Conservation of energy

Definition

In an isolated system — one exchanging neither matter nor energy with the outside — the total energy stays constant.

It is neither created nor destroyed. It transforms, moves, redistributes itself.

Changing form, not amount

The total is the same on every row. An ideal pendulum exchanges potential and kinetic energy indefinitely; a real pendulum eventually stops, not because energy vanishes, but because it disperses as heat into the air and the pivots.

The word "isolated" does all the work

For an open system, energy varies — and the balance stays exact provided the exchanges are counted:

$\Delta E = W + Q$ΔE=W+Q

The change in the system's energy equals the work received plus the heat received. This is the first law of thermodynamics: conservation, stated for a system that is not isolated.

Widening the boundary until it encloses the surroundings brings us back to the isolated case. Conservation never fails; it merely demands that one declare what is being counted.

Mass and energy

The relation $E = mc^2$E=mc2 does not contradict conservation: it widens its domain. Mass is one form of energy among others, and it is the total energy — mass included — that is conserved.

In a nuclear reaction, part of the reactants' mass reappears as radiant and kinetic energy. Nothing is lost: the balance closes as soon as mass is counted as energy.

Where the law comes from

Conservation of energy is not a fact observed and then raised to a rule. Noether's theorem shows that to every continuous symmetry of the physical laws there corresponds a conserved quantity.

The symmetry at issue here is invariance under translation in time: an experiment run today or tomorrow, all else being equal, gives the same result. It is from this invariance that conservation of energy follows.

Summary

The energy of an isolated system changes form without changing amount, because the laws of physics do not depend on the date.