proposition 29.43 The three integrable heavy tops

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1572 · p. 994

Rests on

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 ¶
depends_on proposition 29.36 Three first integrals ¶
- depends_on equation 29.35 eq:rigid-top-lagrangian ¶
- depends_on theorem 21.36 Conservation of the conjugate momentum ¶
  - depends_on definition 21.35 Cyclic coordinate ¶
  - depends_on equation 21.39 eq:lag-el-conservative ¶
  - depends_on equation 21.43 eq:lag-generalized-momentum ¶
  - proves proof ch:04-lagrangian-mechanics@proof-8 ¶
- depends_on theorem 21.43 Conservation of energy ¶
  - depends_on equation 21.51 eq:lag-energy-dot-cons ¶
  - depends_on equation 21.46 eq:lag-energy-h ¶
  - depends_on theorem 19.32 Conservation of energy ¶
    - depends_on definition 19.31 Energy ¶
      - depends_on definition 19.30 Conservative force and potential energy ¶
      - depends_on definition 19.29 Kinetic energy ¶
    - depends_on equation 19.19 eq:newt-conservative ¶
    - depends_on equation 19.4 eq:newt-fma ¶
    - proves proof ch:02-newtonian-dynamics@proof-9 ¶
  - proves proof ch:04-lagrangian-mechanics@proof-11 ¶
- proves proof ch:12-rigid-body-rotating-frames@proof-23 ¶
proves proof ch:12-rigid-body-rotating-frames@proof-28 ¶

Supports

depends_on remark 29.45 What integrability buys, and where it stops ¶
depends_on remark 29.44 The Kovalevskaya integral ¶

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
`cites`	→	Theoria motus corporum solidorum seu rigidorum	derived	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1579
`cites`	→	Sur le problème de la rotation d'un corps solide autour d'un point fixe	derived	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1586
`cites`	→	Méchanique analitique	derived	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1583
`depends_on`	→	The two integrals of torque-free motion	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1591
`depends_on`	→	Three first integrals	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1591
`depends_on`	←	What integrability buys, and where it stops	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1712
`depends_on`	←	The Kovalevskaya integral	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1620
`proves`	←	ch:12-rigid-body-rotating-frames@proof-28	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1594