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
ndimensional
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
mth
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
secondorder
symmetry
operator
of
a
LaplaceBeltrami
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
threebody
Calogero
and
Wolfes
systems,
the
TremblayTurbinerWinternitz
system
and
ndimensional
anisotropic
harmonic
oscillators.
We
propose
here
a
short
review
of
the
known
results
of
the
theory
and
some
previews
of
new
ones.