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SYMPLECTIC
GEOMETRY,
LECTURE
4
Prof. Denis Auroux
1.
Hamiltonian
Vector
Fields
Recall
from
last
time
that,
for
(
M, ω
)
a
symplectic
manifold,
H
:
M
→
R
a
C
∞
function,
there
exists
a
vector
field
X
H
s.t.
i
X
H
ω
=
dH
.
Furthermore,
the
associated
ﬂow
ρ
t
of
this
vector
field
is
an
isotopy
of
symplectomorphisms.
Example.
Consider
S
2
⊂
R
3
with
cylindrical
coordinates
(
r,
θ, z
)
and
symplectic
form
ω
=
dθ
∧
dz
(
ω
is
the
usual
area
form).
Then
setting
H
=
z
gives
the
vector
field
∂
:
the
associated
ﬂow
is
precisely
rotation
by
angle
∂θ
t
.
Note
also
that
the
critical
points
of
H
are
the
fixed
points
of
ρ
t
, and
ρ
t
preserves
the
level
sets
of
H
,
i.e.
(1)
dt
d
(
H
◦
ρ
t
) =
dt
d
(
ρ
∗
t
H
) =
ρ
∗
t
(
L
X
H
H
) =
ρ
∗
t
(
i
X
H
ω
(
X
H
))
=
ρ
∗
t
(
ω
(
X
H
, X
H
))
=
0
One
can
apply
this
to
obtain
the
ordinary
formula
for
conservation
of
energy.
Definition
1.
X
is
a
symplectic
vector
field
if
L
X
ω
= 0
,
i.e.
i
X
ω
is
closed.
X
is
Hamiltonian
if
i
X
ω
is
exact.
By
Poincar´
e,
we
see
that,
locally,
symplectic
vector
fields
are
Hamiltonian.
Globally,
we
obtain
a
class
[
i
X
ω
]
∈
H
1
(
M,
R
).
∂
∂
∂
∂
Example.
On
T
2
,
∂x
and
∂y
are
symplectic
vector
fields:
since
dy
and
dx
are
not
exact,
∂x
and
∂y
are
not
Hamiltonian.
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 Spring '10
 DenisAuroux
 Geometry, Hamiltonian mechanics, Moser, Symplectic manifold, Symplectic geometry, symplectic vector fields, symplectic forms

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