Normal mode tunes for linear coupled motion in six dimensional phase space

The motion of a particle in 6-dimensional phase space in the presence of linear coupling can be written as the sum of 3 normal modes. A cubic equation is found for the tune of the normal modes, which allows the normal mode tunes to be computed from the 6×6 one turn transfer matrix. This result is similar to the quadratic equation found for the normal mode tunes for the motion of a particle in 4-dimensional phase space. These results are useful in tracking programs where the one turn transfer matrix can be computed by multiplying the transfer matrices of each element of the lattice. The tunes of the 3 normal modes for motion in 6-dimensional phase space can then be found by solving the cubic equation. Explicit solutions of the cubic equation for the tune are given in terms of the elements of the 6×6 one turn transfer matrix