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§6. Rational Character of the Solution of (2.4) 61
−1
On account of the symmetry property K
LK = −L derived in Section 2, f(λ)
is an odd function of λ. Thus, if we order the (always distinct) eigenvalues by
λ
n>λn−1
>... >λ
1
we conclude that
λ
= −λ
k
n−k+1
; rk= r
n−k+1
and f(λ) can be represented by
where κ
n
Since f (λ) ∼ λ
f(λ)=
ν
k=1
2λr
λ2− λ
2
k
2
k
=1for n odd, κn=0if n is even and ν =[n/2].
−1
for |λ|→∞we have
n
2
r
=1.
k
k=1
+ κ
2
r
ν+1
n
λ
and we prefer to free ourselves from the latter restriction by using the r
projective coordinates. Therefore we set
n
2
r
/(λ − λk)
k
f(λ)=
The n variables r
k=1
n
2
r
k
k=1
, r2, ..., rν, κnr
1
describe the Jacobi matrix (2.1) uniquely up to scaling of the r
2
squares a
(k =1, 2, ..., n− 1) of the elements in (2.1) can be expressed
k
rationally in terms of those r
ν
2
2λr
/(λ2− λ
k=1
=
, λj, 1 j ν and r
j
k
ν
2r
k=1
, λ1, λ2, ..., λνcan be used to
ν+1
2
+ κnr
k
2
)+κn(r
k
ν+1
2
/λ)
ν+1
2
ν+1
. In fact, the
k
if n is odd. The reason
. (6.2)
for this fact lies in the representation of f(λ) as a continued function
(6.1)
as
k
f(λ)=
λ −
λ −
1
2
a
1
a
λ − a
(6.3)
2
2
.
.
.
2
n−1
λ
62 Three Integrable Hamiltonian Systems
which goes back to Stieltjes (used also in [12]). Since the computation of the
continued fraction from the partial fraction expression is a rational process one
finds that
2
a
= Rk(r, λ)(6.4)
k
where the R
are rational functions, homogeneous of degree zero in the rjand
k
homogeneous of degree two in λ.
Moreover, (6.4) can be viewed as mapping which takes the domain
D = {(λ, r),λ
>λ2>... >λν 0; rj> 0(j =1, 2, ..., n− ν)}
1
into the domain onto
D = {a
> 0,j=1, 2, ..., n− 1}
j
in such a way that the pre-image of each point is precisely one ray (ρr, λ) with
a scalar ρ>0.
We will show that in these homogeneous coordinates the differential equa-
tions take the simple form
˙
λ
=0; ˙rk= −λ
k
2
so that, via (6.4) the a
e
−λ
2
t
1
, ..., e
−λ
2
t
ν
.
appear as rational functions of exponentials
k
To prove this assertion we introduce the eigenvectors ϕ(λ
2
rk, (6.5)
k
) of L which we
j
normalize by
)=(e1,ϕ(λj)) > 0; |ϕ(λj)| =1. (6.6)
ϕ
1(λj
If L is a solution of (2.3) these eigenvectors become functions of which
evolve according to
,t)=U(t)ϕ(λj, 0),
ϕ(λ
j
where U(t) is the unitary matrix of Section 2. Thus the eigenvectors satisfy the
differential equation
,t)
dϕ(λ
j
= −Bϕ(λ
dt
We compute the resulting differential equation for the first component ϕ
=(e
,ϕ)
1
˙ϕ
= −a1a2ϕ
1
,t).
j
=
1
3
§6. Rational Character of the Solution of (2.4) 63
and use the equations resulting from (L − λ)ϕ =0
λϕ
+ a1ϕ2=0
1
− λϕ2+ a2ϕ3=0
a
1ϕ1
to express ϕ
in terms of ϕ1. One finds readily
3
a
1a2ϕ3
=(λ2− a
2
)ϕ
1
1
so that the differential equations for ϕ1become
˙ϕ
Finally, to show that the ϕ
= −(λ2− a
1
1(λk
2
)ϕ1. (6.7)
1
) are proportional to the rkwe write the
resolvent R(λ) in terms of the eigenvectors obtaining
2
k
f(λ)=(R(λ)e
1,e1
)=
(ϕ(λk),e1)
k
λ − λ
so that
r
ϕ
1
=(λk)
k
n
j=1
.
1/2
2
r
j
Thus the differential equations (6.5) give
2
2
r
j
j
ϕ
1(λk
).
˙ϕ
1(λk
)=−λ
2
−λ
k
It is easy to verify that
2
a
=
1
2
2
λ
r
j
j
j
j
−1
2
r
j
and the second equations of (6.5) have been verified. The first equations of (6.5)
are clear from the derivation.
2
Thus the solutions a
of (2.4) are rational functions of exponential functions.
k
We describe the solution for n =4. Computing the continued fraction of f (λ)
explicitly one finds
2
2
2
2
)2r
r
1
1
2
+ r
,
2
)
2
2
2
r
)(r
2
1
Inserting r
2
a
1
= rj(0)e
j
2
2
2
λ
r
1
1
=
2
r
1
−λ
2
+ λ
r
2
2
2
,a
2
+ r
2
a
2
t
j
we obtain the explicit solutions of (2.4).
2
2
λ
λ
1
2
=
3
2
λ
1
(λ
− λ
(λ
2
1
+ λ
2
r
1
+ r
2
2
2
1
r
2
+ λ
2
)
2
2
2
2
2
.
=
2
(r
2
2
r
1
64 Three Integrable Hamiltonian Systems
§ 7. The Scattering Problem Associated with the Equation of
Kac and Van Moerbeke
In order to study the asymptotic behavior of the solution of (2.5) we consider
= xk− x
u
k
as the difference between the positions x
,k=1, 2,...,n− 1(7.1)
k+1
of n particles on the line. If the x
k
satisfy the differential equations
1
u
u
k
˙x
k
where we formally set e
= −
(e
2
u
0
=0=e
clearly (2.5) follows. Conversely the x
and for any solution x
(t) of (7.2) also xk(t)+c is a solution giving rise to the
k
+ e
k−1
),k=1, 2, ..., n (7.2)
u
n
,orx0= −∞, x
are determined only up to translation
k
=+∞ then
n+1
same solution of (2.5), provided c is a constant. For simplicity we will assume
that n =2ν is even.
We ask for the asymptotic behavior of the solution of (7.2) for t →±∞and
the relation between the scattering data. We will show that any solution of (7.2)
behaves linearly for large t:
±
k
= α
t + β
−
n−2j+1
±
as t →∞
k
,j=1, 2, ..., ν, (7.3)
where
(±t) ∼±α
x
k
+
α
+
= α
2j
2j−1
= α
−
n−2j+2
i. e., the particles travel asymptotically in pairs, while the different pairs have
negative and different velocities, in fact, it turns out
where the λ
1>λ2
+
α
2j
>... are the eigenvalues of L.
= −2λ
2
,j=1, 2, ..., ν, (7.3)
j
We will also determine the relation between the phases. First of all, for the
neighbors we have the asymptotic distances
−
β
n−2j+1
β
+
2j−1
− β
− β
−
n−2j+2
+
= log(−2α
2j
= log(−2α
+
) = log(4λ
j
+
)
j
2
)
j
(7.4)
and for the phases of pairs with the same velocities
+
β
−
− β
2j
n−2j+2
= −
k<j
log 4(α
+
2k
− α
+
2j
)2+
k>j
log 4(α
+
2k
− α
+
)2. (7.5)
2j
k
7. The Scattering Problem 65
Thus the particles undergo a scattering in which the pairs behave as if they
interacted pairwise at a time.
The results (7.3), (7.3
), (7.4) are easily derived and we begin with their
proof. We recall the differential equation (2.4)
= ak(a
˙a
k
with a
=0=anfrom which we see that
0
along solutions. Thus a
we conclude that
Since ˙a
is bounded this implies that
2j
2
k+1
are bounded. Since
k
d
dt
a
2
− a
log(a
(t) → 0 as t → +∞. (7.6)
2j
),k=1, 2, ..., n− 1
k−1
n−1
2
a
=const
k
k=1
0
∞
1a3
a
···a
2
dt < ∞.
2j
2j−1
)=a
2
2j
Thus, the Jacobi matrix L(t), given by (2.1), is asymptotic to a matrix blocked
into two by two matrices with eigenvalues ±a
on the other hand the eigenvalues λ
that the limits a
2j−1
(t) → a
2j−1
are distinct and independent of t it follows
k
(∞) exist and agree with these eigenvalues in
(t), j =1, 2, ...,ν. Since,
2j−1
some order. From the differential equations
˙a
2j
= a
a
2j
and from (7.6) it follows that
2
a
(∞) <a
2j+1
and thus, if we order the eigenvalues λ
a
(t) → λj, (j =1, 2, ..., ν). (7.7)
2j−1
2
2j+1
2
− a
2j−1
2
(∞)
2j−1
of L according to (6.1) we conclude
k
66 Three Integrable Hamiltonian Systems
Using the relation
4a
2
u
xk−x
k
= e
k
= e
k+1
(7.8)
we conclude from (7.1), (7.2), (7.7), (7.8) that
(+∞)= ˙x
˙x
2j
(+∞)=−2λ
2j−1
2
j
proving (7.3) and the first part of (7.3). The other part follows by considering
the asymptotic behavior for t →−∞analogously.
Moreover, (7.7) and (7.8) implies that
− x2j→ log(4λ
x
2j−1
2
) for t → +∞
j
proving the first part of (7.4). The second follows similarly.
It remains to prove (7.5). This will be done by relating the first order
differential equations (7.2) to a second order system related to the Toda lattice, for
which the scattering problem has been solved [12]. We notice that differentiation
of (7.2) yields
1
u
k
(e
= −
˙x
k
2
1
u
k
{e
= −
4
1
xk−x
(e
= −
4
˙uk+ e
u
k+1
(e
k+2
u
k−1
− e
− e
˙u
k−1
u
k−1
x
k−2−xk
)=
)+e
)
u
u
k
− e
u
k−2
)} =
k−1
(e
where we set the undefined exponential terms equal to zero. Thus with
ξ
= x2j; τ = t/2(7.9)
j
we have
2
d
ξ
where
dτ
j
ξ
j−1−ξj
= e
2
− e
ξj−ξ
U =
j+1
=
ν−1
j=1
∂U
, (j =1, 2, ..., ν)(7.10)
∂ξ
j
ξj−ξ
j+1
e
.
This Hamiltonian system has already been established as an integrable one [13].
For the scattering one has again
ξ
(+∞)=ξ
j
(−∞),j=1, 2, ..., ν
ν+1−j
References 67
which is consistent with (7.3) as ξ
(±∞)=2α
j
±
2j
,and
ξ
(τ) − ξ
j
(−τ) − 2γjτ →
ν+1−j
k=j
δ
kj
(7.11)
where
− γj)2,k>j
log(γ
γ
= ξ(+∞)=2α
j
+
; δkj=
2j
−log(γ
k
− γj)2,k<j.
k
With (7.9) the relation (7.11) translates readily into the statement (7.5).
We conclude with a comment on the relation between the differential equa-
tion (2.5) by Kac and v. Moerbeke and the equations (7.10) for the Toda lattice.
The first one corresponds to an isospectral deformation the Jacobi matrix L given
by (2.1), with zeros in the diagonal, while the second-order differential equation
corresponds to such deformations of such Jacobi matrices with arbitrary diagonal
elements (see [4, 12]). To establish the connection between the two we form L
which is not any more a tridiagonal matrix, but is similar to one. In fact, with e
(α =1, 2, ..., n) denoting the unit vectors, one finds that L2leaves the spaces
E
=span{e1,e3, ..., e
1
reduces in each of these spaces to a symmetric Jacobi matrix. This explains why
the solutions of (2.4) are rationally expressible in terms of e
of the corresponding equations for the Toda lattice are rational in e
illustrates in a simple example how the operation L → L
} and E2=span{e2,e4, ..., en} invariant and
n−1
2
−λ
t
j
while solutions
−λjt
2
and more generally
.This
L → f (L) plays a role in these problems.
2
α
References
[1] V. I. Arnold, Sur la g´eom´etrie diff´erentielle des groupes de Lie de dimension
infinie et ses applications `a l’hydrodynamique, Ann. Inst. Fourier, Grenoble,
1966, 16, 319–361.
[2] F. Calogero, Solution of the one-dimensional n-body problems with
quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 1971,
12, 419–436.
[3] F. Calogero, C. Marchioro, Exact solution of a one-dimensional three-body
scattering problem with two-body and/or three-body inverse square potential, J. Math. Phys., 1974, 15, 1425–1430.
68 Three Integrable Hamiltonian Systems
[4] H. Flaschka, The Toda lattice, I, Phys. Rev., 1974, B9, 1924–1925.
[5] H. Flaschka, The Toda lattice, II, Progr. Theor. Phys., 1974, 51, 703–716.
[6] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Korteweg –
de Vries equations and generalizations,VI,Methods for exact solutions,
Comm. Pure Appl. Math., 1974, 27, 97–133.
[7] M. H´enon, Integrals of th e Toda lattice, Phys. Rev., 1974, B9, 1921–1923.
[8] M. Kac, P. van Moerbeke, On an explicitly soluble system of non-linear
differential equations related to certain Toda lattices, to appear.
[9] M. Kac, P. van Moerbeke, Some probabilistic aspects o f scattering theory,
to appear.
[10] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves,
Comm. Pure Appl. Math., 1968, 21, 467–490.
[11] C. Marchioro, Solution of a three-body scattering problem in one dimension,
J. Math. Phys., 11, 1970, 2193–2196.
[12] J. Moser, Finitely many mass points on the line under the influence of
an exponential potential — An integrable system. (See the article in this
collection.)
[13] M. Toda, Waves in nonlinear lattice, Progr. Theor. Phys. Suppl., 1970, 45,
174–200.
[14] B. Sutherland, Exact results for a quantum many-body problem in one di-
mension, II, Phys. Rev., 1972, A5, 1372–1376.
[15] V. E. Zakharov, and L. D. Faddeev, Korteweg – de Vries equations: Acom-
pletely integrable Hamiltonian system, Funk. Anal. i Pril., 1971, 5, 18–27.
Various Aspects of Integrable Hamiltonian Systems
1
§ 1. Integrable Hamiltonian Systems
a) In these informal lecture notes we discuss a number of integrable
Hamiltonian systems which have surfaced recently in very different connections.
It is our goal to discuss various aspects underlying the integrability of a system like that of group representation, isospectral deformation and geometrical
considerations. Since this subject is still far from being understood or being
systematic we discuss a number of examples which are seemingly disconnected.
In fact, there are some rather unexpected connections like between the inverse
square potential of Calogero (Section 4) and the Korteweg de Vries equation.
Here we show a surprising new connection between the geodesics on an ellipsoid
and Hill’s equation with finite gap potential.
b) The differential equations of mechanics can be written in Hamiltonian
form
where x =(x
phase space R
field X
H
, ..., xn) ∈ Rn, y =(y1, ..., yn) ∈ Rn, are coordinates in the
1
2n
defined by
For any function F the expression
is antisymmetric in F , H. It is called the Poisson bracket of F and H.The
Hamiltonian systems form a Lie algebra and
∂H
=
˙x
k
, ˙yk= −
∂y
k
∂H
(k =1, 2, ..., n)(1)
∂x
k
or an open subset of R2n. Thus a function H defines a vector
n
X
H
X
F =
H
n
k=1
=
[X
k=1
∂H
∂y
H,XG
k
∂H
∂y
k
∂F
∂x
k
]=−X
∂x
−
∂H
∂
−
∂x
k
∂H
∂F
∂x
∂y
k
{H, G}
∂
∂y
k
k
= {F, H}
k
.
.
1
Various aspects of integrable Hamiltonian systems. Proc. CIME Conf., Bressanone, 1978.
70 Various Aspects of Integrable Hamiltonian Systems
A nonconstant function F is called an integral of XHif
X
F = {F, H} =0.
H
In particular, H is an integral. If F is an integral of X
of X
.
F
A set of functions F
, F2, ..., Frare said to be «in involution» or to
1
then H is an integral
H
commute, if
{F
This implies clearly that the vector fields X
If ϕ = ϕ(ξ
{ϕ(F
Thus, if F
D ⊂ R
, ..., Frare in involution, so are any functions of F1, ..., Fr.
1
Definition 1. A Hamiltonian system (1), defined in an open domain
2n
is called «integrable» if there exist n integrals F1, F2, ..., Fnin
, ..., ξr), ψ = ψ(ξ1,ξ2, ..., ξr) then
1,ξ2
, ..., Fr),ψ(F1, ..., Fr)} =
1
} =0 for k, j =1, 2, ..., r.
k,Fj
commute.
F
k
k, j
∂ϕ
∂ξ
∂ψ
∂ξ
k
{Fk,Fj}.
j
involution with linearly independent gradients, i. e. in D we have
(i) {H, F
} =0; (ii) {Fk,Fj} =0, (iii) dF1, ..., dFnlinearly indepen-
j
dent.
n
XAMPLE 1. H =
E
with Fk= x
E
with F
2
+ y
k
XAMPLE 2. If H = H(y) is independent of x then the system is integrable
= yk.
k
1
2
2
k
k=1
(k =1, 2, ..., n).
αk(x
2
2
+ y
) defines an integrable system in R
k
k
2n
Locally, that is near any point where dH =0any system is integrable; in
fact, in appropriate canonical coordinates H agrees with y
which is a special
1
case of Example 2.
Generally it makes sense to speak of a system being integrable in a domain
which is invariant under the flow generated by X
.
H
It is highly exceptional for a Hamiltonian system to be integrable globally in
an invariant open domain — or even locally near a stationary point (where dH =
=0). However many systems occurring in application are closely approximated
by integrable systems. For example, the n-body problem becomes integrable
in the limit when all but one mass tends to zero. The resulting system is a
decoupled system of Kepler problems.
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