By Anthony D. Blaom
ISBN-10: 0821827200
ISBN-13: 9780821827208
The perturbation concept of non-commutatively integrable platforms is revisited from the perspective of non-Abelian symmetry teams. utilizing a coordinate approach intrinsic to the geometry of the symmetry, we generalize and geometrize famous estimates of Nekhoroshev (1977), in a category of platforms having nearly $G$-invariant Hamiltonians. those estimates are proven to have a average interpretation when it comes to momentum maps and co-adjoint orbits. The geometric framework followed is defined explicitly in examples, together with the Euler-Poinsot inflexible physique.
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Additional info for A geometric setting for Hamiltonian perturbation theory
Example text
That one arrives at the following estimate for the action variables is clear: |t| |pt − p0 | r0 b (gt , pt ) ≡ φ−1 (xt ) . 3, we have J(xt ) = (JG ◦ φ−1 )(xt ) = gt · ϕ−1 (pt ) and compute |J(xt) − O| J(xt ) − gt g0−1 · J(x0) = gt · ϕ−1 (pt ) − gt · ϕ−1 (p0 ) = ϕ−1 (pt ) − ϕ−1 (p0 ) = |pt − p0 | , since the isomorphism ϕ : t → t is isometric. This proves that the estimate claimed in the corollary indeed holds. Part 2 Geometry CHAPTER 8 On Hamiltonian G-spaces with regular momenta In this chapter we assume some familiarity with the structure of compact Lie groups, say with what has been summarized in Chap.
Arrows indicate basis roots. The chamber t0 is shaded. Bounds on λp Recall from Chap. 2 that λp (p ∈ t0 ) is defined as the inverse of adp : t⊥ → t⊥ . 14, we will need to estimate the operator norm of λp before proceeding to verify Assumptions B and C. 14 Remark. Since the operator adp : t⊥ → t⊥ becomes singular as p approaches the walls ∂t0 of t0 , the norm of λp is unbounded as p → ∂t0. We will see later that this results in a deterioration in the estimates of Assumptions B and C as p¯ → ∂t0.
1 says that we have a natural way of identifying the abstract quotient (G × U)/T with V ≡ JG (G × U) ⊂ g∗ , the momentum map JG : G × U → V being a realization of the natural projection G × U → (G × U)/T . , Marsden and Ratiu (1994, Chapter 10)). If u is a smooth function on g∗ , then by definition its corresponding Hamiltonian vector field Xu on g∗ is the vector field satisfying the equation Xu df = {f, u}+ for all smooth f : g∗ → R. Hamiltonian vector fields on a Poisson manifold are always tangent to the symplectic leaves, which in this case are the co-adjoint orbits.
A geometric setting for Hamiltonian perturbation theory by Anthony D. Blaom
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