On the structure of Hamiltonian impact systems

Abstract

Tools for analyzing dynamics in a class of 2 degrees-of-freedom Hamiltonian impact systems with underlying separable integrable structure are derived. Integrable, near-integrable and far-from integrable cases are considered. In particular, a generalization of the energy momentum bifurcation diagram, Fomenko graphs and the hierarchy of bifurcations framework to this class is constructed. The projection of Liouville leaves of the smooth integrable dynamics to the configuration space allows to extend these tools to impact surfaces which produce far from integrable dynamics. It is suggested that such representations classify dynamically different regions in phase space. For the integrable and near integrable cases these provide global information on the dynamics whereas for the far from integrable regimes (caused by finite deformations of the impact surface), these provide information on the singular set and on the non-impact orbits. The results are presented and demonstrated for the Duffing-center system with impacts from a slanted wall.

Non-uniform ergodic properties of Hamiltonian flows with impacts

“…On the other extreme, at the high energy limit, for compact billiard tables, mechanical HIS limit to the billiard flow in the specified billiard table. The theory for intermediate energy values includes local analysis near periodic orbits [5] and near smooth convex boundaries [22,2], and, for some specific classes of HIS, hyperbolic behavior [19], Liuoville integrable [11,6,3,16,14] and near-integrable [15] dynamics were established. A class of quasi-integrable HIS, which is related to the quasi-integrable dynamics in families of polygonal right angled corners, was introduced in [1].…”

“…The smooth flow without reflection is trivially integrable and oscillatory. The projection of S E,E 1 to the configuration space is the projected rectangle [14]:…”

“…The union of all isoenergy rectangles is the Hill region: D Hill (E) = 0≤E 1 ≤E R (E,E 1 ) = {(x, y)|V 1 (x) + V 2 (y) E} (see [14] for more general formulation).…”

“…Similarly, if a trajectory meets a horizontal segment at (p x , x, p y , y) then it jumps to (p x , x, −p y , y) and continues its movement with (1.3), see [1] for the general construction, a mechanical example and the description of the resulting dynamics on energy surfaces and [14] for global structure of energy surfaces of such systems.…”

“…Figures 5 and 6 demonstrate the division of the intervals E 1 ∈ [0, E] of iso-energy level sets to a finite number of intervals, each having level sets with a fixed genus. The figures show how this division depends on the energy E. In these plots, called Impact Energy-Momentum bifurcation diagrams (IEMBD, see [15,14]) the regions in the (E, E 1 ) plane at which level sets include impacts with certain parts of the boundaries are shown. Figure 5 shows these plots for symmetric polygons and Figure 6 shows it for the asymmetric polygon of Figure 4.…”