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1. CJM Online first
Motion in a Symmetric Potential on the Hyperbolic Plane We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.
Keywords:Hamiltonian systems with symmetry, symmetries, non-compact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 |
2. CJM 2012 (vol 66 pp. 760)
Regularization of the Kepler Problem on the Three-sphere In this paper we regularize the Kepler problem on $S^3$ in several
different ways. First, we perform a Moser-type regularization. Then, we
adapt the Ligon-Schaaf regularization to our problem. Finally, we show
that the Moser regularization and the Ligon-Schaaf map we obtained can be
understood as the composition of the corresponding maps for the Kepler problem
in Euclidean space and the gnomonic transformation.
Keywords:Kepler problem on the sphere, Ligon-Shaaf regularization, geodesic flow on the sphere Category:70Fxx |
3. CJM 2012 (vol 65 pp. 1164)
Partial Differential Hamiltonian Systems We define partial differential (PD in the following), i.e., field
theoretic analogues of Hamiltonian systems on abstract symplectic
manifolds and study their main properties, namely, PD Hamilton
equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in
standard multisymplectic approach to Hamiltonian field theory, in our
formalism, the geometric structure (kinematics) and the dynamical
information on the ``phase space''
appear as just different components of one single geometric object.
Keywords:field theory, fiber bundles, multisymplectic geometry, Hamiltonian systems Categories:70S05, 70S10, 53C80 |
4. CJM 2003 (vol 55 pp. 247)
Differential Structure of Orbit Spaces: Erratum This note signals an error in the above paper by giving a counter-example.
Categories:37J15, 58A40, 58D19, 70H33 |
5. CJM 2001 (vol 53 pp. 715)
Differential Structure of Orbit Spaces We present a new approach to singular reduction of Hamiltonian systems
with symmetries. The tools we use are the category of differential
spaces of Sikorski and the Stefan-Sussmann theorem. The former is
applied to analyze the differential structure of the spaces involved
and the latter is used to prove that some of these spaces are smooth
manifolds.
Our main result is the identification of accessible sets of the
generalized distribution spanned by the Hamiltonian vector fields of
invariant functions with singular reduced spaces. We are also able
to describe the differential structure of a singular reduced space
corresponding to a coadjoint orbit which need not be locally closed.
Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds Categories:37J15, 58A40, 58D19, 70H33 |
6. CJM 1998 (vol 50 pp. 134)
On critical level sets of some two degrees of freedom integrable Hamiltonian systems We prove that all Liouville's tori generic bifurcations of a
large class of two degrees of freedom integrable Hamiltonian
systems (the so called Jacobi-Moser-Mumford systems) are
nondegenerate in the sense of Bott. Thus, for such systems,
Fomenko's theory~\cite{fom} can be applied (we give the example
of Gel'fand-Dikii's system). We also check the Bott property
for two interesting systems: the Lagrange top and the geodesic
flow on an ellipsoid.
Categories:70H05, 70H10, 58F14, 58F07 |