In particular, the following result was proved, following an approach generally called the Weinstein construction: Let O be a coadjoint orbit lying in the image of the standard moment map µ. Then every smooth symplectic layer (T∗Q//OK)(L) of the reduced space can be realized globally as. In Section 5 we compute the symbolic leaves of the reduced space W/K, and in Section 6 we give examples of our constructions.
The dimension of a collector consisting of a finite number of connected components is defined as the largest dimension of the components of the collector. The maps (U, ψ) and (V, φ) are called compatible if they are such at every point of intersection U ∩ V. Two singular atlases are said to be compatible if all maps of the first are compatible with all maps of the second atlas.
By the Tube Theorem there is a K-invariant open neighborhood U of the orbit K · x0 such that K ×HV ∼= U as smooth K-spaces where V is an H-invariant open neighborhood of 0 in Norx0(K · x0) . We can also think of the tube neighborhood as drawn back onto Si via the projection πi. Assume we are given a system of control data related to the stratification of X, and identify the tube neighborhoods of the strata with the corresponding inner product bundles.
Now we can equip X ×M Y with the product stratification given by strata of the form S ×M Y, which is the smooth fiber product of a layer S of X with Y over M.
Mechanical connection and Weinstein construction
Furthermore, X ×MY inherits a smooth structure in the sense of subsection 2.1 from the canonical topological inclusion X ×M Y ,→ X × Y. The smooth-structured singular space thus obtained is called the fiber product of X and Y over M .Since π : X → M is a set of unitary fibers, therefore, τ∗π : X ×M Y → Y is a set of unitary fibers, as well as with the same typical fiber F.
Since the K action on Q has only a single isotropy type, the orbital space Q/K is a smooth manifold and the projection π : Q → Q/K is a surjective Riemann immersion with compact fibers. However, the lifted action of K on T∗Q is already much more complicated, and the quotient space (T∗Q)/K is just a layered space in general. We define the dual horizontal subbundle of T∗Q as the subbundle Hor∗ consisting of the co-vectors that vanish on all vertical vectors.
Similarly, we define the double vertical subbundle of T∗Q as the subbundle Ver∗ consisting of those covectors that vanish on all horizontal vectors. The minus in CurvA's definition comes out, as we are dealing with leftist actions. Using this relation again, it follows that CurvA(v, ζZ2)(q) ∈ kq and this is already sufficient for the purpose of this article.
However, using [2, Proposition 4.7] and Feh's theorem for Riemann actions, the stronger result (ii) is also true. Using the horizontal lift map which identifies Hor ∼= (Q ×Q/KT (Q/K)) and the mechanical relation A we obtain an isomorphism. Through the Riemannian structure there is a dual version of this isomorphism, and to save on printing we will abbreviate it.
To put some notes on the following proposal and clarify the picture, consider the following collection of pull diagrams. This isomorphism can be used to induce an asymptotic form in the connection-dependent realization of T∗Q, namely σ = ψ∗Ω where Ω = −dθ is the canonical form in T∗Q. However, the proof can be simplified considerably by using the relation σ = ψ∗Ω = −ψ∗dθ, and we present this simplification.
Gauged Poisson reduction
However, it is also tricky on the other hand, as it involves a choice of signs in the definition of the fundamental vector field from section 1, and there are many possibilities of getting confused. The point here is the choice of sign in the definition of the fundamental vector field mapping ζ in Section 1. Note also that the momentum map of the cotangent bundle µ : T∗S → h∗ is given by hµ(s, p), Xi = θ(ζXa)( s, p) where θ is the Liouville form of T∗S and Ω = −dθ is the cotangent bundle symplectic form.
Then E|U is K-invariant, and if (L) is an element of the isotropy lattice for the K action on E, then the corresponding layer is downplayed as. Here L0⊆ H is an isotropy subgroup for the H action conjugated to L in K, and (L0)H is the conjugation class of L0 in H. In the case that K acts freely on Q, the first statement of the above can be theorem also found in Cendra, Holm, Marsden, Ratiu [8].
According to Davis [10] or also subsection 2.5, withdrawals are well defined in the category of stratified spaces, and it therefore makes sense to define T (Q/K) ×Q/K(F . q∈Qk⊥q)/K as a stratified space with smooth structure. Again, it follows from the definition of the smooth structures on the respective spaces that ϕ−10 is smooth. Since ξ takes values in the vertical subbundle V (%) and vertical bundles are integrable, its flow preserves fibers of %, that is, Flξt(q0, p0) = (q0, pt) for some curve ptin Tq∗0Q.
For the purpose of the lemma, we can assume that the vertical part of the first entry vanishes, i.e. A.T (eτ ◦ρ).∇e σF = 0 with notation as in diagram (1). Here T∗R(H) is the cotangent raised action of the right multiplication of H on K, and we use left multiplication to trivialize T∗K = K × k∗. Note in particular that (as in Hochgerner [14, section 4]) the isomorphism is symplectic since it comes as the cotangent lifting of a diffeomorphism of the base spaces.
Since we already know that the part of the Hamiltonian vector field of F which is tangent to T∗S is given by the local coordinates of (v(q)hor(π), η0([q], η)), we can Possibly. reduce the problem further to consider a function F ∈ C∞(K ×HAnn h)K = C∞(Ann h/H). Finally, CurvA0 is the induced form of Q/K associated with the mechanical connection A from Proposition 3.1. In the case where K acts freely on Q, the Poisson fitting of the reduced Poisson manifold T∗Q/K is determined in Zaalani [34] and in Perlmutter and Ratiu [23].
The first part of the theorem has already been checked in the proof of Theorem 4.4. Because of the striking similarity of the symbolic form Ωλ to that which appears in electromagnetism, Ωλ is called a magnetic symplectic form, and we think of the totally isotropic value of the momentum λ as the charge of the test particles moving in the electromagnetic field in the reduced configuration space. N/A
Symplectic leaves
In the universal reduction scheme of Arms, Cushman and Gotay [3], this morphism is actually used to endow µ−1A (λ)/Kλ with a Poisson structure. Therefore, it is reasonable to expect that the smooth symplectic leaves of W are the connected components of the smooth symplectic layers of W//OK. Here, L0 is an isotropy subgroup of the induced H action on O, and (LO)H denotes its isotropy class in H.
Here, L0 is an isotropic subgroup of the induced action H on O that is conjugate in K on L, and (L0)H denotes its isotropic class in H. iii). Finally, B is the form introduced in Proposition 3.2. where i = 1, 2 we have explicit formulas. iv). As before, (L) denotes the isotropic type of K-action on T∗Q, and (L0)H denotes the isotropic type of Ad∗(H)-action on O.
We want to make use of the Witt-Artin decomposition and thus denote the symplectic section of the H-action on O by λ by. where ΩO denotes the positive KKS form on O. The H action on O is Hamiltonian with momentum map O → h∗ given by restriction to h.). AeF (q, η, λ) as an element of T([q],η)∗ (T∗(Q/K)) through the isomorphism given by the dual of the horizontal lift with respect to the mechanical connection A on Q Q /K. The statement about the integrability of the characteristic distribution is tautological, since it is described as the tangent bundle of a smooth manifold.
Examples
The difference in sign comes from a different convention in defining the Hamiltonian vector field of a function.). We denote by Q the open and dense subset of elements (v, w) ∈ S9 ⊆ R5× R5 such that v and w are linearly independent. It is clear that Q is preserved by the SO(5) action, and moreover constitutes the regular stratum with respect to this action.
Writing K as a matrix group, we enclose H in the lower right corner in the usual way. The path space Q/K can be identified with the open disk B2 of radius 1 in R2, and the projection Q Q/K is a (non-main) fiber bundle with typical fiber K/H. According to Theorem 4.11, the simple Poisson is reduced space with respect to the canceled K-action of the form.
Thus, the singular Poisson reduced space with respect to the raised K-action is of the form. However, the induced Poisson structure is not obvious at all (if we did not have Theorem 4.11). Note also that the induced shape CurvA0 on B2 resulting from mechanical curvature is R-valued and t-factor-valued by Proposition 3.1.
Using a little more linear algebra, we calculate that is the simple normal space (see especially Subsection 5.3) for the H-action on O at λ. Gotay, A universal reduction procedure for Hamiltonian group actions, Geometry of Hamiltonian Systems (Berkeley, CA, 1989), Math. Ratiu, Lagrange reduction, Euler-Poincar equations and semidirect products, Geometry of differential equations, Amer.
