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§6. Geodesics on an Ellipsoid 101
For this purpose one introduces the family of confocal quadrics, defined by
(x)=(a − z)−1x, x =1 (2)
Q
z
for real z = a
z<a
etc. they are hyperboloids; in fact one has n +1different types of such
2
quadrics corresponding to the intervals (−∞,a
Through every point x ∈ R
quadratics (2). Indeed for a given such x the rational function 1 − Q
.Forz<a0these are ellipsoid, but for a0<z<a1or a1<
ν
), (a0,a1), ..., (a
n+1
with
n
ν=0
0
xν=0 pass n +1 such confocal
n−1,an
(x) has
z
).
n +1distinct roots, one in each of the above intervals and one can write
⎧
⎪
⎨
⎪
⎩
1 − Q
u
0<a0
(x)=
z
n
z −u
ν
,
z −a
ν=0
ν
<u1<... <un<an.
(3)
The equation z = uνdefines the desired n+1 hyperquadrics intersecting at x;it
is well known that they intersect orthogonally. Actually they possess a stronger
orthogonality property which is less well known — and which will be of interest
to us. The following theorem is due to Chasles [0].
Theorem. If L is a straight line which touches two confocal quadric Q
(1)
=1, Q
=1at the points x
z
2
points of tangency are perpendicular to each other, provided z
P
ROOF.
(2)
, x
then the normal of the quadrics at these
1
= z2.
=
z
1
Let the points of L be given by x + ty where t varies over R,andlet
(j)
x
= x + tjy. The condition of tangency of L at x
(j)
(x
∇Q
),y =0,Q
z
j
z
(xj)=1 (j =1, 2). (4)
j
(j)
is given by
102 Various Aspects of Integrable Hamiltonian Systems
We want to show that
∇Q
(1)
(x
), ∇Q
z
1
(2)
(x
) (5)
z
2
vanishes. For this purpose we use the identity
− z2)(a − z1)−1ξ, (a − z2)−1η = (a − z1)−1ξ, η−ξ, (a − z2)−1η.
(z
1
Therefore, if Q
(x, y) is the symmetric bilinear form belonging to Qz(x) we
z
have for (5)
4(z
1−z2
)−1(Q
+(t
(x
z
1
− t1)Q
2
(1),x(2)
z
(x
1
) −Q
(1)
(1),x(2)
(x
z
2
,y) − Q
)) = 4(z1−z2)−1(Q
(2)
(x
) − (t1− t2)Q
z
2
(1)
(x
)+
z
1
(2)
(x
,y))
z
2
which vanishes on account of (4).
This remarkable property implies that a line — which generally touches n
confocal quadrics — has n mutually perpendicular normals associated with it.
These normals are clearly also perpendicular to the line, so that we have an
orthogonal (n +1)frame associated with each such line.
c) To determine the differential equations of the geodesics of the ellipsoid
we generalize this question in two ways: (i) We replace the ellipsoid by any of
the confocal quadrics Q
(x)=m and (ii) determine the extended «line flow» of
z
the previous section.
For this purpose let u
(x)=z be any of these quadrics and consider all
j
lines x + ty which are tangential to it. Since for the given value z we have
(x)=m the condition for x+ ty to represent a tangent to Qz(x)=m is that
Q
z
for the point of contact ξ = x + t
∗
y
(ξ)=m, Qz(ξ, y)=0
Q
z
or, if Q
z
∗
t
= −Qz(x, y)/Qz(y).
Inserting this into the first we see that x + ty is tangential to Q
only if
(x, y)=mQz(y) − (Qz(x)Qz(y) − Q
Φ
z
vanishes, always under the assumption Q
(y) =0.
z
2
(x, y))
z
(x)=m if and
z
§6. Geodesics on an Ellipsoid 103
Thus we can take Φ
to Φ
=0.
z
(x, y) as the Hamiltonian for the desired flow restricted
z
As a consequence of the theorem and the proposition of the previous section
we have
, Φ
{Φ
z
1
} =0 if Φ
z
2
=0, Φ
z
1
=0,
z
2
i. e. the flows commute.
For large values of m we have
1
Φ
(x, y)=
z
m
and it follows that the zeroes of Φ
n
ν=0
z
2
y
ν
aν− z
are distinct if m is large enough. For Φ
+ O
1
m
with distinct zeroes one can show now that the above relation
{Φ
, Φ
} =0
z
z
1
2
, Φ
holds without further restriction, i. e. the Φ
z
are in involution (see Exer-
z
1
2
cises 1, 2). This holds for large m, but the above expression being quadratic
in m, it holds identically for all m, in particular for m =1.
If we express
one computes the F
Φz(for m =1) in partial fractions
n
Φ
z
as residues to be
ν
2
= y
F
ν
+
ν
Fν(x, y)
=
ν=0
(xνyμ− xμyν)
μ=ν
aν− z
aν− a
μ
(6)
2
. (7)
z
Being functions of Φ
Thus the fact that the F
, ..., Φ
z
0
for n +1distinct z0, ..., znwe also have
z
n
{F
are in involution is a reflection of the geometrical
ν
} =0. (8)
ν,Fμ
proposition of Section 5. Of course, this fact can also be verified by direct
calculation (Exercise 3).
The Hamiltonian of the line flow for the ellipsoid Q
expressed in terms of the F
ν
as
Φ0=
n
ν=0
−1
a
Fν.
ν
(x)=1can be
0
104 Various Aspects of Integrable Hamiltonian Systems
Theorem. The above Φ0-flow is integrable and possesses the rational
integrals F
given by (7) which are in involution.
ν
Corollary. The geodesic flow on the ellipsoid is integrable and has the
integrals F
. The energy takes the form
ν
1
2
|y|
2
n
1
=
Fν.
2
ν=0
Corollary. The real components of the algebraic manifold M : F
= c
ν
(ν =0, 1, ..., n) is a torus times a line R1,ifdFνare linearly independent
of M .
XERCISE 1. Define the polynomials
E
n
P
(x, y)=|y|−2A(z)Φzwhere A(z)=
z
1
m
(aν− z).
ν=0
Show that
{P
,P
using the identity
XERCISE 2. Let P (z, x, y)=z
E
} = |y|−4A2(z){Φ
z
z
1
2
{|y|
2
, Φz} =0.
n
+ p1(x, y)z
, Φ
}
z
z
1
2
n−1
+ ... + pn(x, y) be a
polynomial with rational coefficients. If P (z, x, y) has distinct zeroes somewhere then
,P
{P
z
1
} =0 if P
z
2
=0 and P
z
1
=0
z
2
implies that
,P
} =0.
z
z
1
2
INT. Factorize P =
H
that at a point (
{P
n
(z −uν(x, y)) in some neighborhood and show
◦
x0,◦y0) and for z1= uν(◦x0,◦y0), z2= uμ(◦x0,◦y0) = z1one has
ν=0
ν
{P
,P
} = c{uν,uμ} =0
z
z
1
2
with c =0. Thus show that the zeroes u are in involution, hence also P
for al z1, z2, x, y.
, P
z
z
1
2
§6. Geodesics on an Ellipsoid 105
Note that the assertion of Exercise 2 does not hold if double roots occur: if
z
z
= Q
Q
1
2
z
{Q
,Q
1
and |y|
} =0!
z
2
2
z
z
2
2
μ
P
then for P
E
E
= P
z
1
XERCISE 3. Show the functions (7) are in involution by direct calculation.
XERCISE 4. Show that the n +2functions
=0we always have
z
2
{P
,P
} =4Q
z
z
1
2
=
μ=ν
(xνyμ− xμyν)
G
ν
aν− a
are in involution. Therefore they must be functionally related; indeed
n+1
Gν≡ 0.
ν=0
E
(x)=0,z1= z
Q
z
j
2
or tangential to such a cone and the sphere
|x| =1
has perpendicular normals at the points of contact.
(This fact leads to the expressions G
E
XERCISE 6. The tangents of any geodesics on an ellipsoid are tangent to
of Exercise 4.)
ν
the same set of confocal quadrics, i. e. independently of the point on the given
geodesic (see [5]).
With minor changes one can show that the motion on an ellipsoid under
the influence of a potential |x|
2
is also integrable. This was shown already by
Jacobi [1].
106 Various Aspects of Integrable Hamiltonian Systems
References
[0] L. Bianchi, Vorlesungen ¨uber Differential geometrie, 2, deutsche Auflage,
Teubner 1910, 345 ff.
[1] C. Jacobi, Vorlesungen ¨uber Dynamik, Gesammelte Werke, Supplement
band, Berlin, 1884.
[2] H. Sch¨uth, Stabilit¨at von periodischen Geod¨atischen auf n-dimensionalen
Ellipsoiden, Dissertation, Bonn, 1972.
[3] A. Thimm, Integrabilit ¨at bien geod¨atischen Fluss, Diplomarbeit, Bonn,
1976. (In this paper (in Theorem 4.1) it is shown that the geodesic flow
on the ellipsoid admits «global» integrals in involution. This fact is evident
from our representation of the integrals.)
[4] K. Weierstrass, Math. Werke I, 257–266.
[5] D. Hilbert, and Cohn Vossen, Auschauliche Geometrie, Dover, 1955, 197.
§ 7. An Integrable System on the Sphere
a) A point moving on the sphere Sn: |x| =1under the influence of a
1
quadratic potential U (x)=
system. For n =2this was shown by C. Neumann in 1859 [1] using Ja-
cobi’s approach of separating variables in the Hamilton– Jacobi equations. First
we proceed differently and show that this system is obtained by reduction of
another integrable system in R
ables.
The equations of motion are
ax, x, a =diag(a0, ..., an) is an integrable
2
2(n+1)
. Then we will apply separation of vari-
= −aνxν+ λx
¨x
ν
ν
(1)
where λ is determined in such a way that the particle says on the sphere. This
leads to
2
λ = ax, x−|˙x|
. (2)
Inserting (2) into (1) we obtain the desired nonlinear system of differential
equations which we want to establish as an integrable one.
§7. An Integrable System on the Sphere 107
b) We compare the above system with the Hamiltonian system
= H
˙x
ν
, ˙yν= −H
y
ν
(ν =0, 1, ..., n)(3)
x
ν
with
1
H =
ax, x+
2
Note that this system has the integral |x|
symplectic action (x, y) → (x, y +2xs) generated by |x|
1
2
(|x|
|y|2−x, y2). (4)
2
2
since it is invariant under the
2
. Therefore it is
sensible to reduce the above system by this integral, which we fix at |x| =1.
The isotropy group is G = R and given by the action (x, y) → (x, y +2xs).
We characterize the quotient manifold
{(x, y) ||x| =1}/G
by picking the point of minimal distance in the line y +2xs, obtaining as reduced
manifold
M = {(x, y) ||x| =1, x, y =0}. (5)
To determine the reduced flow we determine
0
= H − μ(|x|2− 1)
H
in such a way that
0
, |x|2− 1} = {H0, x, y} =0.
{H
The first condition is, of course, satisfied as |x|
2
is an integral of H, hence of H0,
but the second yields
1
μ =
ax, x
2
and therefore
1
=
2
1
ax, x(|x|
2
ax, x|x|
H
0
=
1
ax, x−
2
Dropping the term x, y
1
0
H
=
2
2
− 1) +
2
−ax, x(|x|2− 1) +
2
whose gradient vanishes on M we obtain
2
(|y|
+ ax, x)|x|2−ax, x(|x|2− 1)
1
2
(|x|
|y|2−x, y2)=
2
1
2
(|x|
2
|y|2−x, y2).
108 Various Aspects of Integrable Hamiltonian Systems
and the reduced differential equations (on M)
0
ν
= H
νxν
= yν|x|2= y
y
ν
+(ax, x−|y|2)x
ν
ν
˙yν= −H
˙x
= −(|y|2+ ax, x)xν− aνxν+2ax, xxν=
x
ν
= −a
which is precisely the system (1), (2).
Thus we have shown: The system (1), (2) is obtained from (3), (4) by
reduction to the manifold M.
c) The system (3), (4) is integrable, and possesses the rational integrals
2
. (6)
μ
F
(y, x)=x
ν
+
μ=ν
(xνyμ− xμyν)
aν− a
2
ν
These are the functions of the previous section with x, y exchanged — and
therefore are in involution.
To prove this assertion it suffices to show that H of (4) is a function
, ..., Fn. But it is readily verified that
of F
0
n
1
H =
2
ν=0
aνFν(y, x).
Thus this system and the geodesic flow and the ellipsoid are closely re-
lated — although not equivalent. If one makes the «hodograph transformation»
(x, y) → (y, −x) then the integrals of one go into those of the other. One is
governed by the Hamiltonian
n
−1
a
ν=0
F
ν
ν
(7)
the other by
n
1
aνFν. (8)
2
ν=0
Although this problem was treated already in 1859 by C. Neumann he did
not arrive at the algebraic integrals. They were found by K. Uhlenbeck [2] and
Devaney [3] a few years ago.
§7. An Integrable System on the Sphere 109
d) Hamilton Jacobi equations. We show how this problem was solved by
C. Neumann following the pattern of Jacobi, who used separation of variables of
the Hamilton – Jacobi equation. This technique requires an: appropriate choice
of variables. In this case these variables are «elliptical spherical coordinates»
(elliptische Kugel koordinaten) which are defined as follows:
For given a
define the u
ν=0
where the u
<a1< ... < anand x =(x0,x1, ..., xn),
0
= uj(x) as the solutions of the equation
j
n
2
x
ν
z −a
interlace the aνas follows:
j
n
=
ν
(z −uj)/A(z); A(z)=
j=1
n
ν=0
(z −aν)(9)
n
ν=0
xν=0
a
0<u1<a1
According to (9) we have for z = u
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Thus the u
(x) can be viewed as «coordinates on the sphere». z = ujdefines
j
<... <un<an. (9)
(x)
j
n
ν=0
2
x
ν
z −a
n
ν=0
=0,
ν
2
x
=1.
ν
the intersection of the sphere with one of a family of confocal cones.
2
From (9) one can express the x
rationally in terms of the uj—e.g. by
ν
computing the residue
n
U(a
)
=
ν
A(aν)
where U (z)=
n
in terms of u1, u2, ..., un.
2
x
ν
These formulae express the x on S
Similarly like the elliptical coordinates the u
(z −uj). (10)
j=1
form an orthogonal system
j
of coordinates. Indeed, one computes
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
ds
n
2
=
(dxν)2=
ν=0
U
(uj)
1
= −
g
j
4
A(uj)
n
j=1
gjdu
2
,
j
.
(11)
110 Various Aspects of Integrable Hamiltonian Systems
To prove this we compute the coefficient of dujdukin ds2. Since by (10)
n
j=1
du
uj− a
j
.
ν
2 dx
x
ν
=
ν
The coefficient of du
n
1
4
(uj− aν)(uk− aν)
ν=0
2
x
ν
is found to be
jduk
=
4(uj− uk)
n
1
ν=0
2
x
ν
uk− a
n
−
ν=0
uj− a
ν
2
x
ν
=0
ν
if j = k and for j = k equal to
n
1
g
=
j
4
ν=0
1
= −
4
dz
x
(uj− aν)
U(z)
d
A(z)
2
ν
z=u
2
j
= −
= −
1
4
dz
1
4
d
ν=0
U
(uj)
A(uj)
n
2
x
ν
z −a
=
ν
z=u
j
which establishes (11) — and the orthogonal character of these coordinates. To
describe the differential equation we use the variational principle
δ(T − V ) dt =0
where, by (11),
n
2
=
aνx
1
2
j=1
2
ν
2
gj˙u
,
j
n
1
=
aν−
2
ν=0
j
by
n
1
uj.
2
j=1
−2
in (9). If we
1
T =
|˙x|
2
n
1
V =
2
ν=0
The second equality follows by comparing the coefficients of z
introduce the canonically conjugate variables v
the Hamilton function becomes
H = T + V =
∂T
v
=
j
∂ ˙u
1
2
j
j=1
n
= gj˙u
(g
j
−1
2
v
− uj) (12)
j
j
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