Generally speaking, as a consequence of a famous Liouville’s theorem, the volume of the phase space under the Hamiltonian flow is constant. This also means that the Jacobian of the mapping determined by the Hamiltonian flow is equal to 1 and, therefore, this mapping is never singular. However, in many important physical systems one needs to consider a specific Hamiltonian flow, which is defined by specific initial conditions. It can be shown that in this way obtained the reduced mapping can be singular. As a consequence, caustics and rainbows occur, in the configuration and angular space, respectively. Importance of the study of the singular dynamics is that the family of particles along the singularities produces a strong focusing effect. It should be also stressed that the singularities represent important global characteristic of a dynamical system. In the scattering physics of photons off a rain droplet this corresponds to the well-known rainbow effect.
The singularities in 2D Hamiltonian dynamical systems described by symmetric potentials will be presented. It is assumed that the initial motion is at rest i.e. that the initial values of particles in the 2D momentum space are equal to zero. Motivation for the choice of the initial condition is related, but not restricted, to the scattering theory. Moreover, dynamic of the system can be treated independently of the scattering theory. It will be shown that a rich complex behavior of the singularities occurs. Their evolution includes qualitatively change of their form, which is accompanied with their “interactions”. Main result of this presentation is that by introducing the singularities in the impact parameter plane, which are not observable, one can explain their complex behavior in the configuration and the momentum spaces.