Conservation of energy

The conservation of energy is a fundamental principle of physics stating that the total energy of an isolated system remains constant over time. In thermodynamic terms, the energy content of a closed system may only change if energy enters or leaves the system; within the system, it can neither be created nor destroyed but may be transformed from one form to another. A familiar example is the conversion of chemical energy into kinetic, thermal and acoustic energy in the explosion of a stick of dynamite, where the total of all resulting energy forms equals the chemical energy lost during combustion.

Mass–energy equivalence and modern interpretation

Historically, the conservation of energy was considered distinct from the conservation of mass. However, the theory of special relativity established that mass and energy are interconvertible, related by the equation E = mc². As a result, mass–energy taken together is conserved. This equivalence is particularly evident in nuclear processes, where mass defects correspond to binding energies, and becomes increasingly significant in extreme environments such as the early universe or near black holes emitting Hawking radiation.
The conservation of energy can be formally derived from Noether’s theorem, which states that continuous symmetry under time translation leads to the conservation of energy. Because the fundamental laws of physics do not vary over time, the total energy associated with a system remains constant. A direct implication is the impossibility of perpetual motion machines, since such devices would violate this conservation principle.
Certain interpretations of general relativity suggest that, depending on how energy is defined, conservation may not straightforwardly apply on cosmological scales. In quantum mechanics, Noether’s theorem applies to expectation values, preserving global consistency, although discussions continue regarding whether individual events might appear to violate conservation at microscopic scales.

Historical development

Ideas resembling energy conservation can be traced back to ancient natural philosophers. Thales of Miletus proposed that a single underlying substance formed the basis of all matter, though his identification of water differs from the modern concept of energy. Empedocles maintained that the four classical elements did not come into being or perish but only rearranged. Epicurus later argued for the eternal constancy of the total quantity of matter in the universe.
By the early seventeenth century, Simon Stevin demonstrated that perpetual motion was impossible in systems governed by static equilibrium. Galileo’s work on pendulum motion in 1639 identified that a body rises to the same height from which it falls, interpreted today as the conversion of potential energy into kinetic energy and back. This observation held irrespective of the shape of the path taken, provided no friction was present.
Christiaan Huygens’s 1669 analysis of collisions introduced the idea that both total linear momentum and the sum of kinetic energies remained unchanged under specific idealised conditions, although distinctions between elastic and inelastic collisions were not fully understood. Huygens’s later work connected the height of a body’s ascent to the impossibility of perpetual motion, establishing that the centre of mass of a system cannot lift itself without an external force.
Gottfried Leibniz, drawing on Huygens’s findings, introduced the concept of vis viva (living force), defined by the expression Σmᵢvᵢ², which represented the approximate conservation of kinetic energy in frictionless systems. Debate during the seventeenth century centred on whether momentum or vis viva was the more fundamental conserved quantity. Later work demonstrated that both are conserved under the correct physical conditions.
Isaac Newton’s Principia Mathematica (1687) formalised the laws of motion and momentum. However, the analysis of rigid and fluid bodies required additional principles beyond Newton’s formulations. Leibniz argued that the conservation of vis viva challenged dualist philosophical doctrines, an argument revived in the nineteenth century as energy conservation became better understood.
Johann and Daniel Bernoulli championed the vis viva principle. Daniel Bernoulli’s Hydrodynamica (1738) developed the Bernoulli principle, linking decreases in fluid pressure to energy transformations. His studies also formulated early notions of mechanical work and efficiency. This emphasis on conserved quantities helped lay the foundation for stationary principles in mechanics, including D’Alembert’s principle and the later Lagrangian and Hamiltonian frameworks.
The eighteenth-century thinker Émilie du Châtelet significantly advanced the concept of energy conservation. Drawing inspiration from Leibniz, she repeated experiments in which balls dropped into clay demonstrated that deformation increased with the square of velocity, supporting the expression for kinetic energy Eₖ = ½mv². Her conclusion that energy retains consistent dimensions across all forms paved the way for unifying potential, kinetic and thermal energy descriptions.
Engineers such as John Smeaton, Peter Ewart, Gustave-Adolphe Hirn and Marc Seguin later recognised that momentum alone was insufficient for practical engineering calculations. Their work contributed to the formulation of the conservation of energy as a central principle for analysing engines and machinery, supporting the development of thermodynamics during the nineteenth century.

Energy transformations and thermodynamic context

In thermodynamics, energy conservation is embodied in the first law: the internal energy of a closed system changes only when heat is added or work is done. Mechanical, thermal, chemical, electrical and nuclear energies may transform from one form to another, but their total sum remains constant. These transformations underpin common physical processes, such as:

• mechanical work converted into heat by friction
• potential energy converted to kinetic energy during free fall
• chemical energy transformed into electrical energy in batteries
• nuclear binding energy released as heat and radiation during fission