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
^
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