Student Seminar: Classical and Quantum Integrable Systems

are called the Euler-Arnold equations. They are equivalent to the spectral-dependent
Lax equations
d
(I 2 + 1 )
dt λ J = [λI + Ω, I 2 + 1 λ J] .
The later are known as the Manakov equations.
The Kepler problem. Another interesting Lax pair can be found for the Kepler
problem (M.Antonowicz and S.Rauch-Wojciechowski). Introduce the following L
and M matrices which depend on three different parameters λ 1 , λ 2 , λ 3 :
⎛
L = 1 ⎜
⎝
2
− ∑ 3
i=k
− ∑ 3
i=k
x k ẋ k
λ−λ k
ẋ k ẋ k
λ−λ k
∑ 3
i=k
∑ 3
i=k
⎞
x k x k
λ−λ k
x k ẋ k
λ−λ k
⎟
⎠ , M =
( ) 0 1
,
k
0
r 3
where r = √ x 2 1 + x 2 2 + x 2 3 and x k are coordinates of the particle, while p k = x˙
k
are the corresponding conjugate momenta. Newton’s equation for x k arises as the
condition of vanishing of the residue of the pole λ = λ k .
3.3 The Zakharov-Shabat construction
There is no general algorithm how to construct a Lax pair for a given integrable
system. However, there is a general procedure of how to construct consistent Lax
pairs giving rise to integrable systems. This is a general method how to construct
the spectral dependent matrices L(λ) and M(λ) such that
˙L(λ) = [M(λ), L(λ)]
are equivalent to the eoms of an integrable system.
The basic idea of the Zakharov-Shabat construction is to specify the analytic properties
of the matrices L(λ) and M(λ) for λ ∈ C.
Let f(λ) be a matrix-valued function which has poles at λ = λ k ≠ ∞ of order
n k . We can write
f(λ) = f 0 }{{}
const
+ ∑ k
f k (λ),
f k (λ) =
} {{ }
polar part
∑−1
r=−n k
f k,r (λ − λ k ) r .
Around any λ k this function can be decomposed as
f(λ) = f + (λ) + f − (λ) ,
where f + (λ) is regular at λ = λ k and f − (λ) = f k (λ) is the polar part.
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