By Anthony D. Blaom

The perturbation idea of non-commutatively integrable structures is revisited from the perspective of non-Abelian symmetry teams. utilizing a coordinate procedure intrinsic to the geometry of the symmetry, we generalize and geometrize famous estimates of Nekhoroshev (1977), in a category of platforms having virtually $G$-invariant Hamiltonians. those estimates are proven to have a normal interpretation by way of 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**

We INEQUALITIES FOR POISSON BRACKETS (ASSUMPTION C) Ê (p∗ ), there exists pÊ 0 ∈ Br ρ¯(p∗ ) such that |p0 − p0 | (γ−δ)r ρ¯ Since p0 ∈ Brρ¯ compute |pst0 − Brρ¯| |pst0 − pÊ 0| 43 (γ − δ)rρ¯. We |pst0 − p0 | + |p0 − pÊ 0| 4k2 (e − 1) u p∗ ,r γ δ¯ σ t0 + (γ − δ)rρ¯ . 36 1 8k2 (e − 1) t0 σ ¯ ρ¯δ 2 r . 36 holds. 37 c2 ≡ 1 . 8k2 max{ζ, e − 1} Arguments like the above also show that |pt − p0 | |t| t0 δ rρ¯ 2 (|t| t0 ) . With the above choice of c2 , Assumption B of Chap. 5 is thus established.

2πi αk¯ , p ) . Consequently λp = ad−1 p diagonalizes with diagonal −1 1 1 −1 ,... , , ,... 2πi αk¯ , p 2πi α1, p 2πi α1, p 2πi αk¯ , p . BOUNDS ON λp 38 It follows that for any ξ ∈ t⊥ , 1 2π |λp ξ| 1 ¯ | αj , p | k sup 1 j |ξ| . Since αj , a ∈ R for all a ∈ t, | αj , p | | αj , Re p |. Also, every positive root αj ¯ (1 j k) can be written as a linear combination of the basis roots α1, . . , αk with positive coefficients. 15 1 2π |λp ξ| 1 k | αj , Re p | sup 1 j |ξ| (ξ ∈ t⊥ ) . 16 αj = 2κ(α∗j ) .

As the proof is, for the most part, a rewriting of Lochak’s (1992) proof of Nekhoroshev’s theorem for conventional action-angle coordinates (see also (Lochak, 1993)), it is relegated to Appendix A. In Chap. 6 we will demonstrate that our abstraction applies to action-group coordinates. Throughout this chapter, the reader may find it helpful to keep the example of conventional action-angle coordinates in mind. In what follows this corresponds to the choice G = Tn and t = Rn . Notation Let M be a compact C 0-manifold with (possibly void) boundary whose interior int M is a complex (resp.