1. CJM 2017 (vol 70 pp. 426)
 PérezChavela, Ernesto; SánchezCerritos, Juan Manuel

Eulertype relative equilibria in spaces of constant curvature and their stability
We consider three point positive masses moving on $S^2$ and $H^2$.
An Eulerianrelative equilibrium, is a relative equilibrium where
the three masses are on the same geodesic, in this paper we analyze
the spectral stability of these kind of orbits where the mass
at the middle is arbitrary and the masses at the ends are equal
and located at the same distance from the central mass. For the
case of $S^2$, we found a positive measure set in the set of
parameters where the relative equilibria are spectrally stable,
and we give a complete classification of the spectral stability
of these solutions, in the sense that, except on an algebraic
curve in the space of parameters, we can determine if the corresponding
relative equilibria is spectrally stable or unstable.
On $H^2$, in the elliptic case, we prove that generically all
Eulerianrelative equilibria are unstable; in the particular
degenerate case when the two equal masses are negligible we get
that the corresponding solutions are spectrally stable. For the
hyperbolic case we consider the system where the mass in the
middle is negligible, in this case the Eulerianrelative equilibria
are unstable.
Keywords:curved space, relative equilibrium, spectral stability Categories:70F07, 70G60 

2. CJM 2017 (vol 70 pp. 173)
3. CJM 2016 (vol 69 pp. 961)
 Andrade, Jaime; Dávila, Nestor; PérezChavela, Ernesto; Vidal, Claudio

Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature
We classify and analyze the orbits of the Kepler problem on surfaces
of constant curvature (both positive and negative, $\mathbb S^2$ and
$\mathbb H^2$, respectively) as function of the angular momentum and
the energy. Hill's region are characterized and the problem of
timecollision is studied. We also regularize the problem in
Cartesian and intrinsic coordinates, depending on the constant
angular momentum and we describe the orbits of the regularized
vector field. The phase portrait both for $\mathbb S^2$ and $\mathbb H^2$
are pointed out.
Keywords:Kepler problem on surfaces of constant curvature, Hill's region, singularities, regularization, qualitative analysis of ODE Categories:70F16, 70G60 

4. CJM 2013 (vol 67 pp. 450)
 Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian

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, noncompact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 

5. CJM 2012 (vol 65 pp. 1164)
 Vitagliano, Luca

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 

6. CJM 2012 (vol 66 pp. 760)
 Hu, Shengda; Santoprete, Manuele

Regularization of the Kepler Problem on the Threesphere
In this paper we regularize the Kepler problem on $S^3$ in several
different ways. First, we perform a Mosertype regularization. Then, we
adapt the LigonSchaaf regularization to our problem. Finally, we show
that the Moser regularization and the LigonSchaaf 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, LigonShaaf regularization, geodesic flow on the sphere Category:70Fxx 

7. CJM 2003 (vol 55 pp. 247)
8. CJM 2001 (vol 53 pp. 715)
 Cushman, Richard; Śniatycki, Jędrzej

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 StefanSussmann 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 

9. CJM 1998 (vol 50 pp. 134)
 Médan, Christine

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 JacobiMoserMumford 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'fandDikii'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 
