What does Noether's theorem state about the relationship between continuous symmetries and conservation laws in physics?

How does the Noether's theorem relate the conservation of energy to the symmetries of Lagrangian functions in classical mechanics?

Noether's theorem links conserved energy to time-invariance of the Lagrangian

Why spatial translation symmetry gives momentum conservation — if physics doesn't change with position, momentum is conserved

Spatial translation symmetry implies uniform physical laws, thus conserving momentum

Why rotational symmetry gives angular momentum conservation — if physics doesn't change with direction, angular momentum is conserved

Rotational symmetry implies invariance under spatial rotations, leading to conservation of angular momentum

Why time translation symmetry gives energy conservation — if physics doesn't change with time, energy is conserved

Time translation symmetry implies unchanging physics, thus conserving energy

What the Goldstone theorem says — every spontaneously broken continuous symmetry produces a massless boson

Goldstone theorem: Spontaneously broken continuous symmetries yield massless Goldstone bosons

What CPT symmetry is — the combination of charge, parity, and time reversal is always conserved

CP-symmetry implies charge, parity, and time reversal conservation