  It is well-known that classical linear elasticity equations are not
form-invariant under local transformations. This is intrinsically related to
the inhomogeneity of elastic media. However, the reported new linear elasticity
equations for inhomogeneous media may appear in different forms. This paper
tries to clarify this issue by investigating the form-invariance of the
Lagrangian under local temporal or spatial gauge transformations. In this way,
these new equations in different forms can be easily understood as the results
from different choices of gauge fixing schemes. It recommends to choose
appropriate gauges with clear physical meanings to simplify calculations.
