Conservation laws of a decagonal quasicrystal in elastodynamics

A special version of Noether's theorem for the sake of absolute invariance on invariant variational principles is used to obtain conservation laws of a three-dimensional solid periodically stacked in a two-dimensional quasiperiodic structure with decagonal symmetry. By applying Lie's infinitesimal criterion on the invariance to the Lagrangian density function under a continuous transformation group, it is found that there exist fourteen conservation laws (nine in physical space and five in material space). Conservation laws in physical space correspond to the known physical facts. Influence of the phason displacement field on the conservation laws in material space is presented. The infinitesimal symmetry-transformations from scale change and translation of coordinates as well as one coordinate rotation are verified to be true. Especially, although the angular momentum conservation laws do not contain the physical quantities in phason field, and the conservation of angular momentum of the phason displacement field itself does not hold, the physical quantities in phason field play an important role in the conservation law in material space derived from an infinitesimal coordinate rotation.