Conservation of Energy
The law of conservation of energy states that the total energy of an isolated system remains constant over time; energy can be transformed from one form to another — kinetic, potential, thermal, and others — but cannot be created or destroyed.
Curious
Drop a ball from your hand. As it falls, it speeds up — the "stored" energy it had from being held up high converts into the energy of motion. When it hits the ground and bounces, some of that energy of motion turns into sound and heat at the point of impact, which is why a bouncing ball never bounces back quite as high as it started.
Nothing was destroyed in that process. Every joule of energy present at the start is still present at the end — some of it just moved into forms that are harder to notice or recover, like the faint warmth in the floor and the sound wave that spread through the room and faded.
This is one of the most tested ideas in all of science. Every attempt to build a machine that produces more energy than it consumes — a "perpetual motion machine" — has failed, and the law of conservation of energy is the reason physicists are confident, in advance, that every future attempt will fail too.
Exploring
In mechanical systems, the most commonly used statement of the law tracks the sum of kinetic energy (KE) and potential energy (PE):
| Symbol | Quantity | SI Unit |
|---|---|---|
| KE | Kinetic energy, ½mv² | joule (J) |
| PE | Potential energy (e.g., gravitational: mgh) | joule (J) |
When friction, air resistance, or other dissipative forces act, mechanical energy (KE + PE) is not conserved on its own — but total energy still is, once thermal energy, sound energy, and other forms are included. This is why "energy is lost to friction" is a useful shorthand for engineers, but a misleading phrase for physicists: the energy isn't lost from the universe, only from the mechanical (KE + PE) bookkeeping that's convenient to track.
Worked example: A 2 kg ball is dropped from a height of 5 m. Just before hitting the ground, ignoring air resistance, all of its initial potential energy has converted to kinetic energy: PE_initial = mgh = 2 × 9.8 × 5 = 98 J. So KE_final = 98 J, and since KE = ½mv², solving gives v = √(2 × 98 / 2) = √98 ≈ 9.9 m/s.
Deep Dive
Derivation from Newton's laws (mechanical case): Consider a particle of mass m moving under a conservative force F = −dU/dx, where U is potential energy. Newton's second law gives m(dv/dt) = −dU/dx. Multiplying both sides by v = dx/dt:
The left side is d/dt(½mv²) = d(KE)/dt. So d(KE)/dt = −d(PE)/dt, which rearranges to d(KE + PE)/dt = 0 — mechanical energy is constant in time whenever the only forces present are conservative (derivable from a potential).
The deeper justification — Noether's theorem: In 1918, Emmy Noether proved that every continuous symmetry of a physical system's action corresponds to a conserved quantity. Conservation of energy corresponds specifically to time-translation symmetry — the fact that the laws of physics do not change from one moment to the next. This is a far more general result than the Newtonian derivation above: it holds in Lagrangian and Hamiltonian mechanics, in field theory, and (in a locally modified form) in general relativity, wherever the underlying dynamics do not explicitly depend on time.
Conditions of validity: Strict global energy conservation as stated by Noether's theorem requires the system's dynamics to be genuinely time-translation invariant. In general relativity, energy conservation for the universe as a whole becomes subtle because spacetime itself is dynamical and not fixed — there is no single, unambiguous global time-translation symmetry for an expanding universe, a fact that has generated ongoing discussion in the literature on energy conservation in cosmology.
Historical note: The idea that "vis viva" (an early precursor to kinetic energy, ∝ mv²) is conserved in certain interactions dates to Gottfried Leibniz in the late 17th century, in direct dispute with Descartes' conservation of momentum (mv). The modern, general concept of energy — unifying mechanical, thermal, and other forms under one conserved quantity — was established through the work of James Prescott Joule, Julius Robert von Mayer, and Hermann von Helmholtz in the 1840s, cementing the first law of thermodynamics.
Concept Relationship Map
Requires: kinetic energy, potential energy, Newton's Second Law
Enables: work-energy theorem, thermodynamic first law, orbital mechanics energy analysis
Commonly confused with: conservation of mechanical energy specifically (which requires the absence of non-conservative forces)
Sources
- Noether, E. (1918). "Invariante Variationsprobleme." Nachr. Ges. Wiss. Göttingen, Math-Phys. Klasse, 235–257.
- Goldstein, H., Poole, C., Safko, J. (2002). Classical Mechanics, 3rd ed. Addison Wesley, Ch. 2, 13.
- Feynman, R.P. et al. The Feynman Lectures on Physics, Vol. I, Ch. 4. feynmanlectures.caltech.edu
- Landau, L.D., Lifshitz, E.M. (1976). Mechanics, 3rd ed. Butterworth-Heinemann, §6.