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

Willy Sarlet; Tom Mestdag

doi:10.3934/jgm.2021019

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2022, Volume 14, Issue 1: 91-104. Doi: 10.3934/jgm.2021019

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

Willy Sarlet^†, and
Tom Mestdag^†,‡, ,

†.
Department of Mathematics, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium
‡.
Department of Mathematics, University of Antwerp, Middelheimlaan 1, 2020 Antwerpen, Belgium

^* Corresponding author: Tom Mestdag
^* Corresponding author: Tom Mestdag

Dedicated to Professor Tony Bloch on the occasion of his 65th birthday

Received: March 31, 2021

Published: March 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 34A26, 34A30, 53Z30, 70J10.

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