A geometric setting for Hamiltonian perturbation theory by Anthony D. Blaom

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 process intrinsic to the geometry of the symmetry, we generalize and geometrize recognized estimates of Nekhoroshev (1977), in a category of structures having virtually $G$-invariant Hamiltonians. those estimates are proven to have a ordinary 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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Extra info for A geometric setting for Hamiltonian perturbation theory

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We will then describe the dynamics associated with an averaged form of the perturbed Hamiltonian H. This will highlight the role played by the symmetry in determining the nature of the perturbed dynamics, as well as motivate the Nekhoroshev estimates developed in subsequent sections. 10 the equations of motion for a Hamiltonian H0 (g, p) = h(p) are g˙ = gΩ(p) p˙ = 0 , where Ω(p) ≡ ∇h(p) ∈ t. We conclude that all integral curves t → (gt , pt ) of XH0 are of the form gt = g0 etΩ0 pt = p0 , where Ω0 ≡ Ω(p0 ).

5 Lemma. 1. The fibers of JG are the T -orbits. 2. If O ⊂ g∗reg is a co-adjoint orbit, then (JG )−1 (O) is some G-orbit G × {p} (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.

1 H(g, p) = h(p) + F (g, p) . The first task of this section is to describe the dynamics associated with the unperturbed part H0 (g, p) = h(p) of such a Hamiltonian. We will then describe the dynamics associated with an averaged form of the perturbed Hamiltonian H. This will highlight the role played by the symmetry in determining the nature of the perturbed dynamics, as well as motivate the Nekhoroshev estimates developed in subsequent sections. 10 the equations of motion for a Hamiltonian H0 (g, p) = h(p) are g˙ = gΩ(p) p˙ = 0 , where Ω(p) ≡ ∇h(p) ∈ t.

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