I am reading the book by Brian Cox & Jeff Forshaw: "Why does E=MC^2". It is getting to the point of explaining what invariance is. According to Cox & Forshaw, invariance states that the laws of nature does not change if we spin around and determine them while we face different directions is called rotational invariance. Also, the laws of nature does not change if we move from place to place is called translational invariance.
Then it goes on to explain invariance in the situation of why the moon is moving away from earth at 4cm/year. Conserved quantity is angular momentum, and translational invariance conserved quantity is called momentum.
Can someone please shed some light on this because the concept sounds like it should be "common sense", but I am having such difficulty understanding the concept of invariance.