A symmetry is a transformation that leaves the form of a physical law unchanged. Examples include translations in time, translations in space, and rotations. When a Lagrangian is invariant under a continuous transformation, the corresponding system usually has a conserved quantity.

Noether's theorem states that every continuous differentiable symmetry of the action corresponds to a conservation law. Time-translation symmetry implies energy conservation, spatial translation symmetry implies momentum conservation, and rotational symmetry implies angular momentum conservation. This is one of the deepest links between mathematics and physics.

For a central force problem, rotational symmetry reduces the dynamics to motion in a plane and yields conservation of angular momentum $\mathbf{L}=\mathbf{r}\times\mathbf{p}$. If the Lagrangian does not depend explicitly on time, then the Hamiltonian is conserved under suitable conditions.