  The conservation of the recently formulated relativistic canonical helicity
[Yoshida Z, Kawazura Y, and Yokoyama T 2014 J. Math. Phys. 55 043101] is
derived from Noether's theorem by constructing an action principle on the
relativistic Lagrangian coordinates (we obtain general cross helicities that
include the helicity of the canonical vorticity). The conservation law is,
then, explained by the relabeling symmetry pertinent to the Lagrangian label of
fluid elements. Upon Eulerianizing the Noether current, the purely spatial
volume integral on the Lagrangian coordinates is mapped to a space-time mixed
three-dimensional integral on the four-dimensional Eulerian coordinates. The
relativistic conservation law in the Eulerian coordinates is no longer
represented by any divergence-free current; hence, it is not adequate to regard
the relativistic helicity (represented by the Eulerian variables) as a Noether
charge, and this stands the reason why the "conventional helicity" is no longer
a constant of motion. We have also formulated a relativistic action principle
of magnetohydrodynamics (MHD) on the Lagrangian coordinates, and have derived
the relativistic MHD cross helicity.
