3Hamiltonian vector fields
III Symplectic Geometry
3.2 Integrable systems
In classical mechanics, we usually have a fixed H, corresponding to the energy.
Definition
(Hamiltonian system)
.
A Hamiltonian system is a triple (
M, ω, H
)
where (
M, ω
) is a symplectic manifold and
H ∈ C
∞
(
M
), called the Hamiltonian
function.
Definition
(Integral of motion)
.
A integral of motion/first integral/constant of
motion/conserved quantity of a Hamiltonian system is a function
f ∈ C
∞
(
M
)
such that {f, H} = 0.
For example,
H
is an integral of motion. Are there others? Of course, we
can write down 2
H, H
2
, H
12
, e
H
, etc., but these are more-or-less the same as
H
.
Definition
(Independent integrals of motion)
.
We say
f
1
, . . . , f
n
∈ C
∞
(
M
) are
independent if (d
f
1
)
p
, . . . ,
(d
f
n
)
p
are linearly independent at all points on some
dense subset of M.
Definition
(Commuting integrals of motion)
.
We say
f
1
, . . . , f
n
∈ C
∞
commute
if {f
i
, f
j
} = 0.
If we have
n
independent integrals of motion, then we clearly have
dim M ≥ n
.
In fact, the commuting condition implies:
Exercise.
Let
f
1
, . . . , f
n
be independent commuting functions on (
M, ω
). Then
dim M ≥ 2n.
The idea is that the (d
f
i
)
p
are not only independent, but span an isotropic
subspace of T M.
If we have the maximum possible number of independent commuting first
integrals, then we say we are integrable.
Definition
(Completely integrable system)
.
A Hamiltonian system (
M, ω, H
)
of dimension
dim M
= 2
n
is (completely) integrable if it has
n
independent
commuting integrals of motion f
1
= H, f
2
, . . . , f
n
.
Example.
If
dim M
= 2, then we only need one integral of motion, which we
can take to be
H
. Then (
M, ω, H
) is integrable as long as the set of non-critical
points of H is dense.
Example.
The physics of a simple pendulum of length 1 and mass 1 can be
modeled by the symplectic manifold
M
=
T
∗
S
1
, where the
S
1
refers to the
angular coordinate
θ
of the pendulum, and the cotangent vector is the momentum.
The Hamiltonian function is
H = K + V = kinetic energy + potential energy =
1
2
ξ
2
+ (1 − cos ω).
We can check that the critical points of
H
are (
θ, ξ
) = (0
,
0) and (
π,
0). So
(M, ω, H) is integrable.
Example.
If
dim M
= 4, then (
M, ω, H
) is integrable as long as there exists an
integral motion independent of
H
. For example, on a spherical pendulum, we
have
M
=
T
∗
S
2
, and
H
is the total energy. Then the angular momentum is an
integral of motion.
What can we do with a completely integrable system? Suppose (
M, ω, H
)
is completely integrable system with
dim M
= 2
n
and
f
1
=
H, f
2
, . . . , f
n
are
commuting. Let
c
be a regular value of
f
= (
f
1
, . . . , f
n
). Then
f
−1
(
c
) is an
n-dimensional submanifold of M. If p ∈ f
−1
(c), then
T
p
(f
−1
(c)) = ker(df)
p
.
Since
df
p
=
(df
1
)
p
.
.
.
(df
n
)
p
=
ι
X
f
1
ω
.
.
.
ι
X
f
n
ω
,
we know
T
p
(f
−1
(c)) = ker(df
p
) = span{(X
f
1
)
p
, . . . (X
f
n
)
p
},
Moreover, since
ω(X
f
i
, X
f
j
) = {f
i
, f
j
} = 0,
we know that T
p
(f
−1
(c)) is an isotropic subspace of (T
p
M, ω
p
).
If
X
f
1
, . . . , X
f
n
are complete, then following their flows, we obtain global
coordinates of (the connected components of)
f
−1
(
c
), where
q ∈ f
−1
(
c
) has
coordinates (
ϕ
1
, . . . , ϕ
m
) (angle coordinates) if
q
is achieved from the base point
p
by following the flow of
X
f
i
for
ϕ
i
seconds for each
i
. The fact that the
f
i
are
Poisson commuting implies the vector fields commute, and so the order does not
matter, and this gives a genuine coordinate system.
By the independence of
X
f
i
, the connected components look like
R
n−k
×T
k
,
where
T
k
= (
S
1
)
k
is the
k
torus. In the extreme case
k
=
n
, we simply get a
torus, which is a compact connected component.
Definition
(Liouville torus)
.
A Liouville torus is a compact connected compo-
nent of f
−1
(c).
It would be nice if the (
ϕ
i
) are part of a Darboux chart of
M
, and this is
true.
Theorem
(Arnold–Liouville thoerem)
.
Let (
M, ω, H
) be an integrable system
with
dim M
= 2
n
and
f
1
=
H, f
2
, . . . , f
n
integrals of motion, and
c ∈ R
a regular
value of f = (f
1
, . . . , f
n
).
(i)
If the flows of
X
f
i
are complete, then the connected components of
f
−1
(
{c}
)
are homogeneous spaces for
R
n
and admit affine coordinates
ϕ
1
, . . . , ϕ
n
(angle coordinates), in which the flows of X
f
i
are linear.
(ii)
There exists coordinates
ψ
1
, . . . , ψ
n
(action coordinates) such that the
ψ
i
’s
are integrals of motion and ϕ
1
, . . . , ϕ
n
, ψ
1
, . . . , ψ
n
form a Darboux chart.