  We present the theory of time-dependent point transformations to find
independent dynamical normal modes for 2D systems subjected to time-dependent
control in the limit of small oscillations. The condition that determines if
the independent modes can indeed be defined is identified, and a geometrical
analogy is put forward. The results explain and unify recent work to design
fast operations on trapped ions, needed to implement a scalable
quantum-information architecture: transport, expansions, and the separation of
two ions, two-ion phase gates, as well as the rotation of an anisotropic trap
for an ion are treated and shown to be analogous to a mechanical system of two
masses connected by springs with time dependent stiffness.
