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§2. Perturbation of Rank 2 141
The vector field generated by H is generally denoted by X
= X
− ΔXG.
F
The differential equations for X
=
x
d
dt
y
−a −s
are given by
H
sd
βx
βy
+Δ
Γx
Γy
From these one computes readily the following relations:
R = −[Γ,R],XGS = −[Γ,S];
X
G
hence
X
L = −[Γ,R+ S] − [Γ,L]+[Γ,A].
G
Similarly one finds
X
R = r[β, S],
F
S = −(ad −s2)[β, x ⊗ y − y ⊗x];
X
F
hence
X
L = XH(R + S)=XF(R + S) − ΔXG(R + S)=
H
= r[β, S] − (ad − s
2
)[β, x ⊗ y − y ⊗x]+
+Δ[Γ,L] − Δ[Γ,A].
Since
[Γ,A]=−[β, x ⊗ y −y ⊗ x],
,sothatXH=
H
.
we obtain
X
L = r[β, S]+(−(ad − s2)+Δ)[β, x ⊗y − y ⊗x]+Δ[Γ,L]=
H
2
= r[β, S]+r
[β, x ⊗ y −y ⊗x]+Δ[Γ,L]=
=[rβ, S + R]+Δ[Γ,L]=
=[rβ +ΔΓ,L]=[B, L],
where we have used that A and β commute. This proves Theorem 2, and hence
Theorem 1.
c. The Action of Gl(2,R)
The matrices L can be simplified if one subjects them to the linear transformation (x, y) → (αx + βy, γx+ δy) with αδ − βγ =1. This last condition
142 Geometry of Quadrics and Spectral Theory
assures that the transformation is symplectic. If we set
V =
αβ
γδ
; C =
C =Σ+r
ab
cd
, Σ=
01
−10
as
,
sd
,
then this transformation replaces C by
T
V
CV = VTΣV + r
01
−10
.
Thus r is invariant as well as the rank ρ and the determinant of Σ. One finds the
following normal forms:
(i) ρ =2: Σ=
(ii) ρ =1: Σ=
(iii) ρ =0: Σ=
a 0
01
00
01
00
00
, a =0,
,
.
The corresponding form for the matrices L are
(i) L = A +(ax ⊗ x + y ⊗ y)+r(x ⊗ y −y ⊗ x), a =0,
(ii) L = A + y ⊗y + r(x ⊗ y −y ⊗x),
(iii) L = A + r(x ⊗y − y ⊗x).
All these cases occur in examples of mechanics and geometry, as we will
see in the following sections. Case (i) describes the motion of a particle on an
−1
ellipsoid A
x, x =1under the influence of the central force ax. Case (ii)
is related to the geodesic flow on an ellipsoid, to the motion of a particle on the
sphere |x| =1under the influence of the external force Ax, and to the Toda
lattice in the periodic case (see van Moerbeke [22]), as was shown recently by
P. Deift and E. Trubowitz. Case (iii) describes special orbits of the geodesic flow
on the orthogonal group O(n) with a left-invariant metric. In the next section
we describe the connection with the geodesic flow on the ellipsoid, which is the
oldest candidate for an integrable system.
d. Trace Formulae
We present another representation of the basic functions (2.9) which are in
involution for all choices of the diagonal matrix β. For this purpose we introduce
§2. Perturbation of Rank 2 143
a parameter ε and consider the matrix
L
= A + εR + ε2S
ε
so that the matrix
is replaced by
C = ε
2
C =
as
sd
ab
cd
+ εr
01
−10
.
Set
= aQz(x)+(b + c)Qz(x, y)+dQz(y)−
Φ
z
2
(Qz(x)Qz(y) − Q
− r
and for any polynomial g(z) with g
(αj)=βjdefine
2
(x, y)),
z
1
Φ
2H =
2πi
g(z) dz =
z
= aβx, x+(b + c)βx, y + dβy, y−
2
βi− β
αi− α
j
(xiyj− xjyi)2.
j
The functionsΦ
,H are obtained from Φz, H (see (2.7), (2.9)) by replacing
z
Δ=ad−bc be with r
r
−
2
i=j
2
in the coefficient of the last term. This is an insignificant
change, since this coefficient can always be normalized to 1 by rescaling if it is
not zero. Therefore these functionsH are also in involution for any two choices
of the diagonal matrix β.
The following formula is due to M. Adler. For any polynomial g(z) one has
tr g(L
or
)=trg(L0)+2ε
ε
1
d
4
dε
To prove this result we use the formula
tr(z −L
)−1− tr(z −L0)−1= ω
ε
2
tr g(Lε)
2
H + O(ε
=H. (2.14)
ε=0
3
),
dω
−1
, (2.14)
dz
144 Geometry of Quadrics and Spectral Theory
where
ω(z)=det(I −W
).
z
This follows from (2.2) and the identity
log det X =trlogX
for any nonsingular matrix X with appropriate definition of the branch (see
Kato [7]). Hence
tr g(L
ε
) − tr g(L0)=
1
2πi
g(z)ω
−1
dω
dz
dz,
and since in our case
ω(z)=det(I − W
=1−ε
2r2
+(ε
=1−ε
)=
z
2
(aQz(x)+(b + c)Qz(x, y)+dQz(y))+
+ ε4(ad − s2))(Qz(x)Qz(y) − Q
2
Φ
+ O(ε4),
z
2
(x, y)) =
z
we have
2
tr g(L
) − tr g(L0)=−
ε
ε
2πi
g(z)
d
Φ
dz + O(ε4)=ε22H + O(ε4),
z
dz
proving the formula (2.14).
§ 3. Connection with Confocal Quadrics
a. Integrals for the Geodesic Flow on the Ellipsoid
n
In the n-dimensional Euclidean space R
consider a positive definite symmetric matrix A with distinct eigenvalues.
Without loss of generality we can assume A =diag(α
0 <α
defines an ellipsoid. The quadrics U
equation
<α2<... <αn. Then the equation
1
−1
x, x =1 (3.1)
A
confocal to this ellipsoid are given by the
z
−1
(z −A)
x, x +1=0.
with inner product x, y we
, ..., αn),where
1,α2
§3. Connection with Confocal Quadrics 145
We introduce the bilinear form
Q
(x, y)=(z −A)−1x, y,Qz(x)=Qz(x, x), (3.2)
z
so that U
and (3.1) is the quadric U
is defined by
z
Q
(x)+1=0, (3.3)
z
. These confocal quadrics have a number of
0
well-known properties, for which we will give a new interpretation. For example, through any point x with x
···xn=0pass precisely n confocal
1x2
quadratics, which moreover intersect each other perpendicularly.
For any given point x
∈ Rnwe ask for the cone of lines which are tangent
0
to a quadric Q(x)+1=0 (we supress the subscript z, since it is irrelevant for
this question). By an elementary calculation one finds the equation of this cone
to be
det
= Q(x) −2Q(x, x
1+Q(x)1+Q(x, x
1+Q(x, x
)+Q(x0)+Q(x)Q(x0) − Q2(x, x0)=0.
0
Alternatively, if we set y = x −x
det
Q(y) Q(x
Q(x
0
which for fixed x
,y)1+Q(x0)
describes a cone with vertex at the origin.
0
0
,y)
)1+Q(x0)
0
this equation becomes
0
= Q(y)+Q(x
)
0
=
)Q(y) − Q2(x0,y)=0,
0
This equation agrees with
Φ
,y)=0
z(x0
if (in the notation of the previous section) we take a =0, b = −c =1, d = −1
(or a =0, b = −c = i, d =1), showing that the latter equation can be interpreted
geometrically as defining the set of lines x = x
Φ
(x, y)=0
0
describes the tangents of the ellipsoid U
0
+sy tangent to Uz. In particular,
0
; we nave changed the notation x0to x.
The Hamiltonian differential equations
˙x = −
∂
∂y
Φ
0
(x, y), ˙y =
∂
(x, y)(3.4)
Φ
0
∂x
146 Geometry of Quadrics and Spectral Theory
restricted to Φ0=0describe the motion of such tangent lines, which is easily
interpreted: The point of contact with U
moves along a geodesic while the
0
point x moves perpendicularly to this tangent.
Indeed, if the line through x in direction y =0has the point of contact
x + sy = ξ with U
,wehave
0
Q(x + sy, y)=0, or s = −
Q(x, y)
Q(y)
,
and one computes
d
d
ξ =
(x + sy)= ˙x + s ˙y +˙sy =
dt
2Φ
(x, y)
0
=
Q(y)
−1
x +2Q(x, y)A−1y = −2Q(y)A−1ξ.
−1
A
y +˙sy =˙sy,
since Φ
(x, y)=0,and
0
dy
= −2Q(y)A
dt
dt
If we introduce the parameter τ by dτ/dt =˙s,then
dξ
dt
= y,
d
dτ
2
dy
ξ
=
2
dτ
= −
2Q(y)
˙s
−1
A
ξ,
which shows that the point of contact ξ = ξ(τ) moves on a geodesic. From the
relations
˙x, y =2Φ
(x, y)=0, y, ˙y =0
0
it follows that ˙x is perpendicular to the direction of the line, and that y, y is a
constant.
Thus (3.4) can be viewed as an extension of the geodesic flow on the
∗
ellipsoid to a flow in T
Rn= R2n. The geodesic flow is obtained by constrain-
ing (3.4) to the symplectic manifold
(x, y)=0, |y|2=const> 0,
Q
0
and to the energy manifold Φ
(x, y)=0are equivalent to
Q
0
Q
(x)+1=0,Q0(x, y)=0, if Q0(y) =0
0
(x, y) (see Section 5(c)). The relations Φ0=0,
0
§3. Connection with Confocal Quadrics 147
which describes the tangent bundle of the ellipsoid, where the constrained flow
takes place.
To establish the geodesic flow as an integrable one it suffices to show that the
extended flow (3.4) is integrable (see Section 5). This follows from Theorem 1
and the formula
hence G
, G2, ..., Gnare integrals of (3.4) which are in involution.
1
(x, y)=
Φ
z
n
j=1
G
j
z − α
;
j
b. Isospectral Deformation
Since the zeros of 1 − Φ
(x, y) are the eigenvalues of the matrix
z
L(x, y)=A + x ⊗ y − y ⊗x − y ⊗ y,
they also can be viewed as integrals of the motion, and L(x, y) remains similar to
itself along the orbits of (3.4). Since the tangents of U
are given by Φz(x, y)=
z
=0, it is more natural to construct a matrix L(x, y) whose eigenvalues are the
zeros of Φ
(x, y). Such a matrix is
z
L(x, y)=P
(A − x ⊗ x)P
y
y
(3.5)
where
= I −
y ⊗y
y, y
P
y
is for |y| =1the projection into the orthogonal complement of y.
To see this we observe that the eigenvalue of
2
y ⊗ y
are given by Φ
L(x, νy)=A + ν(x ⊗y − y ⊗x) − ν
(x, νy)=1,or
z
Φ
(x, y)=
z
1
.
2
ν
This suggests letting ν tend to infinity. Of course, L(x, νy) has no limit; in
fact one of its eigenvalues λ = ν
2
(|y|2+ O(ν−2)) tends to infinity, and the
corresponding eigenvector
−1
ϕ = y + ν
Pyx + O(ν−2)
148 Geometry of Quadrics and Spectral Theory
tends to y.Now,ifwetakeν purely imaginary, then L(x, νy) is Hermitian, and
on the orthogonal complement of ϕ this matrix becomes
ϕ ⊗
L(x, νy) −
|ϕ|
ϕ
,
2
which has a limit for ν →∞the matrix (3.5). But it is also easy to verify
directly that for the matrix (3.5) one has
det(z −L)
2
|y|
det(z −A)
= −zΦ
(x, y). (3.6)
z
Thus the n−1 roots of Φ
(x, y) are eigenvalues of L and the nth eigenvalue
z
is λ =0, which corresponds to the eigenvector y. It is clear, then, that the
matrix (3.5) undergoes an isospectral deformation under the flow (3.4). We
make this more explicit by writing the differential equations (3.4) in the Lax
form
d
L =[B, L]
dt
with an appropriate matrix B. We generalize the setup right away and replace
the Hamiltonian Φ
1
H =
2
with arbitrary constants β
by
0
n
βi− β
αi− α
j
(xiyj− xjyi)2=
j
1
2
y
+
β
j
j
2
i<j
, β2, ..., βn.Forβj=2α
1
1
βjG
2
j=1
−1
we obtain H =Φ0.As
j
j
(3.7)
in the previous section, we obtain
Theorem 3. The Hamiltonian system
˙x = H
, ˙y = −Hx,
y
with H given by (3.7), can be written in the matrix form
d
L =[B, L], (3.8)
dt
where L is given by (3.5) and
β
− β
i
B = −
αi− α
j
(xiyj− xjyi), (3.9)
j
with zero diagonal elements.
§3. Connection with Confocal Quadrics 149
Corollary. If H = H (G
, ..., Gn) then the vector field XHcorre-
1,G2
sponds to the isospectral deformation (3.8), (3.9) where
j
=2
∂H
∂G
.
j
β
This statement requires a calculation similar to that in the proof of Theorem 2. The differential equations have the form
˙x = βy + Bx, ˙y =+By,
which implies
d
(x ⊗ x)=−[B, A − (x ⊗ x)] + βx ⊗ y + y ⊗βx,
dt
d
(y ⊗ y)=[B, y ⊗ y]
dt
with β = diag(β
, ..., βn).Sincey, y is an integral, the last relation
1,β2
implies
d
P
=[B, Py]
y
dt
and the first gives for M
Combining the last two relations, one finds for L = P
= A −x ⊗ x the equation
x
d
M
=[B, Mx] − βx ⊗ y −y ⊗ βx.
x
dt
(A−x⊗x)Py= PyMxPy,
y
after a short calculation,
d
L =[B, P
dt
=[B, P
y]MxPy
yMxPy
+ Py[B, Mx]Py+ PyMx[B, Py]=
]=[B, L],
which proves the statement.
c. Interpretation of the Eigenvalues and the Frame of L
According to the last theorem the symmetric matric L given by (3.5) undergoes an isospectral deformation, that is, its eigenvalues are constant and the
eigendirections are moved by an orthogonal transformation. We give a geometrical interpretation of the eigenvalues and eigenvectors of L.
150 Geometry of Quadrics and Spectral Theory
For this purpose we introduce the eigenvalues μ1, μ2, ..., μnof the ma-
trix M
and the eigenvalues λ1, λ2, ..., λnof L,whereλ1=0. Moreover, we
x
set
m(z)=
n
(z −μj),l(z)=
j=1
n
(z −λj), and a(z)=
j=1
n
(z −αj).
j=1
Then the formulae of the previous section give
(x)=
det(z −M)
det(z −A)
det(z −L)
2
det(z −A)
1+Q
−zΦ
z
(x, y)=|y|
z
The first equation shows that the eigenvalues μ
m(z)
=
, (3.10)
a(z)
l(z)
2
= |y|
of M = Mxare the elliptic
j
. (3.11)
a(z)
coordinates of x. Indeed, the relation
1+
defines elliptic coordinates, so that x ∈
that the eigenvalues λ
through x with direction y touches the quadric U
equation Φ
confocal quadrics U
as Q
(x, y) defines the tangents of Uz. The «general» line touches n − 1
z
λ
(y) =0.
z
j
Let the eigenvalues λ
for λ
=0is ϕ1= y and the eigenvectors for λj=0are the normals to the
1
confocal quadric U
λ
j
2
x
j
z − α
(j =2, ..., n) are those values of z for which the line
j
=0 for z = μ1,μ2, ..., μ
j
n
%
U
. The second equation shows
μ
j
j=1
. Indeed, we saw that the
z
n
where the λjare singled out as the roots of (3.11), as long
of L = L(x, y) be distinct. Then the eigenvector
j
at the point of contact ξj= x + sjy of the line through x
with direction y.
To prove this statement, we note that the point of contact ξ
= x + sjy is
j
determined by
(ξj,y)=0, 1+Q
Q
λ
j
(ξj)=0.
λ
j
Indeed, the first relation expresses the tangency of the line with U
second follows from
2
(ξj,y)
λ
j
if Q
0=Φ
(y) =0.
λ
j
(x, y)=Φ
λ
j
(ξj,y)=Q
λ
j
(y)(1 + Q
λ
j
(ξj)) − Q
λ
j
λ
j
and the
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