  In this paper we study a Hamiltonian system with a spatially asymmetric
potential. We are interested in the effects on the dynamics when the potential
becomes symmetric slowly in time. We focus on a highly simplified non-trivial
model problem (a metaphor) to be able to pursue explicit calculations as far as
possible. Using the techniques of averaging and adiabatic invariants, we are
able to study all bounded solutions, which reveals significant asymmetric
dynamics even when the asymmetric contributions to the potential have become
negligibly small.
