proposition 24.41 The free rigid body is a Lie–Poisson system

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1328 · p. 864

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 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 definition 18.10 Velocity ¶
      - depends_on definition 19.3 Mass ¶
  - 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 definition 18.8 Inertial frame of reference ¶
      - depends_on definition 19.5 Linear momentum ¶ ↺
- 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 definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - 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 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 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      - 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 5.37 Linear transformation ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on lemma 5.38 Exchange and completion ¶
      - 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 ¶
- 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 equation 13.223 eq:mfd-epsilon-density ¶
      - depends_on example 13.94 The two densities this chapter uses ¶
      - 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 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:07-symplectic-geometry@proof-13 ¶

Supports

depends_on remark 24.48 Reduction is what physicists do without saying so ¶
- depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
depends_on remark 24.42 The intermediate axis ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Casimir function	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1353
`depends_on`	→	Lie–Poisson bracket	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1353
`depends_on`	→	Euler's equations	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1353
`depends_on`	←	Reduction is what physicists do without saying so	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1655
`depends_on`	←	The intermediate axis	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1448
`proves`	←	ch:07-symplectic-geometry@proof-13	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1357