Rigid fibers of integrable systems on cotangent bundles

Morimichi Kawasaki, Ryuma Orita
Journal of the Mathematical Society of Japan, vol. 74 (2022), no. 3, pp. 829–847
·

Overview

#

A fiber of an integrable system is a common level set of its conserved quantities. We study which fibers of classical integrable systems on cotangent bundles cannot be moved away from themselves by Hamiltonian diffeomorphisms. Such fibers may be singular, so the question concerns more than the geometry of smooth invariant tori.

Using the notion of superheaviness, we identify a non-displaceable fiber for a broad class of systems that includes the Lagrange top, the Kovalevskaya top, and the C. Neumann problem. In each case, the detected fiber is also the unique fiber that cannot be displaced from the zero-section. A further application gives a singular energy level that cannot be displaced from the zero-section for Hamiltonians given by kinetic energy plus a potential.

@article{kawasaki2022rigid,
  author        = {Kawasaki, Morimichi and Orita, Ryuma},
  title         = {{Rigid fibers of integrable systems on cotangent bundles}},
  journal       = {Journal of the Mathematical Society of Japan},
  volume        = {74},
  number        = {3},
  pages         = {829--847},
  year          = {2022},
  doi           = {10.2969/jmsj/84278427},
  eprint        = {1905.13112},
  archivePrefix = {arXiv}
}