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First Principles of PhysicsSymmetry & Conservation · Rule 10

Symmetry & Conservation

Rule 10Chapter II

Energy is bookkeeping that always balances

Why this rule exists

Energy is not a substance you can point to but a number, computed by fixed rules, whose total stays constant as a closed system evolves. Its power is precisely that it is a scalar accountancy: kinetic, potential, thermal, and every other form are entries in one ledger that must balance. Because the total is fixed by time-translation symmetry, you can equate it at two instants and never track the tangled path between them.

First Principles of Physics

In practice

Identify every form energy takes at the start and the end — kinetic ½mv², gravitational mgh, spring ½kx², thermal, electrical — and set the total equal across the two states. Choose one reference level for potential energy and hold it fixed. If the ledger fails to balance, energy crossed your boundary as work or heat, so widen the system or add that term. Solve the resulting algebraic equation instead of the motion.

When it doesn't apply

Bookkeeping balances only for a closed system; friction and other dissipation do not destroy energy but move it to a thermal form you must include. In an open system, account for work and heat crossing the boundary. Relativistically, mass is itself an energy term (E=mc²), and only the combined total is conserved.

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