A Geometric Setting for Hamiltonian Perturbation Theory by Anthony D. Blaom

By Anthony D. Blaom

The perturbation concept of non-commutatively integrable structures is revisited from the perspective of non-Abelian symmetry teams. utilizing a coordinate method intrinsic to the geometry of the symmetry, we generalize and geometrize recognized estimates of Nekhoroshev (1977), in a category of platforms having virtually $G$-invariant Hamiltonians. those estimates are proven to have a usual 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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Because Dγ−δ/2 (p∗ , r) is compact, the solution (gt , pt ) ∈ Dγ−δ/2 (p∗ , r) is well-defined until it reaches the boundary of Dγ−δ/2 (p∗ , r). 32 |gst0 − G|S = (γ − δ/2)¯ σ , |pst0 − Brρ¯(p∗ )| = (γ − δ/2)rρ¯ . 28, st0 gst0 − g0 = gt ξu (gt , pt )dt . 2 1) . 33 2k2 k3 (e − 1)¯ ρ . 34 Remark. Notice that ζ → ∞ as the real ball B2ρ¯(¯ t0 . Since g0 ∈ G(γ−δ)¯σ , there exists g0Ê ∈ G such that |g0 − g0Ê|S compute |gst0 − G|S |gst0 − g0Ê|S |gst0 − g0 |S + |g0 − g0Ê|S u pγ∗ ,r t0 + (γ − δ)¯ 4k2 ζ σ .

2, these generators are given ξP (g, p) = (Adg−1 ξ, 0)g,p . One therefore computes JξG (g, p) = p · (Adg−1 ξ) = (Adg p) · ξ = ϕ−1 (Adg p), ξ . Whence a momentum map JG : P → g∗ is given by JG (g, p) = ϕ−1 (Adg p) = Ad∗g−1 ϕ−1 (p) . 3 Recall (see p. 9) that ϕ maps the Weyl chamber W in g∗ bijectively onto the chamber t0, so that ϕ−1 (p) ∈ W ⊂ t∩g∗reg for all p ∈ t0. In particular, JG (P ) ⊂ g∗reg , so that (G × t0, ωG , G, JG ) is a Hamiltonian G-space satisfying our orbit type conditions. 2 in the special case G = Tn .

T0 2 Assumption C (The existence of a ‘Cauchy inequality’ for Poisson brackets). There exist constants c3 > 0 and c4 > 0, independent of m, M, γ, p∗ and r, such that (gt , pt ) ∈ Dγ−δ/2 (p∗ , r) with |pt − p0 | NEKHOROSHEV’S THEOREM 31 for all 0 < δ γ and all u, v, W ∈ A ((Dγ (p∗ , r)) with W of the form W (g, p) = w(p) one has the estimates u γp∗,r v γp∗,r c3 p∗ ,r {u, v} γ−δ σ¯ ρ¯ δ 2r {W, v} p∗ ,r γ−δ v ∇w(p) sup c4 γrρ ¯ p∈Brρ¯ (p∗ ) (Recall that {u, v} ≡ Xv p∗ ,r γ δ2 . ) Nekhoroshev’s theorem Write Ω(p) = ∇h(p) ∈ t and define the ‘time scales’ Tm ≡ 1 ρ¯m TM ≡ 1 ρ¯M −1 TΩ ≡ sup |Ω(p)| −1 Th ≡ p∈B ρ¯ ρ¯ sup sup |D h(p)(u, v, w)| 2 3 p∈B u,v,w∈Ø and the associated ‘dimensionless constants’ c5 ≡ Th Tm c6 ≡ TM TΩ c7 ≡ TM m .

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