remark 29.27 Euler's equations as a Lie–Poisson system

Rests on

• depends_on definition 24.37 Casimir function ¶
  • depends_on definition 24.36 Poisson manifold ¶
    • 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 equation 22.46 eq:ham-jacobi ¶
  • depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
    • depends_on definition 22.29 Poisson bracket ¶ ↺
    • depends_on definition 24.14 Hamiltonian vector field ¶
      • depends_on definition 24.8 Symplectic manifold ¶
        • depends_on definition 13.105 Closed form ¶
        • depends_on definition 13.98 $k$-form ¶
        • depends_on definition 24.2 Symplectic vector space ¶
      • depends_on equation 22.4 eq:ham-pdot ¶
      • depends_on equation 22.3 eq:ham-qdot ¶
    • depends_on equation 22.46 eq:ham-jacobi ¶ ↺
    • proves proof ch:07-symplectic-geometry@proof-4 ¶
• depends_on definition 24.40 Lie–Poisson bracket ¶
  • depends_on definition 5.127 Lie algebra ¶
    • depends_on definition 5.121 Algebra ¶
      • depends_on definition 4.4 Internal binary operation; magma ¶
        • depends_on definition 4.3 Algebraic structure ¶
        • depends_on definition 4.2 Binary operation ¶
      • depends_on definition 4.33 Vector space ¶
        • depends_on definition 4.8 Commutativity; abelian structure ¶
        • depends_on definition 4.32 Field ¶
        • depends_on definition 4.31 Module ¶
  • depends_on definition 24.36 Poisson manifold ¶ ↺
• depends_on proposition 29.26 The two integrals of torque-free motion ¶
  • depends_on equation 29.12 eq:rigid-kinetic-energy ¶
  • depends_on theorem 29.25 Euler's equations ¶
    • depends_on postulate 19.22 Newton's second law, rotational case ¶
      • depends_on definition 19.18 Angular momentum ¶
        • depends_on definition 18.3 Particle ¶
        • depends_on definition 19.5 Linear momentum ¶
      • depends_on postulate 19.8 Newton's second law ¶
        • depends_on definition 19.5 Linear momentum ¶ ↺
        • depends_on postulate 19.6 Newton's first law ¶
    • depends_on theorem 29.16 Principal axes ¶
      • depends_on definition 29.13 Inertia tensor, continuum form ¶
        • depends_on definition 7.125 Multiple integral ¶
        • depends_on definition 13.5 Tensor under orthogonal transformations ¶
        • depends_on definition 19.42 Inertia tensor ¶
      • depends_on proposition 29.15 Positivity ¶
        • depends_on definition 29.13 Inertia tensor, continuum form ¶ ↺
        • proves proof ch:12-rigid-body-rotating-frames@proof-9 ¶
      • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
        • depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
        • depends_on definition 5.41 Adjoint ¶
        • depends_on theorem 5.40 Rank–nullity ¶
        • proves proof ch:03-linear-algebra-representations@proof-31 ¶
      • proves proof ch:12-rigid-body-rotating-frames@proof-10 ¶
    • depends_on theorem 29.7 Transport theorem ¶
      • depends_on definition 29.4 Angular velocity ¶
        • depends_on definition 14.26 The rotation group ¶
        • depends_on lemma 14.62 The Levi-Civita symbol is invariant ¶
        • depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
      • depends_on proposition 29.5 The velocity field of a rigid body ¶
        • depends_on definition 29.4 Angular velocity ¶ ↺
        • depends_on equation 29.1 eq:rigid-constraint ¶
        • proves proof ch:12-rigid-body-rotating-frames@proof-3 ¶
      • proves proof ch:12-rigid-body-rotating-frames@proof-5 ¶
    • proves proof ch:12-rigid-body-rotating-frames@proof-16 ¶
  • proves proof ch:12-rigid-body-rotating-frames@proof-17 ¶

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