In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2n-dimensional symplectic manifold for which the following conditions hold:
(i) There exist n ≤ k independent integrals F i of motion. Their level surfaces (invariant submanifolds) form a fibered manifold
F
:
Z
→
N
=
F
(
Z
)
over a connected open subset
N
⊂
R
k
.
(ii) There exist smooth real functions
s
i
j
on
N
such that the Poisson bracket of integrals of motion reads
{
F
i
,
F
j
}
=
s
i
j
∘
F
.
(iii) The matrix function
s
i
j
is of constant corank
m
=
2
n
−
k
on
N
.
If
k
=
n
, this is the case of a completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on action-angle coordinates of completely integrable Hamiltonian system as follows.
Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold
F
is a fiber bundle in tori
T
m
. Given its fiber
M
, there exists an open neighbourhood
U
of
M
which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates
(
I
A
,
p
i
,
q
i
,
ϕ
A
)
,
A
=
1
,
…
,
m
,
i
=
1
,
…
,
n
−
m
such that
(
ϕ
A
)
are coordinates on
T
m
. These coordinates are the Darboux coordinates on a symplectic manifold
U
. A Hamiltonian of a superintegrable system depends only on the action variables
I
A
which are the Casimir functions of the coinduced Poisson structure on
F
(
U
)
.
The Liouville-Arnold theorem for completely integrable systems and the Mishchenko-Fomenko theorem for the superintegrable ones are generalized to the case of non-compact invariant submanifolds. They are diffeomorphic to a toroidal cylinder
T
m
−
r
×
R
r
.
