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Search: MSC category 70 ( Mechanics of particles and systems )

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1. CJM Online first

Hakl, Robert; Zamora, Manuel
Periodic solutions of an indefinite singular equation arising from the Kepler problem on the sphere
We study a second-order ordinary differential equation coming from the Kepler problem on $\mathbb{S}^2$. The forcing term under consideration is a piecewise constant with singular nonlinearity which changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

Keywords:singular differential equation, indefinite singularity, periodic solution, Kepler problem on $\mathbb{S}^1$, degree theory
Categories:34B16, 34C25, 70F05, 70F15

2. CJM Online first

Andrade, Jaime; Dávila, Nestor; Pérez-Chavela, 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 time-collision 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

3. 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, non-compact symmetry groups, singular reduction
Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20

4. 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

5. CJM 2012 (vol 66 pp. 760)

Hu, Shengda; Santoprete, Manuele
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

6. CJM 2003 (vol 55 pp. 247)

Cushman, Richard; Śniatycki, Jędrzej
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

7. 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 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

8. 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 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

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