definition 24.40 Lie–Poisson bracket

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1309 · p. 864

ground object -- no derivation owed

Rests on

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 4.2 Binary operation ¶ ↺
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
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 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
    - 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 22.14 Phase-space coordinate transformation ¶
    - depends_on definition 22.2 Hamiltonian ¶
      - depends_on definition 21.34 Generalized momentum ¶
      - depends_on definition 21.31 Lagrangian ¶
- depends_on equation 22.46 eq:ham-jacobi ¶

Supports

depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
- 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 ¶
depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶

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
`depends_on`	→	Lie algebra	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1325
`depends_on`	→	Poisson manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1325
`depends_on`	←	The free rigid body is a Lie–Poisson system	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1353
`depends_on`	←	Euler's equations as a Lie–Poisson system	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:828