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Presently, there are two most frequently used parameterizations of linear x-ycoupled motion used in the accelerator physics. They are the Edwards-Teng andMais-Ripken parameterizations. The article is devoted to an analysis of closerelationship between the two representations, thus adding a clarity to theirphysical meaning. It also discusses the relationship between the Eigen-vectors,the beta-functions, second order moments and the bi-linear form representingthe particle ellipsoid in the 4D phase space. Then, it considers a furtherdevelopment of Mais-Ripken parameterization where the particle motion isdescribed by 10 parameters: four beta-functions, four alpha-functions and twobetatron phase advances. In comparison with Edwards-Teng parameterization thechosen parameterization has an advantage that it works equally well foranalysis of coupled betatron motion in circular accelerators and in transferlines. Considered relationship between second order moments, Eigen-vectors andbeta-functions can be useful in interpreting tracking results and experimentaldata. As an example, the developed formalism is applied to the FNAL electroncooler and Derbenev's vertex-to-plane adapter.

Betatron motion with coupling of horizontal and vertical degrees of freedom