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§2. Classical Integrable Systems and Isospectral Deformations 191
But generally this condition is violated and we wish to define a new vec-
torfield on M from the given X
example by replacing X
by
H
. This can be done in many ways [23], for
H
2r
X
−
j=1
λj(x)X
G
j
H
where the multipliers λjare defined so that this vectorfield is tangential to M .
This requires that
2r
{H, G
which by (2.12) defines the λ
}−
k
= λj(x) uniquely on M If we set
j
∗
H
λj{Gj,Gk} =0
j=1
2r
= H −
j=1
λjG
j
(2.13)
then the «constrained vectorfield» on M is given by
2r
X
∗
H
= XH−
j=1
λjX
.
G
j
One verifies readily that this vectorfield does not depend on the extension of
the λ
(x) off M .
j
is integrable one can not expect in general the constrained vectorfield
If X
H
also to be integrable. We describe a special situation where this is the case. For
this purpose assume that X
is integrable with F1F2, ..., Fnas commuting
H
integrals. Moreover, assume that M is given by the equations
= ...= Gr=0,F1= F2= ...= Fr=0
G
1
where G
, G2, ..., Grare some other functions satisfying
1
det({F
j,Gj})ij=1, ..., r
In other words, in the notation of (2.11) we take G
We claim that in this case the restriction of F
strained system. Moreover, the F
not any more independent.
also commute. Of course, the dF
j
M
to M are integrals to the con-
j
=0. (2.14)
= Fj(j =1, 2, ..., r).
j+r
are
j
M
192 Integrable Hamiltonian Systems and Spectral Theory
Indeed with
it follows from
and (2.14) that μ
where the λ
are defined by {H∗,Gk} =0(k =1, 2, ...,r). Hence the
j
∗
H
= H −
0={H
= ...= μr=0on M. Therefore we can set
1
H
∗
,Fk} =
∗
= H −
r
(λjFj+ μjGj)
j=1
r
j=1
r
j=1
constrained vectorfleld
r
X
∗
H
= XH−
j=1
certainly annihilates the Fk, proving the claim.
We illustrate this construction for the Hamiltonian
H =
n
where q, ∈ R
and A is a symmetric matrix. We will see that there are
n commuting integrals F
1
Aq, q +
2
, ..., Fnwhich are polynomials in q, p of which
1
1
2
(|q|
2
μj{Gj,Fk}
λjF
λjX
j
r
j
(2.15)
|p|2−q, p2)(2.16)
1
2
(|q|
=
F
1
− 1)
2
is one. We constrain this system to the tangent bundle of the sphere
= |q|2− 1=0; G1= p, q =0.
2F
1
Note that {F
} =1=0. We find for the constrained Hamiltonian
1,G1
∗
= H − λ1F1with λ1= {H, G1} = Aq, q.
H
Therefore the differential equations become
∗
˙q = H
˙p = −H
= Hp− λ1Fp= |q|2p = p
p
∗
= −Hq+ λ1Fq= −Aq +(λ1−|p|2)q
q
§3. Geodesics on an Ellipsoid and the Mechanical System of C. Neumann 193
where |q| =1, q, =0. With ν = λ
d
dt
−|p|2this can be written as
1
2
q
= −Aq + νq. (2.17)
2
This equation can be interpreted as constraining the linear system ¨q = −Aq to
the unit sphere and ν ·q as the normal constraining force.
This is the system mentioned in Section 1 which was studied by C. Neumann. Its solution and its relation to the geodesic flow on the ellipsoid will be
the object of the next section.
§ 3. Geodesics on an Ellipsoid and the Mechanical System of
C. Neumann
1. Geodesic flow on the ellipsoid
In this section we discuss two classical integrable systems which will play
a central role for the inverse spectral theory see Section 5). The first is the
geodesic flow on an ellipsoid
1<α2
T
is a positive definite symmetric matrix with distinct eigenvalues
<... <αn. The differential equations are given by
where A = A
0 <α
1
{x ∈ Rn, A−1x, x =1}
2
d
x
= −νA−1x (3.1)
2
ds
where the multiplier ν is determined so that
2
d
i. e.
0=
1
2
A−1x, x = A−1x, x + A−1x,x =
ds
−1x|2
= −ν|A
ν = |A
+ A−1x,x
−1x|−2A−1x,x
.
We will show that this system in integrable and that the integrals can be written
as quartic polynomials in x, x
1
Our notation differs from the previous section: We replaced q, p by x, y.
= dx/ds.
194 Integrable Hamiltonian Systems and Spectral Theory
For this purpose it is useful to represent this system by constraining the
«free» Hamiltonian H =
1
|y|2to the tangent bundle of the ellipsoid
2
−1
A
x, x =1, A−1x, y =0.
Using the formulae of the previous section we find
1
H
∗
2
|y|
=
− λ1(A−1x, x−1) − λ2A−1x, y
2
with
1
λ
= −
1
−1x|−2A−1
|A
2
y, y; λ2= −|A−1x|−2A−1x, y
or
with
H
1
∗
=
2
|y|
2
+
μ
Φ
0
2
μ = |A
(x, y) −
−1x|−2
μ
−1
A
2
x, y
2
(3.2)
;
(3.3)
(x, y)=(A−1x, x−1)A−1y, y−A−1x, y2.
Φ
0
We will drop the last term in (3.2) since it vanishes with its derivatives on the
tangent bundle of the ellipsoid. One verifies readily that the constrained system
is
⎧
⎪
⎪
⎨
⎪
⎪
⎩
dx
ds
dy
ds
= H
y
= −H
∗
= y
∗
= −μ<A−1y, y > A−1x,
x
(3.4)
hence agrees with (3.1).
The advantage of this approach of extending the system from a flow on
the tangent bundle of the ellipsoid to R
local co-ordinates on the ellipsoid. Moreover, the extended system X
2n
is that we avoid the use of awkward
H
has an
interesting geometric interpretation. For this purpose we discuss the significance
of the function Φ
(x, y) of (3.3). One verifies readily that the cone
0
n
{y ∈ R
| Φ0(x, y)=0} (3.5)
when translated by x represents the cone of vectors through the point x which
are tangent to the ellipsoid.
§3. Geodesics on an Ellipsoid and the Mechanical System of Neumann 195
For the discussion of the vectorfield
X
1
∗
=
H
X
2
|y|
μ
2
+
X
Φ
0
2
it is useful to note that the two summands commute since
2
{|y|
, Φ0} =0;
therefore it suffices to discuss the two vectorfields separately. The first summand
describes of course, the free flow
(x, y) → (x + κy, y)
and the second is given by
which we restrict to the energy surface Φ
˙x = μΦ
˙y = −μΦ
0y
0x
=0. By our remark below (3.5) we
0
(3.6)
can associate with (x, y) the tangent of the ellipsoid through x in the direction y.
The equation (3.6) with Φ
=0describes the motion of tangent lines of the
0
ellipsoid, where the point of contact moves along a geodesic while the point x
moves perpendicularly to the tangent; the first summand moves the point along
the line, e.g. moving it back to the point of contact. Thus it suffices to study (3.6)
and after change of the independent variable we may replace μ by 1. For a fuller
discussion see [23].
2. Confocal quadrics, construction of integrals
Basic for the understanding of the geodesics on the ellipsoid is the family
of confocal quadrics Q
z
(zI − A)−1x, x +1=0,z∈ R,z= αk,
which contains the ellipsoid for z =0. For abbreviation we set
Q
(x, y)=(zI − A)−1x, y; Qz(x)=Qz(x, x),
z
and in analogy to (3.3) we introduce
Φ
(x, y)=(1+Qz(x))Qz(y) − Q
z
2
(x, y). (3.7)
z
196 Integrable Hamiltonian Systems and Spectral Theory
The cone of tangent lines through the point x to the quadric Qzis given by
{y ∈ R
n
, Φz=0}. The functions Φz(x, y) are quartic polynomials as far
as x, y are concerned, and rational functions of z with simple poles at the
eigenvalues α
where
of A. The partial fraction expansion takes the form
k
n
Fk(x, y)
z − α
k=1
n
(xjyk− xkyj)
j=1
j=k
αk− α
k
2
. (3.9)
j
(x, y)=y
F
k
Φ
(x, y)=
z
2
+
k
(3.8)
The integrability of our system depends on the remarkable fact that these functions commute with respect to the Poisson bracket
n
{Φ
j=1
(F
G
− F
G
z
y
j
j
, Φ
} =0
z
z
1
2
).
y
z
j
j
, z2one has for the func-
1
{F, G} =
Proposition 3.1. For any two numbers z
, Φ
tions Φ
z
defined by (3.7) the identity
z
1
2
hence also for (3.9)
j,Fk
} =0.
{F
This is verified by a calculation (see also Moser [20], [21]). As a conse-
quence we see that the F
are integrals of the system (3.6). Since
j
n
Fk= |y|
k=1
2
also commutes with the Fkit follows that the Fkalso are integrals for (3.2),
and hence the restrictions of the F
to the tangent bundle of the ellipsoid Q0are
k
integrals to the geodesic problem. By proposition 3.1 they commute. Moreover,
the dF
not be said for the restriction of the F
has the relation
are linearly independent on an open set of R2n. The same can, of course,
j
to the tangent bundle indeed there one
j
n
−1
α
Fj= −Φ0(x, y)=0
j
j=1
but at generic points one has n −1 independent commuting integrals.
§3. Geodesics on an Ellipsoid and the Mechanical System of Neumann 197
This shows that the geodesic problem (3.1) is integrable (on an open and
dense set of the tangent bundle) and the integrals are given by the restriction the
functions (3.9).
3. Isospectral deformations
It is interesting that the system
=
x
j
∂
Φ0,y
∂y
j
∂
=
−
Φ
∂x
0
j
j
(3.10)
can be interpreted as isospectral deformation. The difficulty is to guess the
matrices L and B with which the above equation can be written in the form (2.10).
One finds
(A − x ⊗ x)Py, |y| > 0
y
= δij− yiyj|y|
y)ij
−2
where (x ⊗ x)
L = L(x, y)=P
= xixjis the tensor product and
ij
(P
is the projection onto the orthogonal complement of y. Thus L is a symmetric
matrix with Ly =0,i.e. y is eigenvector for λ =0. With the skew symmetric
matrix
B = −
x
iyj
αiα
where the diagonal elements are equal to 0, the differential equation L
− xjy
j
i
=[B, L]
agrees indeed with (3.10). For the necessary calculation we refer to [23, 1].
It follows then that the eigenvalues of L are integrals for (3.10). They are,
in fact, related to the polynomials F
the identity
2
det(z −L)
|y|
·
z
det(z −A)
Therefore the eigenvalues λ
as functions of the F
k
, λ2, ..., λ
1
and therefore commute also. Of course, these functions
are not globally well defined but only on sets where the λ
The leaves of the foliation F
n
2
|y|
=
ck; λj(x, y)=βj(j =1, ..., n− 1)
k=1
and Φzalready formed. Indeed one finds
k
n
F
=Φ
(x, y)=
z
k=1
, (λn=0)and |y|2can be viewed
n−1
= ckcan also bo defined via
k
k
z −α
are distinct.
j
.
k
198 Integrable Hamiltonian Systems and Spectral Theory
if β1, ..., β
in other words, if the β1, β2, ..., β
are the zeros of
n−1
{x, y ||y|
n
c
k
z −α
k=1
n−1
n
2
=
ck;Φ
k=1
k
are distinct, these leaves are given by
= ...=Φ
β
1
=0}.
β
n−1
These (x, y) correspond to the linen x + sy which are common tangents to the
quadrics Q
We discuss the system (3.10) corresponding to the geodesic flow on Q
which case L must have another zero eigenvalue, say, λ
In this case we can take |y|
(j =1, 2, ..., n− 1).
β
j
2
, λ1, λ2...λ
= β
n−1
as n − 1 commuting integrals
n−2
n−1
=0.
0
,in
on the tangent bundle of the ellipsoid. We may restrict ourselves to the case
|y| =1, i. e. to unit velocity.
The fact that λ
, λ2, ..., λ
1
under the flow (3.10) the lines which are common tangents of Q
=0initially remain common tangent for all time.
β
n−1
are integrals has the interpretation that
n−1
, ..., Q
β
1
β
n−1
It is easily verified that the eigenvectors of L are given by the normal ν
of Q
, j =1, ..., n− 1 at the point of contact of the line x + sy together
β
j
with ν
the normals of a common tangent of Q
perpendicular. This is a well known theorem of Chasles
= y.SinceL is symmetric we conclude the geometrical fact that
n
(j =1, ..., n − 1) are mutually
β
j
1
.
Clearly, the above discussion applies equally well to the geodesic flow on
any of the confocal quadrics. The point is that the extension of these flows
2n
to R
commute with one another, on account of proposition 3.1.
,
j
4. The mechanical problem of C.Neumann
The system in question describes the motion of a mass-point on a sphere
n−1
S
= {q ∈ Rn, |q| =1}
under the influence of the force −Aq,whereA is a symmetric matrix with
distinct eigenvalues. The differential equations are given by
2
d
q = −Aq + νq (3.11)
2
dt
1
L. Bianchi, «Vorlesungen ¨uber Differentialgeometrie». 2. Auflage, Teubner, Berlin, Leiрzig,
1910, р. 658.
§3. Geodesics on an Ellipsoid and the Mechanical System of Neumann 199
where νq is the normal force keeping y on the sphere, if
2
ν = Aq, q−|˙q|
. (3.12)
This also is an integrable system which is seen as follows: We extend also this
system to a system in R
(see (2.16), (2.17)) that these equations are obtained by constraining X
to the tangent bundle of S
2n
. As a matter of fact we showed at the end of Section 2
H =
1
Aq, q +
2
n−1
and it suffices to show that this system is integrable.
1
2
(|q|
|p|2−q, p2)(3.13)
2
with
H
For this purpose we can use proposition 3.1 again: Expanding the rational
function Φ
at z = ∞ we find from 3.7)
z
2
(x, y)=
Φ
z
|y|
1
+
z
{Ay, y + |x|2|y|2−x, y2} + ...
2
z
or
2
2H(q, p)
Φ
(p, q)=
z
|q|
+
z
+ ...
2
z
Comparing this with (3.8) we find
n
1
H =
Hence the functions F
(p, q) being defined in (3.9) with (x, y) replaced
k
by (p, q), are the desired integrals of the system (3.13). Since |q|
commutes with the F
the constrained system is also integrable by the argument
k
αkFk(p, q). (3.14)
2
k=1
2
n
=
Fjalso
j=1
at the end of Section 2.
5. The connection between the two systems via the Gauss mapping
From the above formulae it is appearant that the geodesic flow on the
−1
ellipsoid A
x, x =1and Neumann’s problem are closely related. A different,
more geometrical connection between these problems was found by H. Kn¨orrer [13] and we want to present his result.
200 Integrable Hamiltonian Systems and Spectral Theory
For this purpose Kn¨orrer made use of the usual Gauss mapping of the
ellipsoid Q
onto the unit sphere which takes x ∈ Q0into the exterior unit
0
normal
−1
q = rA
x, where r = |A−1x|−1. (3.15)
Aside from a change of the independent variable this Gauss mapping takes
solutions of (3.1) into solutions of (3.11) where, however, A is to be replaced
−1
by A
.
To make the statement more precise, we will change the independent vari-
able s in equation (3.1) into t via s = ψ(t), so that (3.1) becomes
¨
¨x = −ν˙ψ
2A−1
x +
ψ
˙x
˙
ψ
where the dot indicates differentiation with respect to t. We choose ψ(t) so that
2
=1. For abbreviation we write
ν˙ψ
−1
B = A
so that the system becomes
¨
¨x = −Bx + b ˙x, b = b(t)=
ψ
. (3.16)
˙
ψ
The fact that Bx, x =1for the considered solution yields after two or three
differentiations
B ˙x, ˙x
2
|Bx|
=1; b =2
Bx, B ˙x
2
|Bx|
. (3.17)
Thus (3.16) represents the geodesic flow on the ellipsoid in the new parametrization and the first relation of (3.17) characterizes this parametrization by specifying
the velocity.
Theorem 3.2. The Gauss map of Q
0
→ S
n−1
takes the solutions of (3.16)
satisfying
Bx, x =1, Bx, ˙x =0; B ˙x, ˙x = |Bx|
2
(3.18)
into the solutions of the Neumann problem
¨q = −Bq + νq, ν = Bq, q−|˙q|
2
(3.19)
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