Rule 9 of 36 · Chapter II — Symmetry & Conservation
Symmetry implies a conserved quantity
Why this rule exists
Noether's theorem makes precise an idea that would otherwise be a slogan: every continuous symmetry of a system's dynamics yields a quantity that does not change in time. If the laws are unchanged by shifting position, momentum is conserved; by rotating, angular momentum; by advancing the clock, energy. The conservation is not a separate empirical fact bolted onto the theory but a mathematical consequence of the symmetry, which is why these laws feel deeper than the particular forces at play.
In practice
When you meet a new system, first ask what leaves its dynamics unchanged. Is there a direction along which nothing depends? Then the momentum component that way is conserved. Is the setup indifferent to when you start the clock? Then energy is conserved. Each symmetry you find hands you a conserved quantity and thus one equation for free, often collapsing a differential equation into simple algebra.
When it doesn't apply
The symmetry must be a symmetry of the dynamics, not just the shape: a uniform gravitational field breaks vertical translation symmetry, so vertical momentum is not conserved. Only continuous symmetries give conserved quantities; discrete ones like mirror reflection give selection rules instead. In an expanding universe, time-translation symmetry fails and total energy is not globally conserved.