Thursday, January 30, 2014 3:30 pm
-
3:30 pm
EST (GMT -05:00)
MC 5158
Speaker
Dr. Giovanni Rastelli, Department of Mathematics University of Turin, Turin, Italy
Title
Extensions of natural Hamiltonians
Abstract
Given
a
n-dimensional
natural
Hamiltonian
L
on
a
Riemannian
or
pseudo-
Riemannian
manifold,
we
call
the
"extension"
of
L
the
n+1
dimensional
Hamiltonian
H
=
1/2
pu2
+
a(u)L
+
b(u),
with
new
canonically
conjugate
coordinates
(u,
p_u).
For
a
class
of
L,
the
functions
a(u)
and
b(u)
can
be
chosen
such
that
for
any
integer
value
of
m,
H
admits
an
extra
polynomial
first
integral
in
the
momenta
of
degree
m,
explicitly
determined
in
the
form
of
the
m-th
power
of
a
differential
operator
applied
to
a
suitable
function
G
of
coordinates
and
momenta.
In
particular,
if
L
is
maximally
superintegrable
(MS)
then
H
is
also
MS.
Therefore,
the
extension
procedure
allows
the
creation
of
new
superintegrable
systems
from
old
ones.
For
m=2,
the
extra
first
integral
generated
by
the
extension
procedure
determines
a
second-order
symmetry
operator
of
a
Laplace-Beltrami
quantization
of
H,
modified
by
taking
in
account
the
curvature
of
the
configuration
manifold.
The
extension
procedure
can
be
applied
to
several
Hamiltonian
systems,
including
the
three-body
Calogero
and
Wolfes
systems,
the
Tremblay-Turbiner-Winternitz
system
and
n-dimensional
anisotropic
harmonic
oscillators.
We
propose
here
a
short
review
of
the
known
results
of
the
theory
and
some
previews
of
new
ones.