Abstract
We introduce a family of Hamiltonians with two degrees of freedom, axial symmetry and complete integrability. The potential function depends on coordinates and one control parameter. A fold catastrophe typically occurs in such a family of potentials and its consequences on the global dynamics are investigated through the energy-momentum map which defines the singular fibration of the four-dimensional phase space. The two inequivalent local canonical forms of the catastrophe are presented: the first case corresponds to the appearance of a second sheet in the image of the energy-momentum map while the second case is associated with the breaking of an already existing second sheet. A special effort is placed on the description of the singularities. In particular, the existence of cuspidal tori is related to a second-order contact point between the energy level set and the reduced phase space. The quantum mechanical aspects of the changes induced by the fold catastrophe are investigated with the quantum eigenstates computed for an octic potential and are interpreted through the quantum-classical correspondence. We note that the singularity exposed in this paper is not an obstruction to a global definition of action-angle variables.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.