Hamiltonian Systems

Consider a Hamiltonian system whose potential energy is minimized on a smooth submanifold M and is strictly convex in transverse directions. As the potential becomes increasingly steep, one expects the motion to approach ideal constrained dynamics on M. This holds for sufficiently well-prepared initial data but may fail otherwise: rapid transverse oscillations approximately conserve action, not energy, and their varying frequencies generate an additional effective force along M.

In this talk, I will discuss this mechanism in finite dimensions and then explore its counterparts in continuum mechanics. Viewing elastic threads, compressible fluids, and geophysical fluid models as infinite-dimensional Hamiltonian systems, we obtain ideal constrained models - including inextensible thread and incompressible fluid - as strong-potential limits, together with corrections when the initial data are less well prepared. This is joint work with T.Drivas.

https://arxiv.org/abs/2607.27165