The relationship between symmetries and conservation laws is one of the most profound and far-reaching connections in physics.

The central result is called Noether's theorem, and it says that for every continuous symmetry of the Lagrangian or action for a system, you'll find a corresponding conserved quantity.

Momentum conservation, for example, follows from a symmetry called spatial translation invariance, meaning that you can pick up your system and slide it over without changing anything about the physics. Likewise, angular momentum conservation follows from rotation invariance, and energy conservation from time translation invariance.

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