It's intuitively clear that the energy most accurately describes how much the state of the system is changing with time. So if the laws of physics don't depend on time, then the amount how much the state of the system changes with time has to be conserved because it's still changing in the same way.

In the same way, and perhaps even more intuitively, if the laws don't depend on position, you may hit the objects, and hit them a little bit more, and so on. The momentum measures how much the objects depend on space, so if the laws themselves don't depend on the position on space, the momentum has to be conserved.

The angular momentum with respect to an axis is determining how much the state changes if you rotate it around the axis - how much it depends on the angle (therefore "angular" in the name). So the symmetry is linked to the conservation law once again. 

If your intuition doesn't find the comments intuitive enough, maybe you should train your intuition because your current intuition apparently misses the most important properties of time, space, angles, energy, momentum, and angular momentum. ;-)

Can Noether's theorem be understood intuitively?
Let's take a wire or a rope. I usually do this with a chain or my scarf.

I fixate one end in my hand and apply rotation (by subtle movements of this endpoint like spinning a lasso). The rope gets into rotation and obtains certain bent shape:


Some part is missing because cellphone cameras ain't great for high-speed photos but I hope you can imagine all the scarf.

The question is: how can I calculate/predict this shape?

Although this problem doesn't seem that bizarre, I have never seen any solution. Nor I have found this question asked anywhere on internet... Must be because I just don't know how to formulate it without pictures.

Also by taking longer rope I get more than one bend:


I apoligize for the quality again. It is even harder to rotate this while taking picture. The form is not spiral, it is more like a shape in a plane that's rotating.

I'll be thankful for explanations, solutions, links or at least a correct formulation of this problem.
