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

Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations