Q:

Do translational invariance implies the conservation of both magnitude and direction of the corresponding momentum?

Suppose one is studying transmission in a system having translational invariance along $y$ direction. Should both direction and magnitude of $k_y$ ($y$ component of momentum of the wave) be necessarily same? Can a flip of sign of $k_y$ happen?

A:

The value, including direction/sign, is conserved. The Euler-Lagrange equation makes clear why such a momentum has zero time derivative. The Lagrangian doesn't depend on the spatial coordinate along which translation occurs; this coordinate is define is described as cyclic.

