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

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

Noether's theorem links each continuous symmetry to a corresponding conservation law

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

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

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

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

What gauge symmetry is — local phase transformations that leave physics invariant, the basis of all fundamental forces

Gauge symmetry: Local phase transformations preserving physics, underlying fundamental forces