  With use of a variational principle, we investigate a role of breathing width
degree of freedom in the effective theory of interacting vortices in a trapped
single-component Bose-Einstein condensates in 2 dimensions under the strong
repulsive cubic nonlinearity. As for the trial function, we choose a product of
two vortex functions assuming a pair interaction and employ the amplitude form
of each vortex function in the Pad\'e approximation which accommodates a
hallmark of the vortex core. We have obtained Lagrange equation for the
interacting vortex-core coordinates coupled with the time-derivative of width
and also its Hamilton formalism by having recourse to a non-standard Poisson
bracket. By solving the Hamilton equation, we find rapid radial breathing
oscillations superposed on the slower rotational motion of vortex cores,
consistent with numerical solutions of Gross-Pitaevskii equation. In
higher-energy states of 2 vortex systems, the breathing width degree of freedom
plays role of a kicking in the kicked rotator and generates chaos with a
structure of sea-urchin needles. Byproduct of the present variational approach
includes: (1) the charge-dependent logarithmic inter-vortex interaction
multiplied with a pre-factor which depends on the scalar product of a pair of
core-position vectors; (2) the charge-independent short-range repulsive
inter-vortex interaction and spring force.
