  Particle beam eigen-emittances comprise the lowest set of rms-emittances that
can be imposed to a beam through symplectic optical elements. For cases of
practical relevance this paper introduces an approximation providing a very
simple and powerful relation between transverse eigen-emittance variation and
the beam phase integral. This relation enormously facilitates modeling
eigen-emittance tailoring scenarios. It reveals that difference of
eigen-emittances is given by the beam phase integral or vorticity rather than
by angular momentum. Within the approximation any beam is equivalent to two
objects rotating at angular velocities of same strength and different sign. A
description through circular beam modes has been done already in [A. Burov, S.
Nagaitsev, and Y. Derbenev, Circular modes, beam adapters, and their
applications in beam optics, Phys. Rev. E 66, 016503 (2002)]. The new relation
presented here is a complementary and vivid approach to provide a physical
picture of the nature of eigen-emittances for cases of practical interest.
