definition 24.37 Casimir function

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1249 · p. 863

ground object -- no derivation owed

Rests on

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 ¶
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 13.103 Exterior derivative ¶
      - depends_on definition 13.98 $k$-form ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - 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 ¶

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 ¶
depends_on theorem 24.38 Symplectic foliation, quoted ¶

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`	→	Poisson manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1254
`depends_on`	→	Poisson bracket from the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1254
`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
`depends_on`	←	Symplectic foliation, quoted	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1270