38 P. DEIFT, L. C. LI, AND C. TOMEI

In particular |flr+(i,0)| = I and so £+(t,0) = Q+(t,0). But by (2.112), g+{t,0) is real.

Hence Q

+

(£,0) is real and orthogonal, and

M(t) = Q+(t, 0)TMo Q+(t, 0) . (2.128)

Of course for t = 1, Q+(t,0) = tf+(t,0) = c^0r from (2.122), as it should (see (1.12)).

We identify o(N)* with o(N) through the standard, nondegenerate pairing (i?i, #2) =

tr B1B2. For smooth functions F{, 1 i 2, from G£(n,R) to (T, the Lie-Poisson bracket

on o(N)* is given by

{Fi,F

2

}

o ( J V )

.(M) = tr M[7T

0

(Vf\(m)) , 7r

0

(VF

2

(M))] (2.129)

where VF^(M) = ( f l , ) and TT0A =

(A-AT)/2.

Now as noted above, I - XM0-

A2

J

2

is positive definite on iR, and by (2.114), the same is true for I — XM(t) — A2 J 2 . On the

other hand, for a general matrix B G o(Ar),

/ - A 5 - A 2 J 2 0 on iR

(2.130)

\\u\\2 -~f(7r,iBu) + y2\\Ju\\2 0 for all 7 e R

IM-suP]l^lL2.

u*0 11^11111^111

Thus the dynamics t —• M(t) takes place on the Poisson manifold

o2(NY = {M € o(JV)* : |||M||| 2} . (2.131)

To show that the flow t —• M(t) is Hamiltonian on c2(N)*, we first consider a slightly

more general situation. Let F b e a real analytic function on (0, oo). Then the preceding

calculations show that for initial data

-4°(A ) = l - ^ ° - ^

2

_

M o € o2{Ny

^