Why do symmetries give conservation laws?
Every continuous symmetry hands back a conserved quantity, by a fixed recipe Emmy Noether published in 1918. A symmetry is a change that leaves the laws alone. Take the Lagrangian, the one function of where a system is and how fast it is changing, and find a change you can dial down to nothing that leaves it unaltered. Multiply its sensitivity to each quantity's speed by how far the change moves that quantity, add up, and add one extra term for clocks. That number never changes along the path nature takes. Slide sideways and it is momentum, turn and it is angular momentum, advance every clock and it is energy, shift a phase everywhere and it is electric charge. The recipe also works on a world made of finite steps, with no smooth limit needed.
Also in Chapter 32: Why Does Anything Move?.
This answer comes from OPH, Observer Patch Holography. OPH is a rigorous, consistent and complete description of the unified, self-creating computational Universe we inhabit. Its consistency is machine-checked: the fixed point at its core is constructed explicitly, and a theory with a model contains no contradiction.
To understand OPH and check every step yourself, start with the free book Reverse Engineering Reality. It builds the theory from the ground up, and the OPH Notebook answers your questions while you read.
read the free bookask the notebookor ask sage:read chapter 31: What Is a Lagrangian, and Why Is It the Last Thing?watch a universe make its own particles