  The goal of this paper is to apply the recently developed theory of buckling
of arbitrary slender bodies to a tractable yet non-trivial example of buckling
in axially compressed circular cylindrical shells, regarded as
three-dimensional hyperelastic bodies. The theory is based on a mathematically
rigorous asymptotic analysis of the second variation of 3D, fully nonlinear
elastic energy, as the shell's slenderness parameter goes to zero. Our main
results are a rigorous proof of the classical formula for buckling load and the
explicit expressions for the relative amplitudes of displacement components in
single Fourier harmonics buckling modes, whose wave numbers are described by
Koiter's circle. This work is also a part of an effort to quantify the
sensitivity of the buckling load of axially compressed cylindrical shells to
imperfections of load and shape.
