Nature of Physical Laws

Physicists study everything from sub-atomic particles to the most distant stars, and beyond simply cataloguing facts by observation and experiment, they look for the underlying laws — usually written as mathematical equations — that summarise those facts.

In any physical process governed by forces, many quantities keep changing with time — but a special few remain exactly constant. These are the conserved quantities of nature, and the principles behind them are central to describing physical phenomena quantitatively.

For motion under a conservative force, the total mechanical energy — kinetic plus potential — stays constant. A body falling freely under gravity is the standard example: its kinetic energy and potential energy both change continuously, but their sum stays fixed, and if it is released from rest, all of its initial potential energy has become kinetic energy by the moment it hits the ground. This restricted law (valid only under a conservative force) should not be confused with the far more general law of conservation of energy for an isolated system, which is the basis of the First Law of Thermodynamics: once every form of energy — heat, mechanical, electrical, and so on — is counted, energy is always conserved. Bring air resistance back into the falling-object example, and total mechanical energy is clearly no longer conserved once the object comes to rest on the ground — but the general law still holds, because the lost potential energy has simply become heat and sound (and eventually all of that becomes heat too). The energy of the whole system, object plus surroundings, is unchanged.

This general law of energy conservation is believed to hold everywhere, from atomic and nuclear processes up to the universe as a whole. Before Einstein's relativity, the conservation of mass was thought to be a similarly basic law, since matter seemed indestructible — a principle still routinely used when analysing ordinary chemical reactions, where atoms are merely rearranged among molecules (any energy difference between the reactants' and products' total binding energy shows up as heat, making the reaction exothermic or endothermic, but the total mass is essentially unchanged, because the change in binding energy is far too small to register as a change in mass). Einstein's relativity revealed that mass and energy are in fact equivalent, related by $E = mc^2$, where $c$ is the speed of light in vacuum — and in nuclear processes, where the energy changes are enormous, mass genuinely does convert into energy (and vice versa); this is the energy released in both nuclear power generation and nuclear explosions.

Energy is a scalar, but conserved quantities need not be. The total linear momentum and the total angular momentum of an isolated system — both vector quantities — are conserved as well. These particular laws can be derived from Newton's laws of motion within mechanics, but their validity reaches well beyond mechanics, into domains where Newton's laws themselves no longer apply.

Beyond their elegance, conservation laws are hugely useful in practice, because they let us bypass problems too complicated to solve in full. We may have no idea of the detailed forces at play during a collision between two cars, yet conservation of momentum alone lets us predict or rule out possible outcomes. In nuclear and particle physics, conservation laws are indispensable analytical tools — famously, applying energy and momentum conservation to $\beta$-decay is exactly how Wolfgang Pauli correctly predicted, in 1931, the existence of what we now call the neutrino, years before it was actually detected. …