Symmetry properties of a symplectic transport matrix and Twiss parameterization of a fully coupled motion

Twiss parameterization proposed by Courant and Snyder is a keystone of accelerator physics. Several authors have extended it later to the case of the fully coupled betatron motion and synchrobetatron motion of a coasting beam. A few proposals have been made as well how to construct a parameterization which is fully symmetric with respect to all three spatial degrees of freedom. The key point for each of them is to find a similarity transformation which turns the transport matrix into a block-diagonal form with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>2</a:mn><a:mo>×</a:mo><a:mn>2</a:mn></a:math> cells. In the present work, this approach has been considered in detail. It is based on the idea initially proposed by Mais and Ripken for transport matrix eigenvalue calculation only. It shows how to introduce a symmetric Twiss-like parameterization and an emittance concept for an arbitrary symplectic <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mn>6</c:mn><c:mo>×</c:mo><c:mn>6</c:mn></c:math> matrix. The resulting parameterization can be reduced to the well-known Lebedev-Bogacz parameterization in case of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>4</e:mn><e:mo>×</e:mo><e:mn>4</e:mn></e:math> matrix. Special attention has been paid to the intrinsic ambiguity existing in the Twiss parameterization of a coupled motion.