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§4. The Hyperelliptic Curve 161
are in involution, it follows that ω has the form
n
ω =
akjdλk∧ dμj,
k, j=1
and since dω =0one can — at least locally — find a function S = S(λ, μ) such
that
a
kj
Therefore it suffices to compute this function S. Setting ∂
n
S =
∂
μ
j=1
S
dμj, we write also briefly
μ
j
=
2
S
∂
∂λk∂μ
.
j
n
λ
S =
k=1
S
dλk,
λ
k
ω = ∂
To compute S we use (4.11) to write the 1-form x, dy =
x, dy =
j
&
x,
∂y
∂μ
'
dμ
=
j
j
λ∂μ
1
x, y
2
S.
dμ
μ
n
xkdykas
1
n
1
1
+
1
Q
(x, y) dμj,
μ
2
j
j=2
or with (4.15), (4.16), and b + c =2s,
s
bc
μ
+
#
j
#
P (μj)
a(μj)Δ
P (z) dz
a(z)
.
dμ
j
, (4.17)
x, dy =
1
4s
Since ω = −dx, dy = ∂
4s
1
(λ
−S =
− αk)
(λ
k
S, we read off that
λ∂μ
− αk)logμ1+
k
dμ
μ
n
1
1
+
1
2
j=2
n
1
2Δ
j=2
the integration being taken along paths on the Riemann surface. Of course, S is
determined only up to two additive functions of λ
Instead of the λ
metric functions σ
, λ2, ..., λnwe use as independent variables their sym-
1
, σ2, ..., σndefined by
1
l(z)=z
n
+ σ1z
n−1
+ σ2z
and of μkalone.
k
n−2
+ ···+ σn,
162 Geometry of Quadrics and Spectral Theory
so that
ω =
Thus S
are canonically conjugate to the symmetric functions σk.FortheseS
σ
k
we find
−
which shows that S
S
dσk∧ dμj=
σkμ
j
δ
k1
log μ
=
4s
k
−
1
for k =2, 3, ..., n agrees, up to the factor
∂S
∂σ
σ
k, j
k
n
dσk∧ d(S
k=1
μ
j
n
1
4
j=2
#
n−k
z
P (z)
). (4.18)
σ
k
dz,
Abelian differentials of the first kind (4.3).
It remains to rewrite the Hamiltonian vector fields X
in these variables.
H
Setting
H = H(x, y)=Ψ(σ, μ),
the differential equation takes the form
S
0 −S
σμ
T
σμ
0
=
˙μ
˙σ
Ψ
μ
.
Ψ
σ
In particular, if the Hamiltonian Ψ depends only on the σ,i.e. Ψ
det S
with the solutions
Thus the S
2, ..., n, then the vector field X
Since by (4.15)
=0, then the system reduces to
σμ
˙σ =0,
σ = σ
t=0
vary linearly on M. Hence if we set sk=4S
σ
k
X
H
G = sx, y = −
d
S
=Ψ
σ
σ
σ
k
σ
1
σ
+tΨσ.
t=0
∂
.
∂s
k
+α
σ
k
,
j
dt
,Sσ= S
for a Hamiltonian H(x, y)=Ψ(σ) becomes
H
n
=4
k=1
Ψ
1
2
we have
= −2
∂s
∂
,
1
X
G
σ
1
with the
4
=0and
μ
for k =1,
k
§4. The Hyperelliptic Curve 163
so that
X
H
We apply this remark to H = G
that X
have modulo XGconstant coefficients with respect to ∂/∂sk.This
G
j
n
=4
k=2
, which are functions of the σkalone, to show
j
∂
Ψ
+2Ψ
σ
k
∂s
k
XG.
σ
1
proves Theorem 4 in case (i).
d. Degenerate Case
In cases (ii) and (iii) we consider the normal form of Section 2,
L = A + r(x ⊗y − y ⊗x)+dy ⊗ y, r =0,
where d =1or d =0. The spectrum of L is given by the zeros of 1 −Φ
Φ
= dQz(y) − r2(Qz(x)Qz(y) − Q
z
To describe M and M
we consider again the auxiliary matrix
M = P
yAPy
2
(x, y)). (4.19)
z
,where
z
with the spectrum (0,μ2,μ3, ..., μn) and set μ1= x, y. The function m(z)
is defined by (4.8).
As before, we show that x, y can be expressed in terms of μ
= x, y
1
and of the spectra of L, M . For this purpose we merely have to modify the
expressions for the right-hand sides of (4.13) and (4.14).
−1
In case d =0we have from (4.19), taking the coefficient of z
in the
expansion at z = ∞,
n
(λj− αj)=dy2,
j=1
so that y
tr(L − A)=
2
is a function of the λk, while μ1= x, y is an independent variable
on M. The formulae (4.13), (4.14), and (4.16) are to be replaced by
n
1
y −
−1
g
Q
j
2
j=2
#
(a(z) − l(z))a(z)=
μ
j
(x, y)
∂y
∂μ
j
1
a(z)Δ
,
#
P (z)
Q
(x, y)=
z
x =
x, y
2
y
1
ra(z)
for z = μ
, μ3, ..., μn. This latter formula agrees with (4.16), since Δ=r2.
2
164 Geometry of Quadrics and Spectral Theory
These equations, together with (4.10), again give x, y in terms of λ, μ,
solving the inverse problem for d =1. The mapping of M
variety of w
2
= P(z) is the same as before.
into the Jacobi
We determine the symplectic structure from
n
1
−
2
2
j=2
−
(λ
1
2
k
Q
(x, y) dμj=
μ
j
logy
λ∂μ
− αk)+
2
dμ1−
S with
1
2Δ
1
2Δ
j=2
n
a−1(μj)$P (μj)dμj,
j=2
μ
#
j
n
P (z) dz
a(z)
k
, (4.20)
of L.
= d
y, dy
1
y
μ
1
2
2
logy
x, dy = μ
leading to the symplectic form ω = ∂
μ
1
S =
where P (z)=r
the curve w
2
log
2
2
(a − l)l is a polynomial of degree 2n −1, and the genus of
= P(z) is again n − 1. One branch point of the corresponding
Riemann surface is at ∞,andn of them are the eigenvalues λ
Finally, in the case (iii) with d =0one has
n
(λj− αj)=dy2=0,
j=1
so that only n − 1 of the λ
Similarly M has only n −1 independent eigenvalues μ
As additional variables we could use μ
,sayλ2, λ3, ..., λn, are independent variables.
j
, ..., μn, while μ1=0.
2
= G =
1
1
x, x and F = x, y and
2
proceed as above. This is in accordance with our reduction at the beginning
of this section. Notice that the first term in S in (4.17) and (4.20) reflects the
exponential and linear behavior of μ
1
in t.
e. Limit Cases
The isospectral manifold of L = L(x, y) is defined by
(x, y)=1,j=1, 2, ..., n.
Φ
λ
j
In Section 3 we were interested in the manifold
(x, y)=0,j=1, 2, ..., n− 1,
Φ
λ
j
§4. The Hyperelliptic Curve 165
which represented the spectrum of P
(A − x ⊗ x)Py. This manifold can be
y
obtained as a limit case from the former, as was shown in Section 3. We want to
indicate how one can determine the hyperelliptic curve for this situation, at least
for a =0, b = −c = −d =1.
In Section 3 we saw that the spectrum of
= L(x, νy)=A + ν(x ⊗y − y ⊗x) − ν2y ⊗ y
L
ν
is given by the equation
1
ν
n
(z −λj(ν))
,
2
where Φ
of L
,weset
ν
Φ
(x, y)=
z
(x, y) is the function belonging to L1.Ifλj(ν) are the eigenvalues
z
l
(z)=
ν
j=1
and have the hyperelliptic curve
2
w
= Pν(z)=(r2a(z) − Δlν(z))a(z).
We assume again (a, b, c, d)=(0, 1, −1, −1). In the limit ν →∞one
root, say λ
polynomial of degree n − 1 having the eigenvalues of P
roots. Let l
= λ1(ν), tends to infinity, and ν−2lν(z) has a limit, namely a
1
(A − x ⊗ x)Pyas
(0)
(z) be the polynomial of degree n −1 and highest coefficient 1
y
having these roots. Then we conclude
Φ
(x, y)=
z
1
1 −
2
ν
l
(z)
ν
a(z)
→ k
(0)
l
a(z)
(z)
with some factor k independent of z. This factor is determined by the asymptotic
behavior of Φ
for large z as
z
k =(ax, x +2sx, y + dy, y)=2G(x, y).
Thus
(0)
(z)
l
a(z)
,
so that zl
Φ
(x, y)=2G(x, y)
z
(0)
(z) is the characteristic polynomial of Py(A − x ⊗ x)Py.
166 Geometry of Quadrics and Spectral Theory
We will not repeat the above construction in this case, but note that as
auxiliary variables one could use the eigenvalues of either
M = P
yAPy
or N = A − x ⊗x.
The eigenvalues of the latter equation represent the elliptic coordinates of x,
while the eigenvalues of M represent the orthogonal coordinates of y on the
sphere |y| =1which we used previously. In any event the isospectral manifold M of P
(A − x ⊗ x)Pyleads in this case to the Jacobi variety of the
y
hyperelliptic curve
2
w
= a(z)l
(0)
(z)=z−1det(z −A)det(z −L).
This case is particularly pleasant, since the branch points of this polynomial are
given by the eigenvalues of A and L where the trivial zero eigenvalue of L is
omitted.
1M
In this case the manifold
=(M− M∩K)/Γ=M/Γ can be interpreted
geometrically. It is the manifold of common tangents to n −1 confocal quadrics
, U
U
λ
, ..., U
λ
1
2
into each other by the reflections x
the common tangents to n −1 confocal quadrics form a 2
, where those 2ntangents are to be identified which go
λ
n−1
→±xj. Thus we obtain the result that
j
n
-fold covering of the
Jacobian variety (see Staude [20]).
§ 5. Examples of Integrable Flows
a. Constrained Systems
In the following examples we will have to constrain a Hamiltonian system
, ˙y = −H
y
m
dyj∧dxj, to a symplectic submanifold.
j=1
x
in the symplectic space (R
˙x = H
2n
,ω) ω =
For our purposes it will suffice to describe the submanifold M by 2r equations
(5.1)
M : F
(x, y)=... = F2r(x, y)=0.
1
If
det({F
then the manifold M is symplectic with ω
1
By the remarks at the beginning of this section, we can drop K .
}) =0 (j, k =1, 2, ..., 2r), (5.2)
j,Fk
the two-form restricted to TM.
M
§5. Examples of Integrable Flows 167
The vector field (5.1), which we will denote also by X
tangential to M, but we can construct such a vector field by replacing H with H
, need not be
H
M
the restriction of H to M . Then the function HMon the symplectic manifold (M, ω
field X
) defines a vector field X
M
will be called the constrained vector field.
H
M
There is another way to describe this constrained flow: Since ω
degenerate in TM, there exists a complementary space (TM)
which is tangential to M. This vector
H
M
⊥
of TM in R2n,
M
is non-
which is orthogonal to TM with respect to the symplectic structure. Now it is
easily seen that X
of X
into TM with respect to the above splitting of R2n.
H
H
− X
∈ (TM)⊥.InotherwordsX
H
M
is the projection
H
M
Effectively one can describe the constrained vector field by a Hamiltonian
2r
∗
H
= H −
In order for X
∗
to be tangential to M we have to require that
H
∗
0=X
Fk= {H, Fk}−λj{Fj,Fk} on M,
H
which by (5.2) determines the functions λ
Next we turn to a more special situation where F denotes a class of functions
in involution in (R
2n
,ω), and consider a vector field XHconstrained to a
λjFj(x, y). (5.3)
j=1
uniquely on M .
j
symplectic manifold
M : F
= F2= ...= Fr=0,G1= G2= ...= Gr=0, (5.4)
1
where we assume that
, ..., Fr,H∈ F, (5.5i)
F
1,F2
det{F
} =0 (i, j =1, 2, ..., r). (5.5ii)
i,Gj
Clearly this is a special case of the previous situation since (5.5i, ii) imply (5.2)
if we set F
takes the form
Since the F
= Gjfor j =1, 2, ..., r. The constrained Hamiltonian (5.3)
j+r
r
0=
(λjFj+ μjGj).
j=1
r
μj{Gj,Fk}
j=1
∗
= H −
H
and H are in F and hence in involution, we conclude
j
168 Geometry of Quadrics and Spectral Theory
and hence μj=0. Thus the constrained Hamiltonian has the form
r
H
∗
= H −
λjFj,
j=1
and the corresponding vector field is given by
∗
= XH−λjX
X
H
F
j
on M.
This has the consequence that for any function E ∈ F we have on M
∗
E =0,
X
H
i. e., E
situation (5.4), (5.5) the constrained flows have the restriction of the functions
in F as integrals. It is evident that these functions in the class F
is an integral of the constrained vector field. We conclude that in the
M
, obtained
M
from F by restriction to M, are in involution.
We will apply this simple device of constraining an integrable system to
1
obtain a new integrable system in the following examples
.
b. A Mass Point on the Sphere S
n−1
: |x| =1under the Influence of the
Force −Ax (C. Neumann [14])
The differential equation of this system is
¨x = −Ax + λx,
where λ is chosen so that |x| =1, x, ˙x =0. We obtain this system by
constraining the Hamiltonian
H =
1
Ax, x +
2
1
2
(|x|
|y|2−x, y2)
2
to the symplectic submanifold
1
M : F =
1
Note added in proof : Other integrable systems have been recognized as constrained systems.
See P. Derft, F. Lund and E. Trubowitz, Nonlinear wave equations and constrained harmonic motion,
Comm. Math. Phys. 74, 141–188 (1980).
2
(|x|
− 1) = 0,G= x, y =0.
2
§5. Examples of Integrable Flows 169
2
Observe that {F, G} = |x|
=1on M. Moreover, if F is the class of functions
generated by
Φ
(x, y)=Qz(x)+Qz(x)Qz(y) − Q
z
2
(x, y),
z
then the expansion at z = ∞ takes the form
1
z
Φ
(x, y)=
z
|x|
2
1
+
z
(Ax, x +(|x|2|y|2−x, y2)) + O
2
z
Hence both F, H belong to F, and the flow restricted to M is given by
∗
H
= H − λF, λ = {H, G} = Ax, x.
The equations have on M the form
∗
˙x = H
˙y = −H
or
which is the desired flow if |y|
The functions in F restricted to M are the desired integrals. In this case we
have a = −1, d = −c =1, d =0hence the symmetric part of
of rank 1. If we set
= Hy− λFy= |x|2y = y,
y
∗
= −Hx+ λFx= −Ax −|y|2x + λx,
x
2
¨x = −Ax − (|y|
2
− λ is renamed λ.
n
Φ
=
z
j=1
− λ)x,
Gj(x, y)
z − α
j
,
ab
cd
we easily see that
n
2H =
αjGj(x, y), 2F +1=
j=1
which shows explicitly that H, F are functions of the G
of this system. These integrals G
have been found by K. Uhlenbeck (see [21]
j
n
Gj(x, y),
j=1
, which are the integrals
j
and Devaney [3]).
.
3
is
a 0
c. A Mass Point on the Ellipsoid Q
(x)+1=0under the Influence of
0
the Force −ax (Jacobi [6])
The motion of a free particle under the influence of the force −ax is
described by the Hamiltonian
1
2
(|y|
2
+ a|x|2).
H =
170 Geometry of Quadrics and Spectral Theory
To describe the restricted motion we introduce the class of functions F generated
by
(x, y)=aQz(x)+Qz(y)+Qz(x)Qz(y) − Q
Φ
z
2
(x, y),
z
which are in involution by Section 2. We set
F (x, y)=a +Φ
=(1+Q
G(x, y)=Q
(x, y)=
0
(x))(a + Q0(y)) −Q
0
(x, y),
0
2
(x, y),
0
and restrict the motion to
F (x, y)=0,G(x, y)=0.
Since H, F ∈ F it follows that the restricted flow is described by
∗
H
= H − λF,
and hence has all functions of F as integrals. It remains to identify this flow with
the desired one. For this purpose we note that G(x, y)=0and F (x, y)=0
implies
1+Q
(x)=0 or a + Q0(y)=0.
0
We p ic k a + Q
verifies. Thus we have 1+Q
(y) =0. This condition is invariant under the flow, as one
0
(x)=0, Q0(x, y)=0, which means that x lies
0
on the ellipsoid and y is tangential to it at x. The differential equation becomes
∗
˙x = H
˙y = −H
= Hy− λFy= y,
y
∗
= −Hx+ λFx= −ax −2λ(a + Q0(y))A−1x,
x
or
¨x = −ax −2λ(a + Q
(y))A−1x,
0
which is the equation of the constrained motion. Thus this system is integrable
and G
geodesic flow on the ellipsoid. Notice that the resulting integrals G
are the desired integrals in involution. For a =0we obtain the
j
M
are ob-
j
tained from those of the Neumann system (Section 5 (b)) by the symplectic map
(x, y) → (y, −x).
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