118
B. K. Berntson et al.
For the canonical form we find
tanh
u 1
2
= exp(x), u 2 = φ, f 1 (u 1 ) = (sinh(u 1 ))
−2 , f 2 (u 2 ) = 0,
and we look for solutions such that v 1 (u 1 ) = 0 and v 2 (u 2 ) satisfies only a nonlinear
equation. We obtain the solution
F
0
= −4 ¯
h
2 sinh(u 1 ) sin(u 2 ), G
L
= 8 cosh(u 1 ) cos(u 2 ) + a 4 u 2 + a 3 ,
G
0
= cosh(u 1 ) U 1 (u 2 ) + U 2 (u 2 ), G
H
= a 5 ,
subject to the conditions (14)–(17), again exactly the same as for flat space. Thus the
system on the 2-hyperboloid admits Painlevé VI and special Weierstrass potentials
for third order superintegrability.
For our next example we consider horocyclic coordinates {u 1 , u 2 } on the
hyperboloid s 2
1 − s 2
2 − s 2
3 = 1, e.g. [4, section 7.7]:
s 1 =
1
2
u 1 +
u 2
2 + 1
u 1
, s 2 =
1
2
u 1 +
u 2
2 − 1
u 1
, s 3 =
u 2
u 1
.
(21)
These coordinates are separable and the canonical system is defined by f 1 (u 1 ) =
1/u 2
1 , f 2 (u 2 ) = 0. We look for systems such that v 1 (u 1 ) = 0, in analogy with our
first three examples.
We obtain the solution
F
0
= −
1
2
a 8 ¯
h
2 u 1 , G
L
=
u 2
1 (a 8 u 2 + a 9 )
2
−
a 8 u 3
2
6
−
a 9 u 2
2
2
+ a 10 u 2 ,
G
0
=
u 2
1
2
U 1 (u 2 ) + U 2 (u 2 ), G
H
= a 7 ,
subject to the conditions
0 = a 8
dv 2
du 2
+ 2
d 2 U 1
du 2
2
,
(22)
0 =
1
2
¯
h
2 a 8
d 3 v 2
du 3
2
− 4a 8
dv 2
du 2
v 2 + 4
dv 2
du 2
dU 1
du 2
,
(23)
0 = (2a 10 − 2a 9 u 2 − a 8 u
2
2 )
dv 2
du 2
− 4(a 9 + a 8 u 2 )v 2 + 4U 1 + 4
d 2 U 2
du 2
2
, (24)
0 = 4u
2
2
dv 2
du 2
+ 16u 2 v
2
2 + 8a 9 u 2
dv 2
du 2
v 2 + 16a 9 v
2
2 − 8a 10
dv 2
du 2
v 2
(25)
+ ¯
h
2 a 8
d 3 v 2
du 3
2
− 16v 2 U 1 + 8
dv 2
du 2
dU 2
du 2
.