1. CJM Online first
|Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems|
In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters, in particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated to mechanical Hamiltonian system and Hamiltonian system defined by cubic polynomials. Finally, important differences with the continuous case are pointed out.
Keywords:difference equations, Hamiltonian systems, stability in the Lyapunov sense
2. CJM 2013 (vol 67 pp. 450)
|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
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