The connection between symmetries and conservation laws is one of the deepest relationships in physics. Noether's theorem says that for every continuous symmetry of a Lagrangian, you'll find a corresponding conserved quantity.

But to fully understand the connection between the two, we need to investigate their relationship in Hamiltonian mechanics. Any function on phase space generates a "flow," similar to how the Hamiltonian generates time evolution. Then the rate of change of any other function along the flow is given by its Poisson bracket with the generator. A quantity will be conserved if and only if the flow that it generates leaves the Hamiltonian invariant!

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