on remark 30.36 How the moduli are actually measured ¶
    • depends_on proposition 30.34 Minimum of the potential energy ¶
      • depends_on example 30.35 From single crystals to the isotropic aggregate ¶ ↺
    • depends_on theorem 30.42 Uniqueness in elastodynamics ¶
  • depends_on proposition 5.94 Principal axes of a real quadratic form ¶ ↺
  • depends_on remark 7.107 What the Hessian is for ¶
  • depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
    • depends_on corollary 5.89 Rayleigh–Ritz ¶
    • depends_on proposition 5.88 Rayleigh quotient of a definite pencil ¶
      • depends_on corollary 5.91 The operator norm of a real array ¶
        • depends_on proposition 5.92 The eigenvalues do not control the norm ¶
      • depends_on corollary 5.89 Rayleigh–Ritz ¶ ↺
  • depends_on theorem 28.33 Normal modes ¶
    • depends_on example 28.39 The linear triatomic molecule ¶
    • depends_on proposition 28.40 Modes and dispersion of the loaded string ¶
    • depends_on remark 28.34 Part II owes the simultaneous diagonalization of two quadratic forms ¶
  • depends_on theorem 10.8 The type is a coordinate invariant ¶ ↺