  The so-called method of phase synchronization has been advocated in a number
of papers as a way of decoupling a system of linear second-order differential
equations by a linear transformation of coordinates and velocities. This is a
rather unusual approach because velocity-dependent transformations in general
do not preserve the second-order character of differential equations. Moreover,
at least in the case of linear transformations, such a velocity-dependent one
defines by itself a second-order system, which need not have anything to do, in
principle, with the given system or its reformulation. This aspect, and the
related questions of compatibility it raises, seem to have been overlooked in
the existing literature. The purpose of this paper is to clarify this issue and
to suggest topics for further research in conjunction with the general theory
of decoupling in a differential geometric context.
