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§4. The Inverse Square Potential 91
or
−1
L(x, y)U = K(ξ),
U
−1
K(x)U = −L(ξ, η),
U
(11)
where (x, y) → (ξ, η) is the induced mapping. Notice that this mapping is
a symplectic algebraic mapping of {x
{ξ
<... <ξn,η∈ Rn}.
1<ξ2
It takes our Hamiltonian
1
tr L2(x, y) into
2
1
tr K
2
<x2< ... < xn,y∈ Rn} onto
1
n
(ξ)=
1
2
ξ
k
2
k=1
2
and therefore the differential equations into the linear ones˙ξ =0, ˙η = −ξ.
g) Scattering map. If we observe that in our problem the particles repel
each other one sees that the solution runs apart and has asymptotic behavior,
x
(t)=αkt + βk+ O(t−1) for t → +∞
k
(t)=αk+ O(t−1)
y
k
where the α
are distinct. If we insert this estimate in L(x(t),y(t)) and recall
k
that its eigenvalues are independent of t then we see that α
, ..., αnare the
1
(12)
eigenvalues of L(x(t),y(t)), in particular of L(x, y) where x = x(0), y =
= y(0). Thus in (11) we can identify the ξ
with the asymptotic velocities yj(∞).
j
Similarly, by inserting the flows in the second equation of (11) we get
−1
U
K(x(t))Ut= −L(ξ, η − tξ)=L(−ξ, tξ − η)
t
from which we read off that −η
= βj. Thus with the mapping (x, y) →(ξ, η)
j
defined by (11) we have
(t)=ξkt − ηk+ O(t−1),
x
where x
k
(t)=ξk+ O(t−1)
y
k
, ykare the initial values. Thus (11) is the scattering mapping of the
k
for t → +∞
initial values (x, y) into the asymptotic velocities ξ and phases −η.
For t →−∞one obtains similarly asymptotic formulas, say
−
k
t − η
−
+ O(t−1) for t →−∞.
k
x
= ξ
k
92 Various Aspects of Integrable Hamiltonian Systems
Then it follows at once from the reversibility of the system that
−
ξ
= ξ
k
n−k+1
,ηk= η
−
n−k+1
showing that the scattering of this system is the same as for elastic reflections.
E
XERCISE 1. Show that for any 2 function f
the functions tr fj(X2+ Y2)
j
(j =1, 2) are in involution.
The same holds for tr f
E
XERCISE 2. Show that under the above reduction the Hamiltonian
(XY ).
j
H =
1
2
tr(Y
2
+ a2X2)
goes over into the Hamiltonian
|y|
1
2
2
a
+
2
1
2
XERCISE 3. The time dependent canonical transformation
E
|x|2+
(xk− xj)−2.
k<j
(t, X, Y ) → (τ,X,Y )
t =tanτ,
X =cosτX(tan τ),
Y =(cosτ)
−1
Y (tan τ ) −sin τX(tan τ),
1
=
takes the Hamiltonian system with H
H
INT. Note that the solutions X (t)=X (0) + tY (0), Y (t)=Y (0) go into
X(τ)=cosτX(0) + sin τY (0),
Y (τ )=−sin τX(0) + cos τY (0).
E
XERCISE 4. The above transformation of Exercise 3 satisfies
tr Y2intoH =
0
2
1
tr(Y2+X2).
2
[X,Y ]=[X, Y ].
Therefore reduction of the systems via (8), (9) gives rise to the transformation (2)
for a =1of Perelomov.
(Exercises 2, 3, 4 are personal communications by M. Adler.)
References 93
References
[1] D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and
dynamical systems of Calogero type, to appear, Comm. Pure Appi. Math.
1978.
[2] M. Adler, Completely Integrable Systems and Symplectic Actions, MRC Re-
port # 1830, Univ. Wisconsin, 1978.
[3] M. Adler, On a trace functional for formal pseudo-differential operators and
the symplectic structure of the Korteweg – de Vries equation, preprint, to be
published.
[4] F.Calogero, Solutions of the one dimensional n-body problems with
quadratic and/or inversely quadratic pair potentials, Jour. Math. Phys. 12,
1971, 419–436.
[5] J. Moser, Three integrable Hamiltonian systems connected with isospectral
deformations, Adv. Math. 16, 1975, 197–220.
§ 5. Extension of the Geodesic Flow
a) The geodesic flow on a sphere Sn:
|x| =1
where x =(x
, ..., xn) ∈ R
0,x1
n+1
is described by the differential equation
¨x = λx;
here the Lagrange parameter λ is determined such that |x| =1is compatible
with the differential equation, i. e. x, ˙x =0, x, ¨x + |˙x|
2
=0. Hence taking
the inner product with x gives
λ = −|˙x|
2
and the above equation becomes
2
¨x = −|˙x|
x. (1)
94 Various Aspects of Integrable Hamiltonian Systems
This system can be considered as a Hamiltonian system
˙x =
∂H
, ˙y = −
∂y
∂H
∂x
with
1
2
|x|
|y|
2
2n+2
which after restriction
H =
2
to be restricted to the tangent bundle |x| =1, x, y =0.
Thus we have in (2) a Hamiltonian system in R
to the tangent bundle becomes the geodesic flow on the sphere.
b) We want to generalize this construction to arbitrary n-dimensional
surfaces isometrically embedded into R
Let f (x) be a smooth function in R
n+1
with the usual metric ds2=
n+1
which at first is assumed to be strictly
convex, and such that f →∞as x|→∞.Wedefine
f(x + ty), (3)
t
, ˙y = −
∂F
. (4)
∂x
for x, y ∈ R
F (x, y)=min
n+1
and consider the Hamiltonian system
∂F
˙x =
∂y
We have the obvious
Lemma 1. The system (4) possesses the integrals
2
|y|
and F (x, y).
n
ν=0
(2)
(2)
dx
2
.
ν
We may restrict ourselves to solutions of (4) with |y| =1,andF =0.
We will interpret a point (x, y) in phase space with |y| =1as a line through
the point x ∈ R
n+1
with direction y. Thus (x, y) is represented by an oriented
line L = L(x, y) with a distinguished point x on it. We will assume that the
gradient f
does not vanish if f =0,sothatf =0represents a smooth manifold.
x
Theorem 1. If x = x(t), y = y(t) is a solution of (4) with |y| =1, F =0
then the line L = L(x(t),y(t)) is tangent to the surface
f(x)=0
and the point of tangency ξ(t) of L(x(t),y(t)) and {f =0} moves on a geodesic
of the manifold {f =0}.
§5. Extension of the Geodesic Flow 95
ROOF.
P
Let s = s(x, y) be determined such that
F (x, y)=f (x + sy),
i. e. that f(x + ty) has its minimum at t = s. Then one has clearly
F
,y = fx(ξ),y =0 with ξ = x + s(x, y)y. (5)
x
This implies
= fx(ξ)+fx(ξ),ysx= fx(ξ),
F
x
= sfx(ξ)+fx(ξ),ysy= s(x, y)fx(ξ).
F
y
Hence the differential equation (4) become
˙x = sf
(ξ), ˙y = −fx(ξ).
x
For the point ξ = x + sy we get the equation
˙
ξ =˙x + s ˙y +˙sy =˙sy,
˙y = −f
.
x
If we use s = s(x(t),y(t)) as independent variable we find
d
ds
2
ξ
2
dy
= −˙s
ds
= −˙s−1fx.
−1
f
x
or
dξ
ds
= y,
This is the differential equation for a geodesic of f =constsince the second
derivative of ξ is normal to the surface.
If we restrict ourselves to F =0then f (ξ)=f(x + s(x, y)y)=0and ξ
is the point of tangency of the line x + ty to f =0. This proves the assertion.
Thus we can visualize the solutions of (4) on H =0as the motion of lines:
The lines move in such a away as to be tangent to one and the same geodesic
on f =0. The distinguished point x on the line moves perpendicular to this
line in this process. We will refer to this flow as the «line flow» associated with
f(x)=0.
96 Various Aspects of Integrable Hamiltonian Systems
c) This shows that the system (4) is related to the geodesic flow on f =0.
We make this more formal and consider the submanifold of R
M = {(x, y) ∈R
2n+2
; |y|2=1, fx(x),y =0}. (6)
2n+n
given by
This is a symplectic submanifold, as in general any submanifold given by
=0,F2=0, ..., F2r=0 (7)
F
1
is symplectic provided
det{F
} =0 (8)
k,Fj
where {} are the Poisson brackets. In our case
2
{|y|
,fx(x)y} = −2fxx(x)y, y < 0
since we assumed f to be strictly convex. We restrict H to this manifold and
denote this restriction by H
M
.ThenHMdefines a Hamiltonian vector field X
M
tangential to M.
We consider in general the question of describing the Hamiltonian vector
field X
on a symplectic manifold M defined by (7) and the restriction HMof
M
a function H to M . This vector field can be obtained from a Hamiltonian H
in the embedding space which on M agrees with H, but generally has different
derivatives. Indeed the vector field defined by H need not be tangential to M .
We se t
H
0
= H −
2r
ρ=1
λρF
ρ
(9)
0
where the λ
which assures that the flow defined by H
agrees with X
are determined such that
ρ
0
{H
,Fσ} = {H, Fσ}−
.
M
2r
ρ=1
0
is tangential to M . This vector field
We apply this approach to the symplectic manifold
1
F
=
1
2
(|y|
− 1),F2= fx(x),y
2
λρ{Fρ,Fσ} = 0 (10)
§5. Extension of the Geodesic Flow 97
and the Hamiltonian H = F (x, y)=min
0
H
= H − λ1F1− λ2F
f(x + ty).Then
t
2
and since {H, F1} =0one has λ2=0. Moreover
= {H, F2}/{F1,F2}.
λ
1
The flow X
given by
M
˙x = H
0
, ˙y = −H
y
0
x
restricted to H0=0can be simplified by introducing a new parameter τ by
dτ
= λ
−
dt
.
1
Then the differential equations become
dx
= −λ
dτ
On the energy surface H
0
=0we may replace the Hamiltonian simply by
K = −λ
We observe that on M we have min
−1
1
−1
1
dy
0
H
,
y
dτ
H0= −λ
f(x + ty)=f (x) as fx,y =0,so
t
= λ
−1
H + F1.
1
−1
1
H
0
.
x
that
(x, y)=0,Hx(x, y)=fx(x) on M.
H
y
Thus
∂
2
x
2
M
= −λ
F
1
∂y
reduce to
−1
1
−1
= −λ
1
= y on M
Hx= −λ
f
x
−1
1
f
x
(x, y)=
K
y
and the differential equations for X
dx
dτ
= y,
dy
dτ
or
d
dτ
which again are the differential equations for the geodesics.
Hence, we have shown that the geodesic flow on f =0is obtained from (4)
by restricting the Hamiltonian (3) to the symplectic submanifold M , given by (6).
This holds up to reparametrization of t.
98 Various Aspects of Integrable Hamiltonian Systems
d) Example. If f (x)=
sphere S
n
.Wefind
F (x, y)=min
t
1
(|x|2− 1) then we are dealing with the unit
2
f(x + ty)=
|x|
1
2
2
y|2−x, y
|y|
2
2
− 1.
Next we restrict F to
M = {(x, y) ∈ R
2n+2
; |y| =1, x, y =0}
so that
1
and finally, since {F
Thus on F
0
=0
M
=
F
M
, |y|2} =0, {FM, x, y} =0on M we have
0
˙x = F
y
˙y = −F
2
(|x|
|y|2− 1)
2
0
F
= FM.
= |x|2y = y,
0
= −|y|2x = −x
x
defines the geodesics on the sphere.
e) We observe that the manifold (6) is obtained by «reducing the phase
space R
indeed F
2n+2
by the integral F1=
is an integral which belongs to the one dimensional symplectic group
1
1
(|y|2− 1)» (in the sense of Section 3).
2
action
(x, y) → (x + ty, y), (11)
with the group G being R. Thus ψ = F
−1
ψ
(0) = {(x, y) ∈ R
The group G
To form the quotient manifold ψ
x + ty for which f
leaving F1=0invariant is evidently the whole group G = R.
0
,y =0. Thus ψ−1(0)/G0 M and we arrive at the
x
−1
is the moment map
1
2n+2
; |y| =1}.
(0)/G0we single out the point x on the line
result:
If we reduce (4) by the group action (11) we obtain up to parametrization
the geodesic flow on f(x)=const. Moreover, F =0gives rise to the flow on
the unit tangent bundle of {f =0}.
§5. Extension of the Geodesic Flow 99
Conversely we can view (4) as an extended Hamiltonian system of the
geodesic flow — in the sense that the latter is obtained from (4) by a reduction
via a symplectic group action.
Clearly, there are many such extensions. The use of such an extension is to
be compared with the use of homogeneous coordinates to describe points on the
sphere instead of spherical coordinates. Of course in homogeneous coordinates
one has to take account of the identification of x and λx for λ =0. Similarly,
in the Hamiltonian one has to eliminate the redundancy of the group action.
f) For the following we have to generalize the above considerations somewhat. It is, of course, not necessary that f (x) is a convex function. All that
∗
matters is that for some point ξ = x + t
(x, y)y on the line x + ty
ϕ(x, y)=f(ξ)
0=f
(ξ),y
x
Then ϕ(x, y)=c is the equation for the tangents of f(x)=c.
Since we are not interested in the parameter dependence of the Hamiltonian
system we could take any function H (x, y, c) defining the tangent to f (x)=c
by
H(x, y, c)=0
as a Hamiltonian, as long as
∂
H(x, y, c) does not vanish at these points.
∂c
Indeed since ϕ(x, y) is obtained by solving
H(x, y, ϕ(x, y)) = 0
we have
H
+ Hcϕx=0,Hy+ Hcϕy=0
x
and therefore
˙x = H
= −H
y
−1
ϕy, ˙y = −Hx= H
c
c
−1
ϕx,
which is, up to parametrization, the above system.
g) We consider two surfaces f (x)=0, g(x)=0with nonvanishing
gradient and let ϕ(x, y), ψ(x, y) respectively be the Hamiltonians of the corresponding line flows. We inquire when the corresponding flows commute.
100 Various Aspects of Integrable Hamiltonian Systems
Proposition. If for any line tangent to both f =0and g =0one has
(ξ),gx(η) =0
f
x
for the points of contact ξ, η where f (ξ)=0, g(η)=0th en
{ϕ, ψ} =0 on ϕ = ψ =0,
i. e. both flows commute. Thus if the moments of f =0, g =0are perpendicular
then the flows commute.
P
ROOF.
If x + ty describes the line one has
ϕ(ξ, y)=f (ξ),ϕ
at the point ξ = x + t
{ϕ, ψ} = ϕ
∗
y of contact. Hence
x,ψy
(ξ, y)=fx(ξ),ϕy(ξ, y)=t∗fx(ξ)
x
−ϕy,ψx =(s∗− t∗)fx(ξ),gx(η) =0.
§ 6. Geodesics on an Ellipsoid
a) It was Jacobi who established the geodesic flow on an ellipsoid as
an integrable one. He used separation of variables on the Hamiltonian Jacobi
equation after introducing elliptic coordinates and various tricks. This derivation
can be found in various places in the literature, still Jacobi’s derivation may
be just as well. Here we give a different derivation using the extension of the
geodesic flow to a «flow of lines» in the embedding space as discussed in the
previous section.
b) The
Q(x)=a
be a positive definite quadratic form in R
a
<... <an. We can assume that a =diag(a0, ..., an) and
0
Q(x)=
Then Q(x)=1defines an ellipsoid in R
flow on this ellipsoid.
−1
x, x (1)
n+1
with distinct eigenvalues, 0 <
n
−1
2
a
x
. (1)
ν
ν=0
ν
n+1
, and we will study the geodesic
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