Trouble Understanding the Concept of Invariance 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.
 A: One has to start from what "laws of nature are". They are a distillation of observations.
The basic observation is that objects do not change if thrown or spinned . A golf ball may be spinning , but its shape is invariant , it does not change shape because it is spinning. An airplane travels with supersonic velocity, but in the plane  the objects on the plane have the same shapes  as on the ground. This should give an intuition of what invariance  means.
Newton developed laws that would allow mathematically to describe the kinematic behavior of objects.The invariance observed on the objects has been built into the laws, because if the laws describe the behavior of objects, then they should contain this invariance.
The mathematical description/encapsulation in laws and functions allow predictability for new situations with the need for observations only for unusual situations. All of our technological constructs from roads to ships to airplanes are designed by these functions that contain the laws which contain the invariances that keep the world as we see it.
Thus the statement:

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. 

just states the obvious, that to describe correctly the world around us is enough to say that the laws do not change, because the laws are an encapsulation of the unchanging behavior of matter ( macroscopically) under translations ( airplane) rotations (golf ball).
