Bounded Geodesics for the Atiyah-hitchin Metric

The Atiyah-Hitchin metric has bounded geodesies which describe bound states of a monopole pair. Introduction. The dynamics of two nonrelativistic BPS monopoles was described by Manton [1] as the geodesic flow on the space of collective coordinates of the monopoles M$ with a special metric found explicitly by Atiyah and Hitchin [2]. Gibbons and Manton [3] studied the asymptotic metric (the Taub-Nut metric) and using its additional symmetry they integrated the equations of geodesies. They found in particular quasiperiodic solutions which describe bound states of a pair of monopoles. It is thus natural to treat the Atiyah-Hitchin metric as a small perturbation of the Taub-Nut metric and to apply the KAM theory to establish the existence of quasiperiodic geodesies. In the present note we sketch an implementation of this idea. The detailed exposition will appear elsewhere. 1. Analytic description of the Atiyah-Hitchin metric on M$. The Atiyah-Hitchin metric on the four-dimensional manifold M° admits SO(3) as a symmetry group and the orbits of the action are nondegenerate, i.e., 3-dimensional with only one exception. Hence we can identify the tangent space to the orbit with the Lie algebra so(3) and write the metric in the form (1) ds = fdrj + aa\ + ba\ + c)-^ d, E{k) = f {Ik sin ) d<\> Jo Jo are complete elliptic integrals and k' = v^l — & is the conjugate modulus. Received by the editors May 2, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 58F05, 53C80. Research supported in part by NSF Grant DMS-8601897. ©1988 American Mathematical Society 0273-0979/88 $1.00 + $.25 per page 179 180 M. P. WOJTKOWSKI In the formulas (2) k is assumed to be a function of r\ which can be chosen arbitrarily for the price of changing the function ƒ appropriately. Gibbons and Manton [3] suggested the use of r\ = 2if(fc), 7r < r\ < +oo, which leads to (3) f = {KE)E/K{E Kk'). 2. The reduced hamiltonian system. The SO(3) symmetry of the Atiyah-Hitchin metric (1) allows the reduction of the geodesic equations to the Euler type equations dM1 / 1 1 \ dM2 / 1 1 \ d^_dH_ dp _ dH dt " dp ' , which leads to the system dr\ _dH d4> _ dH dp _ dH dM3 _ dH ( ' dt~ dp' dt~ dM 3 ' dt~ dri' dt d(j> where \ p (1 M) cos d> (1 M) sin <* M 2 j , / a 6 c J Transforming H further, we get H = HQ + Hi, where 1 \p 1 / 1 1 \ „ l i r». M$] ^ i Ka-e) ( | ) c o s ^ The hamiltonian Ho does not depend on so it defines an integrable hamiltonian system, the other integral being M3. Moreover there are bounded quasiperiodic motions in the system. Indeed Ho = ^ + V(V,M3), F(„ ,M 3 ) = ± ( ^ + p ) ( l M f ) + ^ i THE ATIYAH-HITCHIN METRIC 181 and for a fixed value of M3, V has a global minimum Vo{M^), at least for small values of M3. To see this note that 1/a + 1/6 is decreasing to zero and 1/c is increasing to some positive value as r\ —* +00. The manifold {Ho = const, M3 = const} for values of Ho close to Vo{Mz) is compact and hence by the Liouville-Arnold Theorem it must be the torus carrying a quasiperiodic motion. 3. KAM theory. We want to treat the hamiltonian system (5) as a perturbation of the integrable system with the hamiltonian HQ. TO apply the KAM theory we have to find the action-angle variables for HQ and to estimate the perturbation. Expanding K in the conjugate modulus k' we have K = ln(*74)(l + 0(k')) + 0{k') as k' 0. Hence (6) k' = 0(e-") as r? — +00. Also E = 1 + ln(fc74)0(A;) + 0(k') as k' -> 0 so that (7) E = 1 -h Oirje-) as rj-+ +00. Applying (6) and (7) to the formulas (2) and (3), we get ï ( ? + p) = 2 ^ ) + 0 < , " V ' ) ' (8) 2 c 8 f ± = -±^ + 0(r,e-»), and We introduce the integrable hamiltonian *»=te5^ï(^-'-;)By (8) #01 = H0 #00 = Ofae"^) and also #1 = O ^ ^ " ^ ) . We will find the action variables / , J for the hamiltonain .Hbo m the region of the phase space where the motion is bounded. Let Hoo = 5Pw/(w — 2) + W(r/, M3) and let 170 (M3) be the value of r\ at which W attains its minimum, i.e., (dW/drf)(rjo,M3) = 0. We will use e = 1/rjo as a small parameter. We have e e 2 = i M | . We choose the basic cycles for the torus {Ms = const, Hoo = const} to be 71 = {M3 = const, p = const, 77 = const, 0 < <\> < 2TT} and 72 = {^3 = const, 182 M. P. WOJTKOWSKI = const, Hoo = const} and then I = 7T l pdrf + M3d = M3, 2 7 r A i = 2 T T / pdri + M3d=— I pdrj. To evaluate the last integral we make for a fixed Ms the change of variables tf = (1 + fj)/e, p = ep. We have Hoo = \e-e + ±e (ïêîH( i r£s) and J = (l/2ir)fl2pdfj. By straightforward integration