# DIFFERENTIAL GEOMETRY AND AVERAGING

I had discovered an interesting underlying geometry behind some systems with rapid time-dependence. This observation really illuminates some important physical phenomena in a very simple way; previous treatments of these phenomena relied on formal machinery of normal forms but did not explain what is ``really going on".

The papers [36, 41, 42, 43, 46, 49, 50] develop this observation. As an example, it turns out that the effective potentials of Kapitsa arising in averaging rapid vibrations have a geometrical significance related to curvature of family of curves. In its simplest manifestation, the stabilization of an inverted pendulum by the vibration of its suspension point is quantified by the curvature of the tractrix (the pursuit curve). This effect is behind some other counterintuitive phenomena such as laser tweezers (used to move particles inside a living cell), Paul traps, etc. See here for more details.

The geometric observation mentioned above led me to a simple geometric explanation of why the Paul trap works (actually it led me to rediscover the Paul trap before I learned of its existence). The classical explanation given by Paul in his 1989 Nobel Prize acceptance paper is analytic/computational.

With I. Mitkov and V. Zharnitsky we applied some of these ideas to the study of some idealized anisotropic ferromagnets. The analysis in [41] predicts the existence of traveling domain walls if a rapidly oscillating magnetic field is applied. These walls separate the opposite directions of magnetization.