# Download Communications in Mathematical Physics - Volume 193 by A. Jaffe (Chief Editor) PDF

By A. Jaffe (Chief Editor)

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**Extra resources for Communications in Mathematical Physics - Volume 193**

**Example text**

Our estimates for the local dynamics near the manifold M0 in Sect. 1 make no use of the Hamiltonian H1 , hence they hold without change. 2 can also be proved using the function K0 instead of H0 , noting that H1 ≡ 0. Indeed, Eq. (105) ensures that K0 has the same type of Taylor expansion near the resonant circle C as H0 does. Furthermore, the change of K0 along perturbed solutions during passages near the manifold M can be computed as (cf. (67)) N −1 N −1 Ti∗ K0 (qi ) − K0 (pi ) = i=1 i=1 K˙ 0 (x(t)) dt 0 N −1 Ti∗ DK0 · ω (H0 ) + g = i=1 N −1 Ti∗ {K0 , H0 } + DK0 , g = i=1 x(t) 0 x(t) dt dt (108) 0 N −1 Ti∗ DK0 , g = i=1 0 x(t) dt, where we used (104).

The curvature F of D will be called the universal curvature, it is a End(E) valued two form on X × A. With respect to the natural decomposition of two forms on X × A: 2 (X × A) = 2 (X) 1 (X) 1 (A) 2 (A), we decompose F into corresponding three components, F = F2,0 + F1,1 + F0,2 . They are given explicitly by F2,0 (x, A) = FA at x, F1,1 (x, A)(v, B) = B(v) at x, F0,2 = 0. where (x, A) ∈ X × A, v ∈ Tx X and B ∈ 1 (X, End(E)). Now, we want to use A to parametrize certain families of first order elliptic operators on X by coupling different connections with a fixed differential operator.

The circle C is surrounded by periodic solutions in the plane Π. Introducing the actionangle variables (I, φ) ∈ R × S 1 by letting uK = Ieiφ , we can rewrite Eq. (114) in the form I˙ = ( cos φ − αI), φ˙ = 2( 2 − I 2) − I sin φ. (115) For = 0, this system is a one-degree-of-freedom Hamiltonian system with Hamiltonian HΠ ≡ H0 |Π = 2K 2 1 I − 2 (1 + 2 h h 2 2 h ) log(1 + h2 I 2 ) , and with the symplectic form ωΠ ≡ ω|Π = −KI dφ ∧ dI, 2(1 + h2 I 2 ) which is clearly nondegenerate. Linearizing (111) about any point of the circle C, one finds that for π < < ∞, if K = 3, 3 tan K 2π π < < K tan , if K > 3, (116) K tan K K off the plane Π, the linearized system possesses one positive, one negative, and K − 2 pairs of pure imaginary eigenvalues (see Li and McLaughlin [31]).