  Magnetic skyrmions have attracted considerable attention recently for their
huge potential in spintronic applications. Generally skyrmions are big compared
to the atomic lattice constant, which allows for the Ginzburg-Landau type
description in the continuum limit. Such a description successfully captures
the main experimental observations on skyrmions in B20 compound without
inversion symmetry. Skyrmions can also exist in inversion-symmetric magnets
with competing interactions. Here we derive a general Ginzburg-Landau theory
for skyrmions in these magnets valid in the long wavelength limit. We study the
unusual static and dynamical properties of skyrmions based on the derived
Ginzburg-Landau theory. We show that an easy axis spin anisotropy is sufficient
to stabilize a skyrmion lattice. Interestingly, the skyrmion in
inversion-symmetric magnets has a new internal degree of freedom associated
with the rotation of helicity, i.e. the "spin" of the skyrmion as a particle,
in addition to the usual translational motion of skyrmions (orbital motion).
The orbital and spin degree of freedoms of an individual skyrmion can couple to
each other, and give rise to unusual behavior that is absent for the skyrmions
stabilized by the Dzyaloshinskii-Moriya interaction. The derived
Ginzburg-Landau theory provides a convenient and general framework to discuss
skyrmion physics and will facilitate the search for skyrmions in
inversion-symmetric magnets.
