Pergamon
Mechanics Research Communications, Vol. 22, No. 4, pp. 313-317, 1995
Copyright © 1995 Elsevier Science Ltd
Printed in the USA. All rights reserved
0093-6413/95 $9.50 + .(X)
0093-6413(95)00030-5
Q U A D R A T I C H A M I L T O N I A N S ON T H E U N I T S P H E R E
d6rg Frauendiener
Max-Planck-Institut f(ir Astrophysik, Karl-Schwarzschild-Strafie 1, D-85740 Garching,
Germany
(Received 4 January 1995; accepted for print 15 March 1995)
I. Introduction
The purpose of this letter is to point out the general form of a certain class of Hamiltonians
on the unit sphere. We consider Hamiltonians as functions of the cartesian coordinates
(:r y, z) in terms of which the unit sphere is given by the equation x 2 + y2 4- z 2 = 1 The
('lass of Hamiltonians treated is then defined as those functions which are at most quadratic
in these coordinates. Such Hamiltonians occur naturally in various problems of mechanics
[1 3], optics [4] and others. In section 2. we introduce the necessary machinery and apply
ir in section 3. to obtain the classification of quadratic Hamiltonians on the unit sphere
This extends the work in [5] where a classification of biparametric quadratic Hamiltoma~ls
has been given.
II. Preliminaries
Let S 2 be given as the set of unit vectors in If{3, S 2 = {x ~ : xaxb~]~b = 1}, where 7]~b =
diag(1, 1, 1) is the s t a n d a r d metric in IR3 and :ca = (x,y, z)o Define the flmction S(:r" I =
~]~bxOx b - 1, then 5 '2 is the zero-set of S.
Since it is a two-dimensional manifokt all
symplectic structures on the sphere are proportional to the area form of 5 '2 which we take
as the sympleetic form f / o n S 2. So f / = x d y A d z + y d z A d x + z d x A d y
313
= }%bcx~dx b Adz ° .
314
J. FRAUENDIENER
Here eabc is the a l t e r n a t i n g Levi-Civit£ symbol. T h e Poisson a l g e b r a of functions on the
sphere with respect to this s y m p l e c t i c form is g e n e r a t e d by the Poisson brackets between
the c o o r d i n a t e functions
{ ~ , ~ } = ~,
{ ~ , ~ } = ~,
{ ~ , ~ } = y.
(zl/
We now define the class Q of functions on S 2 which are at most q u a d r a t i c in the c o o r d i n a t e s
(x,y,z).
Hence H E Q if a n d only if H = ~=abX
1~
=x b + B a x a + D with A~b a constant
q u a d r a t i c form, Ba a c o n s t a n t (co)vector a n d D an a r b i t r a r y c o n s t a n t . O u r aim is to find
a s t a n d a r d form for H a m i l t o n i a n s in ~ . To this end we r e g a r d H a m i l t o n i a n s as equivalent
if t h e y induce the stone p h a s e d i a g r a m , i.e., if the t r a j e c t o r i e s of their h a m i l t o n i a n vector
fields can be m a p p e d onto each o t h e r by symplectic t r a n s f o r m a t i o n s , where we do not
care a b o u t the p a r a m e t r i z a t i o n .
Hence, two H a m i l t o n i a n s are equivalent if there exists
a s y m p l e c t i c t r a n s f o r m a t i o n which m a p s one h a m i l t o n i a n vector field onto a m u l t i p l e of
the o t h e r
Thus, the first question to be answered is, how big is the group of symplectic
t r a n s f o r m a t i o n s which leave Q invariant. We first consider infinitesimal transformations.
Let X = X~0~ be a vector field in IR:~, where X~(:r ~) are t h r e e functions on R :~ and
(~c
:--
0 . This vector field will be t a n g e n t to the unit sphere iff L x S ~x S, L x being the
Ox ~
Lie derivative along the vector field X . This implies ~]~bxaX b = o~S for some function c~.
F r o m this we get t h a t X can be r e p r e s e n t e d on S 2 in t h e form
X = e~b~:r,,BbOc
for three a r b i t r a r y functions B ~
(2.'2)
Here, the s y m b o l e"5~ denotes the "dual" Levi-Civit5
s y m b o l which is defined by the equation eabce "b~ = 6.
Note~ t h a t X vanishes oi1 S 2 if
Be = Bx~; so t h e r e are only essentially two free functions c o n t a i n e d in X as it should be
for vector fields oi1 a t w o - d i m e n s i o n a l manifold. Next, we ask which of these vector fields
leave Q invariant. This entails the condition L x Q C c2. Let H E Q t h e n
LxH
= A a b x a X b + B a X a.
(2.3)
This will be in O iff X a is at most linear z a which, in view of (2.2) implies t h a t Ba
has to be c o n s t a n t . Hence, t h e r e exists a t h r e e - d i m e n s i o n a l space of vector fields on S ~
which leaves Q invariant.
It is easy to see t h a t this space is g e n e r a t e d by the three
HAMILTONIANS ON THE UNIT SPHERE
315
infinitesimal rigid rotations a r o u n d three orthogonal axes in N 3. Hence, the group which
leaves Q invariant is locally isomorphic to the group SO(3) of proper rotations. The group
O(3) (also locally isomorphic to SO(3)) additionally includes reflections and also leaves Q
invariant. However, while all proper rotations are also sympleetic transformations (they
leave ft invariant), this is not so for reflections, because they m a p f~ to - f / .
Hence, the
subgroup of symplectic transformations leaving Q invariant is exactly the group SO(3)~
This has been pointed out already in [5].
III. The s t a n d a r d form
As we mentioned above we consider Hamiltonians as equivalent iff their hamiltonian vector
fields can be m a p p e d onto multiples of each other by symplectic transformations.
This
implies that any H E Q is equivalent to
H + aS,
bH,
where a , b , c C IR, b ~k 0 and R E SO(3).
H + c,
(3.1)
R'H,
There are ten parameters in H (six in A,~b,
r,hree in Ba and one in C) and alltogether there are six parameters in the equivalence
transformations (3.1). So we expect that we will be left with four essential parameters in
tile s t a n d a r d form for H.
Let us start with the simplest case: A~b = 0 and Ba = 0. T h e n H is constant and thus
equivalent to zero: H ~ 0. Now let Bo be non-vanishing. T h e n we can use the rotational
freedom to align B a with the z-axis and the scaling transformation to achieve H ~ :Let .4~b ¢ 0. T h e n H = 71 ~. .b. .a : :cb + B~:t ~ + C. Let A1, A2, ,k3 be the three eigenvalues
of A°b. We first assume that these are all equal.
Then A,b is diagonal and by adding
an appropriate multiple of S we see that Aabza:c b is equivalent to a constant and hence
to zero.
So this brings us back to the cases already discussed.
Now if X~ = ;k,~ ¢i Aa
then we can use part of the rotational freedom to put A,b into diagonal form.
Thus
A a b z a a b = AI(Z 2 -b !I2) -b )~az2. Since we can still perform rotations a r o u n d the z-axis
we can use these to orient the z-axis so that Ba is in the zz-plane.
Finally, using the
remaining equivalence transformations we find that H ~ ½z 2 + A z + C z.
316
J. FRAUEND1ENER
In the last case all the eigenvalues are different. Thus, p u t t i n g Aab into diagonal form uses
up all the rotational freedom apart from even p e r m u t a t i o n s of the three axes. The effect
of the equivalence transformations on the eigenvalues is
Ai ~ Ai + a,
Ai ~ bAi.
(3.1)
We can use the additive tranformation to put one of the eigenvahes equal to zero. Which
of these we choose depends on the relation between B~ and the eigenvectors of Aab. Let us
first assume that Ba is proportional to an eigenvector which we can choose (by way of an
even p e r m u t a t i o n ) as pointing along the z-axis, Then we use the shift to put A3 = 0 and
the
1 2
scaling transformation to put A1 = 1. Thus we have found that H ~ ~x
+ ~1A y 2 -± Cz.
Now, if B~, is in the subspace spanned by two eigenvectors then we can choose those as
being the x- and the y-axis.
Again, shifting A3 to zero and dividing by A1 (which i~
1 2
1
non-zero) we obtain the equivalence H ~ ~x
÷ ~Ay
2 + Ax + By.
The final case is where B~ is generic, i.e., its components along the eigenvectors are all
non-zero. T h e n we can again put A.3 to zero and scale At to unity to obtain H ~ ~x
I 2+
2Ay 2 + Az + B y + Cz,
To summarize then, we have found the following classification
Aab Ba
(3)
(3)
(21)
(111)
(111)
(111)
[0]
[31
[3]
[1]
[2]
[3]
H
0
2
½x 2 + Ax + Cz
1 2
1
2
gx
+ gay
+Cz
1 2
1
2
f x + flay + A x + B y
] ,2
ffa
+ f 1A y 2 + Ax + B y + C z
(3.3)
Here we have indicated in the column below Aab the degeneracies of the spectrum of
Aah: (111), (21) and (3) indicating that Aab has three, two or one different eigenvalue(s).
Similarly, [7~] m the colmnn below Be mdicate~ that B~ lies m a subspace spanned by /z
eigenvectors of A,b.
IV. Conclusion
In this article we have completed the work started in Ref,
[5] on the classification of
HAMILTONIANSON THE UNIT SPHERE
317
quadratic Hamiltonians on the unit sphere. The basic tool for this was the geometry
underlying the structure of the unit sphere as a symplectic manifold. It would be useful
to have the phase portraits for the various classes and to discuss the bifurcations which
might occur. This will be done in some further article.
References
[1] A. Deprit and A. Elipe, Celest. Mech. 51 227-250, (1991)
[2] B. Miller, Celest. Mech. 51 251-270 (1991)
[3] A. Elipe and V. Lanchares, Bol. Astron. Observ. Madrid 12 56-67 (1990)
[4] D. David, D.D. Holm and M. V. Tratnik, Phys. Rep. 187 281-367 (1990)
[5] A. Elipe and V. Lanchares, Mech. Res. Comm. 21 209-214 (1994)