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§5. Associated Differential Equations 41
we get
2
= −2
a
n−1
Δ
2
(a
Δ
+(b
n−3
n−2
n
n−1
+ bn)Δ
n−2
).
df
dt
Using the recursion formulae (3.7) for k = n − 1, n we find
df
dt
= −2(λ
2
2
− b
− a
n
Comparing the residue of these expressions at λ
2
˙r
k
− b
k
= −λ
2
− a
n
2
k
= −(λ
˙r
k
or using r
and e
as homogeneous coordinates
k
Thus the solutions of (5.2) can be expressed as rational functions of λ
2
−λ
t
k
and the asymptotic behavior of its solutions is also completely under-
Δ
n−1
2
n−1
2
n−1
)
k
.
Δ
n
with those of (5.3) we find
)rk,
rk.
stood from the results of the previous sections.
It is interesting to observe that the differential equations (5.2) possess b
k
=0, k =1, ..., n, as an invariant manifold on which they reduce to
˙a
k
= ak(a
2
k+1
− a
2
),k=1, ..., n− 1(5.4)
k−1
which are the deformation equations for a Jacobi matrix L with a zero diagonal
To understand which of the solutions of (5.3) corresponds to (5.4) we
consider again the continued fraction expansion (3.5), denoting the left-hand
side by f (λ, a, b). One easily verifies that the involution b
→−bk, ak→ a
k
gives rise to
n
2
r
k
λ − λ
.
k
−f(−λ, a, −b)=f (λ, a, b)=
Hence, since the eigenvalues λ
k=1
are ordered according to size, the above invo-
k
lution corresponds to
k
=
1
.
k
→−λ
λ
k
1
These equations for n = ∞ were recently studied by M. Kac and van Moerbeke, according to
a letter from M. Kac.
n−k+1
,rk→ r
n−k+1
.
42 Finitely Many Mass Points on the Line
The fixed points of this involution are the points (a, b) with bk=0in the first
representation and the points (λ, r) with
λ
+ λ
k
n−k+1
=0,rk→ r
. (5.5)
n−k+1
This is evident also from the fact that the symmetric Jacobi matrices with zero
diagonal have a spectrum symmetric with respect to the origin. Thus the solutions
of (5.4) are given by precisely those rational functions in λ
the λ
satisfy (5.5).
j
Using (2.1) it is easy to rewrite the system (5.2) in the variables x
2
−λ
t
j
, e
j
for which
, ykand
k
one finds a Hamiltonian system
∂H
2
with
H
˙x
2
=
k
= −
∂y
1
6
k
n
k=1
, ˙yk= −
3
y
k
Again, in the above system one has to set x
∂H
2
,k=1, 2, ..., n,
∂x
k
n
1
−
2
k=1
yk(e
0
x
k−1−xk
= −∞, x
xk−x
+ e
n+1
k+1
).
=+∞. Although this
system has no physical interpretation one has a full description of the scattering
problem.
If one expresses the above Hamiltonian H
4
3
tr L
=
H
2
3
in terms of a, b one finds readily
2
n
4
=
3
λ
.
k
3
k=1
Since our original Hamiltonian (1.1) is given by
n
H =2trL
2
=2
k=1
2
λ
,
k
one can expect that the further differential, equations are associated with Hamiltonian proportional to tr(L
p+1
).
We will not follow this up but conclude with establishing the existence
of the matrices B
for p =1, 2, ..., n − 1. It is convenient to write the
p
matrices as difference operators. Let ξ stand for a double infinite sequence with
components ξ
(k integers) and let σ denote the shift operator
k
(σξ)
= ξ
k
k+1
.
§5. Associated Differential Equations 43
We will assume that ξ
=0if k 0 or k>nand write the matrix L in the
k
form
Lξ = a(σξ)+bξ + σ
Here a, b stand for sequences with components a
−1
(aξ).
and (aξ)k= akξk. Thus
k
σ(aξ)=σ(a) · σ(ξ).
In this notation B
B
ξ = γσpξ + ...+ β(σqξ)+...− σ−q(βξ)+...− σ−p(γξ), (5.6)
p
where the q
th
The commutator [B
order term indicates a typical term, 1 q<p.
will be presented by
p
,L]=BpL − LBpcontains σ, σ−1to powers up
p
to p +1. In fact, the highest order terms of this commutator are given by
[B
and we determine the γ
,L]={γσp(a) − aσ(γ)}σ
p
, γ2, ..., γ
1
p
(γσ
(a) − aσ(γ))ξ =0 (5.7)
n−1
so that
p+1
+ ...
i. e.
γ
kak+p
− akγ
=0,k=1, 2, ..., n− p − 1.
k+1
This can be satisfied by
γ
= aka
k
k+1
...a
,k=1, 2, ..., n− p.
k+p−1
Now we proceed inductively and determine the coefficients β of σ
remove the terms of order q +1in [B
,L], decreasing q from q = p to q =1.
p
q
in (5.6) to
Analogously to (5.7) this gives an equation of the form
q
(βσ
(a) − aσ(β))ξ = g · ξ
where g is a given sequence. In components
β
kak+q
− β
= gk,k=1, 2, ..., n− q −1.
k+1ak
These are n −q −1 equations for n −q unknowns. The solution is therefore not
unique, but if β
and uniquely, since a
is fixed arbitrarily, these equations can be solved recursively
1
> 0.
k
44 Finitely Many Mass Points on the Line
Thus Bpcan be so determined that in [Bp,L] all coefficients of σ
q+1
for q =1, 2, ..., p vanish. Since [Bp,L] is symmetric it is a Jacobi matrix,
giving rise to the desired differential equation
˙
L =[B
,L]. (5.8)
p
These are clearly the analogues of the higher order Korteweg –de Vries equations.
Finally, it is obvious that the multiplication
(r ⊗ r)
= rkr
k
k
introduces a group structure into the manifolds λk=constmaking the
(n − 1) dimensional manifold of Jacobi matrices L with fixed spectrum into
an Abelian group. This group action commutes with the vector field (2.2), and
more generally with the vector fields (5.8) for p =1, 2, ..., n− 1.
References
[1] H. Flaschka, The Toda Lattice, I, Phys. Rev. B 9, (1974) 1924–1925.
[2] H. Flaschka, The Toda Lattice, II, Prog. of Theor. Phys. 51 (1974) 703–716.
[3] F. R. Gantmacher, and M. G. Krein, Oszillationsmatrizen, Oszillationskerne
und Kleine Schwingungen Mechanischer Systeme, Akad. Verlag, Berlin
(1960). See, Anhang II.
[4] M. Henon, to appear in Phys. Rev. B 9, (1974) 1921–1923.
[5] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves,
Comm. Pure Appl. Math. 21 (1968), pp. 467–490.
[6] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Korteweg– de Vries
Equation and Generalizations VI, Methods for Exact Solutions, Comm. Pure
Appl. Math. 27 (1974) 97–133.
[7] V. I. Arnold, and A. Avez, Probl`emes Ergodiques de la M´ecanique Classique,
Gauthiers-Villars, Paris (1967).
[8] M. Toda, Wave propagation in anharmonic lattices, Jour. Phys. Soc. Japan
23 (1967) 501–506.
Three Integrable Hamiltonian Systems Connected
with Isospectral Deformations
DEDICATED TO STAN ULAM
§ 1. Introduction
(a) Background. In the early stages of classical mechanics it was the ultimate
goal to integrate the differential equations of motions explicitly or by quadrature.
This led to the discovery of various «integrable» systems, such as Euler’s two
fixed center problems, Jacobi’s integration of the geodesics on a three-axial
ellipsoid, S.Kovalevski’s motion of the top under gravity for special ratios of the
principal moments of inertia, to name a few nontrivial examples. These efforts
and their climax with the work of Jacobi who applied skillfully the method
of separation of variables to partial differential equations, the Hamilton – Jacobi
equations associated with the mechanical system, to establish their integrable
character.
However, this development took a sharp turn when Poincar´e showed that
most Hamiltonian systems are not integrable and gave arguments indicating the
nonintegrability of the three-body problem. In the same negative direction lies
Brun’s discovery that the three-body problem has no algebraic integral except
for the well-known classical ones and algebraic functions of these. These results
express, in other words, that integrability of Hamiltonian systems is not a generic
property; it is destroyed under small perturbations of the Hamiltonian.
Therefore it seems an anachronismus to discuss these exceptional integrable
systems nowadays. However, in recent years various phenomena were discovered which are clearly intimately related to integrable Hamiltonian systems yet
they have very different origin. One is related to the discovery by Kruskal and
others [6] of so-called solitons for the Korteweg – de Vries equation. These are
wave solutions of a nonlinear partial differential equation having a strong stability behavior. Originally these phenomena were brought to light by numerical
1
Adv. Math., v. 16 (1975) 197–220.
1
46 Three Integrable Hamiltonian Systems
experiments and later on related to the existence of infinitely many conservation
laws that restrict the evolution of the solutions severely. If one interprets the
partial differential equation, in this case the Korteweg –de Vries equation, as
a Hamiltonian system in an infinite-dimensional function space, with a certain
symplectic structure, and the conservation laws as integrals of this system, one
can view this as an example of an integrable system of infinitely many degrees
of freedom. This was made precise in the work of Zakharov and Faddeev [15].
In an entirely unrelated development Calogero [2, 3] found that the quantum
theoretical problem of n mass points on the line interacting under the influence
of a potential proportional to the inverse square of the distance can be solved
explicitly, and he conjectured that the corresponding classical problem might be
integrable. This was established by Marchioro for the «three-body problem»
by explicit calculation. Moreover, Calogero used his formula to study the scattering problem associated with the n-particle system in the quantum theoretical
framework and found that the scattering is essentially trivial, in the sense that
the particles behave asymptotically like elastically reflected mass points.
(b) Results. It is our goal to show a close algebraic connection between
these so different problems. However, instead of studying the infinite dimensional problems related to the partial differential equation in the one and the
quantum theoretical framework in the other case, we will restrict ourselves to
finite-dimensional systems. The Korteweg – de Vries equation can be discretized
so as to retain the desired integrability, as was shown by Toda [13] and his
collaborators. Another discretization leads to the differential equations
du
k
1
u
=
(e
2
dt
(where we set formally e
k+1
u
0
u
k−1
− e
=0, e
)(k =1, 2, ..., n− 1) (1.1)
u
n
=0) suggested by M. Kac and
P. v. Moerbeke [8, 9]. Although this system does not have the appearance of
a Hamiltonian system, it can be embedded into one, as was shown in [12]. The
u
remarkable fact is that there are [n/2] = ν polynomials P
of uk, e
μ
k
which are
integrals of the motion, i. e.,
/dt =0 (μ =1, 2, ..., ν)
dP
μ
if one inserts a solution of the above differential equations. Moreover, all
solutions can be expressed in the form
u
k
= Rk(η)
e
§1. Introduction 47
where R
are rational functions of
k
η =(η
, ..., ην) and η1= e
1
α1t
, ..., ην= e
ανt
.
These rational functions can, of course, not be explicitly described, but this
representation suffices to give a complete description of the scattering problem
related to this problem (see Section 7).
Instead of Calogero’s quantum theoretical problem we look at the corre-
sponding classical one, described by the equations
2
d
x
k
∂U
where
the coordinates x
= −
2
dt
U =
k
(xk− xl)−2,k,l=1, 2, ..., n, (1.2)
k<l
of the mass points being distinct real numbers. This system
, (k =1, 2, ..., n)
∂x
k
is clearly a Hamiltonian system with
n
1
2
y
+ U
k
2
k=1
where y
=
are the momenta. We will show that this system is an integrable
k
Hamiltonian system, by which we mean that this system possesses n independent
integrals I
this case these functions are, in fact, polynomials in y
= Ik(x, y), globally defined in the phase space and in involution. In
k
and (xk− xl)−1.Using
k
this result it is quite easy to verify Marchioro’s conjecture: The particles have
an asymptotic velocity ˙x
(±∞) satisfying
k
˙x
(+∞)= ˙x
k
n+1−k
(−∞).
Thus after a fairly complicated interaction the particles emerge as free particles
with velocities exchanged, that is, the first particle has for t →∞the velocity
of the last for t →−∞,etc.
As a third example we discuss the equation on the circle
2
x
d
k
∂U
= −
dt
2
∂x
k
48 Three Integrable Hamiltonian Systems
with
Here the x
U =
are considered mod π as distinct points on a circle. These equa-
k
1
2
k=l (mod n)
sin−2(xk− xl). (1.3)
tions are the classical mechanics analog to those of Sutherland [14]. Also this
system will be shown to be an integrable Hamiltonian system with n integrals I
which are polynomials in ykand cot(xk− xl).
In contrast to the previous examples the last problem has a compact energy
surface. On account of this fact the surfaces I
=const(k =1, 2, ..., n) are
k
compact and hence, as is well known, tori on which the solutions are quasiperiodic. However, the function theoretical character of these solutions has not yet
been satisfactorily described.
(c) Lax’s method. The common link between these problems is that they
can be related to deformations of matrices leaving the eigenvalues fixed, that is,
to isospectral deformations. For example, with (1.2) we associate a Hermitean
matrix L having y
as diagonal elements and i(xk− xl)−1as elements in the
k
(k, l)-position if k = l. Then (1.2) gives rise to a differential equation for L =
= L(t) whose solutions have fixed eigenvalues, i.e., the eigenvalues, and hence
their symmetric functions I
are integrals of the motion. The idea of finding
k
integrals of the motion as eigenvalues of an associated linear operator L was
developed by Lax [10] for the Korteweg – de Vries equation, where L is given
by the classical Sturm – Liouville operator
2
d
+ q
−
2
dx
k
and the potential q is to be deformedin such a way that the spectrum is unchanged.
This question is intimately related to the inverse problem of determining the
spectrum from the potential. Instead of developing these ideas in generality we
will illustrate them in the three simple examples mentioned above.
In Section 2 we illustrate this method for Eq. (1.1), although this is in no
way new. Indeed Flaschka [4, 5] observed first that this method can be applied
to the Toda lattice and this example is only a slight variation on this theme. In
Section 3 we put Eq. (1.2) into the same framework and draw the conclusion for
the associated scattering problem in Section 4. The n-particle system (1.3) on
the circle will be studied in Section 5. Finally in Sections 6 and 7 we discuss the
inverse spectrum problem and the scattering problem associated with a special
Jacobi matrix. The latter leads to an interesting motion in which particles separate
§2. Isospectral Deformations 49
in pairs, each pair having a different asymptotic velocity, while the two particles
of one pair have the same asymptotic velocity. The scattering phases can also
be determined by relating the differential equations to those for the Toda lattice
for finitely many particles.
(d) General remarks. These problems have connections with a multitude
of topics besides that of dynamical systems. The fact that they arc related to
isospectral deformation points to the connection with spectral and scattering
theory. The function theoretical nature of the solution and the rational character
of the integrals relates to functions of complex variables. But also Lie algebras
play into the subject; in fact the equations are very similar in nature to those
studied by Arnold [1]. Arnold generalized the Euler equation for the rotation of
a rigid body to dynamical systems in arbitrary Lie algebra.
Many of these connections are still obscure, and we hope that the study
of these simple finite dimensional examples will lead to further investigations
clarifying the many questions left open.
I want to express my thanks to H. Flaschka and G. Galavotti for many
stimulating discussions in the beginning of this work. I am particularly indebted
to Galavotti who pointed out Calogero’s work and insisted that the classical
analog should be integrable.
§ 2. Isospectral Deformations
We begin with an idea that was introduced by P. D. Lax in a different but
closely related connection. Consider a class of matrices, say all Jacobi matrices
of the form
with positive entries a
We ask for all matrices in this class having the same spectrum. One may expect
that there are not enough parameters available, but since
−1
K
⎛
0 a
⎜
⎜
a
⎜
L =
⎜
⎝
0 a
, a2, ..., a
1
00
1
.
.
0 a
1
.
n−1
.
2
.
.
.
.
.
n−1
. Their eigenvalues are real and simple.
a
n−1
0
⎞
⎟
⎟
⎟
⎟
⎠
LK = −L for K = diag(1, −1, +1, ...),
(2.1)
one has for the characteristic polynomial
(λ)=det(λI − L)
Δ
n
50 Three Integrable Hamiltonian Systems
the relation
Δ
(λ)=(−1)nΔn(−λ). (2.2)
n
Therefore, with λ also −λ is an eigenvalue and λ =0is an eigenvalue precisely
if n is odd. Thus fixing the eigenvalues amounts to [n/2] conditions and the
dimensionality of the isospectral matrices of the form (2.1) is n − [n/2].
To get some isospectral deformations, Lax [10] considered differential equa-
tions of the form
d
L = BL − LB (2.3)
dt
where L = L(t), t being the deformation parameter. The matrix B has to be
chosen appropriately, so that the commutator [B,L] has zeros except in the two
off-diagonals, and those should agree. In this example one finds as one possible
choice the skew symmetric matrix
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
B =
⎛
00a
⎜
000 a
⎜
⎜
⎜
−a1a20
⎜
⎜
⎜
⎝
1a2
.
.
0
2a3
.
.
.
.
0 a
n−2an−1
000
−a
n−2an−1
00
for which the differential equation (2.3) takes the form
˙a
= ak(a
k
where we set formally a
2
k+1
=0=an.
0
− a
2
),k=1, 2, ..., n− 1(2.4)
k−1
It is clear that (2.3) gives rise to isospectral deformations: If we solve the
differential equation
d
U = BU, U(0) = I
dt
then (2.3) assures that
d
−1
(U
dt
hence
U
LU)=0,
−1
LU = L(0).
Thus the eigenvalues of L remain constant under this deformation. Also the
coefficients I
of the characteristic polynomial
k
(λ)=λn+ I1λ
Δ
n
n−1
+ ···+ I
n
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