Betatron motion with coupling of horizontal and vertical degrees of freedom

Presently, there are two most frequently used parameterizations of linear x-y coupled motion used in the accelerator physics. They are the Edwards-Teng and Mais-Ripken parameterizations. The article is devoted to an analysis of close relationship between the two representations, thus adding a clarity to their physical meaning. It also discusses the relationship between the Eigen-vectors, the beta-functions, second order moments and the bi-linear form representing the particle ellipsoid in the 4D phase space. Then, it considers a further development of Mais-Ripken parameterization where the particle motion is described by 10 parameters: four beta-functions, four alpha-functions and two betatron phase advances. In comparison with Edwards-Teng parameterization the chosen parameterization has an advantage that it works equally well for analysis of coupled betatron motion in circular accelerators and in transfer lines. Considered relationship between second order moments, Eigen-vectors and beta-functions can be useful in interpreting tracking results and experimental data. As an example, the developed formalism is applied to the FNAL electron cooler and Derbenev's vertex-to-plane adapter.