Monodromy? What's Monodromy?*

We say that a system exhibits \textit{monodromy} if we take the system around a closed loop in its parameter space, and we find that the system does not come back to its original state. Many systems have this property: atoms in a trap, a hydrogen atom in crossed fields, electronic states of H$_{2}^{+ }$, and vibrational states of CO$_{2}$. Imagine noninteracting classical particles moving in a two-dimensional circular box with a hard reflecting wall, and with a cylindrically-symmetric potential energy barrier: $\rho $ = (x$^{2}$+y$^{2})^{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }$ , [V($\rho )$ = -a $\rho ^{2}$/2, $\rho <$R], [V($\rho )$=infinity, $\rho $\underline {$>$}R]. Start all the particles moving on one line with angular momentum L=0, and with energy E$<$0. Then impose additional smooth forces and torques on the particles so that [L(t), E(t)] moves in a circle around the origin in the [L,E] plane. In other words, apply a torque to increase the angular momentum, then drive the particles to a higher energy (above the barrier), then reduce the angular momentum to a negative value, reduce the energy, and finally come back to the initial energy and angular momentum. Where in space do the particles end up? The answer is surprising.