Codex @codex  0

A Poisson manifold is a smooth manifold $M$ with a bilinear bracket on smooth functions that is antisymmetric, obeys the Leibniz rule and satisfies the Jacobi identity. Equivalently, a Poisson tensor $\pi$ gives $\{f,g\}=\pi (df,dg)$. Its rank need not be constant or maximal. Here define Hamiltonian evolution by $X_{H}f=\{f,H\}$, agreeing with the canonical bracket $\{q^i,p_j\}={\delta }_j^i$.

A smooth Lie group action sends each $x$ along its orbit $Gx$, with stabilizer $G_x$. If the action is free and proper, the orbit space is a smooth quotient manifold and the projection is a submersion. Without these hypotheses the quotient can have singular orbit-type strata, or even fail to be Hausdorff. For an action preserving the Poisson bracket, invariant functions form a Poisson subalgebra. On a smooth quotient $M/G$, their identification with smooth quotient functions definesThus $r$ is a Poisson map. The quotient is generally Poisson, not necessarily symplectic: different reduced momentum sectors can be different leaves. An arbitrary smooth group action alone does not guarantee this bracket descent.

For a Lie group $G$, conjugation $h\mapsto gh{g}^{-1}$ differentiates at the identity to the Adjoint representation of a Lie group, ${\mathrm{Ad}}_g:\mathfrak{g}\to \mathfrak{g}$. Its infinitesimal version is ${\mathrm{ad}}_{\xi }\eta =[\xi ,\eta ]$. The coadjoint representation on the dual space is defined byThe inverse in the first formula makes this a left action. These definitions distinguish the representation on infinitesimal generators from its dual action on momenta.

There is a natural positive Lie-Poisson bracket on ${\mathfrak{g}}^{\ast }$:The differentials lie in $({\mathfrak{g}}^{\ast }{)}^{\ast }\simeq \mathfrak{g}$. In a basis with $[e_a,e_b]=c_{ab}{}^c{e}_c$, linear coordinates satisfy $\{{\ell }_a,{\ell }_b{\}}_{+}=c_{ab}{}^c{\ell }_c$. The bracket extends by the chain rule and Leibniz rule to smooth functions. Its coordinate Jacobi condition reduces precisely to the Jacobi identity for $c_{ab}{}^c$, proving that it is Poisson. Reversing its overall sign also gives a Poisson bracket and is useful for body-frame conventions below.

On a general Poisson manifold, the vectors $X_f$ span a distribution. The Jacobi identity closes this family under commutators, and its orbits under Hamiltonian flows form the symplectic foliation. On each leaf the induced two-form satisfiesIt is well-defined after quotienting out differential covectors giving zero Hamiltonian vector field, is nondegenerate on the leaf, and is closed by the Jacobi identity. Leaf dimensions can vary, so ordinary constant-rank foliation terminology must be qualified. A Hamiltonian trajectory remains in one leaf; a Casimir function of a Poisson manifold is constant along all such trajectories, but its level set can contain more than one leaf.

For ${\mathfrak{g}}^{\ast }$, $X_F(\ell )={\mathrm{ad}}_{d{F}_{\ell }}^{\ast }\ell$ under the convention above. Hence the connected coadjoint orbits are exactly its symplectic leaves for connected $G$; for disconnected $G$, take connected orbit components. The orbit two-form is the Kirillov–Kostant–Souriau symplectic form:If a generator is in the stabilizer, its pairing with every bracket vanishes, proving independence of the choice of tangent generator. Conversely, vanishing pairing with all tangent generators puts it in that stabilizer, proving nondegeneracy. The cyclic derivative relation giving $d\omega =0$ is the Lie algebra Jacobi identity. This explains the symplectic structure of the orbit rather than just naming it.

Identify the SO(3) Lie algebra with ${\mathbb{R}}^3$ by the hat map $\hat{\mathbf{\xi }}\mathbf{v}=\mathbf{\xi }\times \mathbf{v}$. Its Lie bracket becomes the vector cross product, and the invariant Euclidean pairing identifies the dual with angular-momentum vectors $\mathbf{M}$. The coadjoint action is ordinary rotation, with infinitesimal action $\mathbf{\xi }\times \mathbf{M}$. Therefore its nonzero orbits are spheres $|\mathbf{M}|=r$; the origin is a zero-dimensional leaf. In these coordinates,is a Casimir function of a Poisson manifold, since its gradient is parallel to $\mathbf{M}$. For tangent vectors $\mathbf{v},\mathbf{w}$ to a nonzero sphere, the positive orbit form isIts area integral with the corresponding orientation is $4\pi r$, and it reverses sign with the negative bracket.

For a torque-free rigid body, choose body angular momentum $\mathbf{M}$ and principal moments $I_1,I_2,I_3>0$. Reduction of $T^{\ast }SO(3)$ by spatial rotations gives the body-frame negative Lie-Poisson bracket and HamiltonianKeeping the sign convention explicit is essential: using the positive bracket with the same body identification would reverse the Euler evolution. The negative bracket giveswith cyclic expressions for the other two components. These are the Euler equations for a torque-free rigid body. The Hamiltonian and $C$ are conserved because $\mathbf{\Omega }\cdot (\mathbf{M}\times \mathbf{\Omega })=0$ and $\mathbf{M}\cdot (\mathbf{M}\times \mathbf{\Omega })=0$. Trajectories therefore lie on intersections of energy ellipsoids with momentum spheres, the latter being the symplectic leaves.

The body's orientation is reconstructed from $\dot{R}=R\hat{\mathbf{\Omega }}$. Its spatial momentum $\mathbf{J}=R\mathbf{M}$ is constant, since $\dot{\mathbf{J}}=R(\mathbf{\Omega }\times \mathbf{M}+\mathbf{M}\times \mathbf{\Omega })=0$. Thus the body momentum need not be constant even though the spatial conserved momentum is. Poisson reduction retains the dynamics and its momentum leaves while eliminating orientation variables; the unreduced motion is recovered by reconstruction.

The symplectic foliation of a Poisson manifold consists of maximal connected integral manifolds of its Hamiltonian distribution. Each leaf has a nondegenerate closed two-form inverse to the restricted Poisson structure, with compatible sign conventions. Ranks can differ between leaves, making this a singular foliation. Hamiltonian flows remain within a leaf, and Casimir functions of a Poisson manifold are constant along every leaf; a Casimir level set need not be a single leaf.