definition 29.35 The Lagrange top

Rests on

• depends_on definition 21.31 Lagrangian ¶
• depends_on proposition 29.11 Angular velocity in Euler angles ¶
  • 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 equation 13.223 eq:mfd-epsilon-density ¶
      • depends_on example 13.94 The two densities this chapter uses ¶
        • depends_on definition 13.92 Tensor density ¶
        • depends_on proposition 13.93 Algebra of densities ¶
      • depends_on notation 14.1 not:lie-indices ¶
      • proves proof ch:12-lie-groups-fibre-bundles@proof-29 ¶
    • depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
      • depends_on definition 14.26 The rotation group ¶ ↺
      • proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
  • depends_on notation 29.9 The $z$–$x$–$z$ Euler angles used in this treatise ¶
    • depends_on definition 14.26 The rotation group ¶ ↺
    • depends_on theorem 29.3 Euler's rotation theorem ¶
      • depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
        • depends_on equation 14.28 eq:lie-expso2 ¶
        • depends_on equation 14.25 eq:lie-so2-matrix ¶
        • depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶ ↺
        • proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
      • depends_on proposition 29.2 Six degrees of freedom ¶
        • depends_on definition 21.9 Degrees of freedom ¶
        • depends_on definition 29.1 Rigid body ¶
        • proves proof ch:12-rigid-body-rotating-frames@proof-1 ¶
      • proves proof ch:12-rigid-body-rotating-frames@proof-2 ¶
  • proves proof ch:12-rigid-body-rotating-frames@proof-7 ¶
• 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 axiom 7.1 Completeness of $\R$ ¶
      • depends_on definition 7.39 Darboux sums and the definite integral ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    • depends_on definition 13.5 Tensor under orthogonal transformations ¶
      • depends_on definition 13.4 Orthogonal coordinate transformation ¶
        • depends_on definition 13.3 Orthogonal coordinate system ¶
    • depends_on definition 19.42 Inertia tensor ¶
      • depends_on definition 13.5 Tensor under orthogonal transformations ¶ ↺
      • depends_on definition 19.41 Moment of inertia about an axis ¶
        • depends_on definition 19.39 Body ¶
        • depends_on definition 19.3 Mass ¶
  • 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 theorem 8.19 Fundamental theorem of algebra ¶
        • depends_on theorem 8.18 Liouville ¶
        • proves proof ch:06-complex-analysis@proof-13 ¶
      • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
        • depends_on definition 5.70 Characteristic polynomial ¶
        • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
        • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • proves proof ch:03-linear-algebra-representations@proof-26 ¶
      • proves proof ch:03-linear-algebra-representations@proof-27 ¶
    • depends_on definition 5.41 Adjoint ¶
      • depends_on definition 5.18 Inner product ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶ ↺
    • depends_on theorem 5.40 Rank–nullity ¶
      • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.14 Linear independence ¶
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
      • depends_on lemma 5.38 Exchange and completion ¶
        • depends_on definition 5.15 Basis ¶ ↺
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
        • depends_on definition 5.14 Linear independence ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-11 ¶
      • proves proof ch:03-linear-algebra-representations@proof-13 ¶
    • proves proof ch:03-linear-algebra-representations@proof-31 ¶
  • proves proof ch:12-rigid-body-rotating-frames@proof-10 ¶

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