34
P. DEIFT, L. C. LI, AND C. TOMEI
Theore m 2.113. Let
e
t l o g ( M o ( A ) )
=
^
+ (
^
A )
^ _ ^
A )
be the factorization (2.110) above. Then
M(t,X)=g+(tA)-1M0{\)g+(t,\)
= g-(t,\)Mo(\)g-(t,\)-1
solves the flow
(2.114)
^ % ^ = [0r.logAf(i,.))(A) ,M(t,A)] ,
M(0,A) = Afo(A)
generated by the Ad* -invariant Hamiltonian
H(A) = ]im [^ ti{A(\)logA(\)-A(\))^f (2.116)
rtoo J_ir 1 Az
on g*. ,and at integer times interpolates the Moser-Veselov algorithm
M(k,\) = Mk(\)/(l-\2) , ke 2Z . (2.117)
For all times t, M(£, A) has the form
M(t, A) = (I - AM(0 - A2 J 2 ) / (1 - A2) , (2.118)
where M(t) is real and skew, and M(k) = Af* for all k £ ZZ .
Proof: Equation (2.115) follows from jR-matrix theory and can be verified directly.
One obtains the formula
and one needs
'-g.(t, A)" 1 = (TT_ log M(t, •) )(A) . (2.119)
dff-(t,A)_ , . _ !
dt
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