Understanding the first-order inhomogeneous linear elasticity through local gauge transformations

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.