location:  Publications → journals
Search results

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

 Expand all        Collapse all Results 1 - 9 of 9

1. CJM Online first

Pérez-Chavela, Ernesto; Sánchez-Cerritos, Juan Manuel
 Euler-type relative equilibria in spaces of constant curvature and their stability We consider three point positive masses moving on $S^2$ and $H^2$. An Eulerian-relative 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 Eulerian-relative 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 Eulerian-relative equilibria are unstable. Keywords:curved space, relative equilibrium, spectral stabilityCategories:70F07, 70G60

2. 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 theoryCategories:34B16, 34C25, 70F05, 70F15

3. 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 ODECategories: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, non-compact symmetry groups, singular reductionCategories: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 systemsCategories:70S05, 70S10, 53C80

6. 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 sphereCategory:70Fxx

7. 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

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 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 manifoldsCategories: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 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
 top of page | contact us | privacy | site map |