# Conservation of energy **Conservation of energy** is the law that the total [[Energy|energy]] of an [[Isolated_system|isolated system]] does not change with time. Energy can be moved and converted — [[Gravitational_energy|gravitational]] to [[Kinetic_energy|kinetic]] to [[Thermal_energy|thermal]], [[Chemical_energy|chemical]] to [[Radiant_energy|radiant]], [[Mass–energy_equivalence|mass]] to everything — but the books always balance. Thomas Murphy's table of the forms that matter for engineering lists them by formula: m·g·h for hydro and tides, ½·m·v² for wind and ocean current, h·ν for sunlight, H − T·S for chemical fuel, c_p·m·ΔT for stored heat, q·V for electricity and m·c² for nuclear.[^murphy-forms] No machine has ever been found that creates the stuff, which is why [[Perpetual_motion|perpetual motion]] is a design constraint rather than a slander. In the microsim below the reader takes a [[Pendulum|pendulum]] whose length is modulated in time, L(t) = L₀·(1 + ε·sin Ω·t), and turns two knobs: the modulation depth ε, from 0 to 0.3, and the drive ratio Ω/ω₀, from 0.5 to 3, where ω₀ = √(g/L₀). At ε = 0 nothing in the equations of motion refers to the clock, and the plotted total energy E = ½·m·L²·θ′² + m·g·L·(1 − cos θ) is a flat line — this is [[Noether's_theorem|Noether's theorem]] in one screen, time-translation symmetry producing a conserved quantity. Move ε off zero and the line stops being flat: the hand that shortens and lengthens the string does [[Work_(physics)|work]] on the pendulum, dE/dt ≠ 0, and near Ω = 2·ω₀ the work accumulates instead of cancelling, which is parametric [[Resonance|resonance]] — the physics of a child pumping a swing. The readouts are E(t) and dE/dt, and the caption marks the drive strength at which the motion period-doubles on its way to [[Chaos_theory|chaos]]. On the [[Energy]] flagship this article is the whole of Part VI — *Conservation of energy* (row E54), the section every other part of the spine is measured against, and the place where the sim shows that the law is not an axiom bolted on from outside but a consequence of a symmetry the reader can switch off. ## History The idea arrived in pieces and took two centuries to assemble. [[Galileo_Galilei|Galileo]] noticed that a ball rolling down one incline rises to nearly the same height on another, and [[Christiaan_Huygens|Huygens]] made the observation exact for colliding bodies and for the compound pendulum. [[Gottfried_Wilhelm_Leibniz|Leibniz]] named the quantity m·v² the *[[Vis_viva|vis viva]]*, the "living force", and argued against the Cartesians that it, and not [[Momentum|momentum]], is preserved in elastic collisions; [[Émilie_du_Châtelet|Émilie du Châtelet]], in her commentary on [[Isaac_Newton|Newton]], supported him with the observation that the depth of a ball's impression in soft clay scales with v² rather than with v. Through the eighteenth century *vis viva* remained a mechanical doctrine with an obvious exception: friction, which destroyed it without visibly producing anything. The nineteenth century closed that hole by identifying the missing quantity as [[Heat|heat]], and by the 1840s three people had said so independently. Julius Robert von Mayer argued from the difference between the specific heats of a gas at constant pressure and constant volume that a fixed amount of mechanical work corresponds to a fixed amount of heat;[^mayer1842] [[James_Prescott_Joule|James Prescott Joule]] measured that correspondence directly;[^joule1850] and [[Hermann_von_Helmholtz|Hermann von Helmholtz]] gave the general statement in 1847, deriving from the impossibility of perpetual motion that the sum of all forms of energy in a closed system is constant.[^helmholtz1847] [[Rudolf_Clausius|Clausius]] then paired the law with the [[Second_law_of_thermodynamics|second law]] in the two sentences that still open most textbooks: the energy of the universe is constant, its [[Entropy|entropy]] tends to a maximum.[^clausius1865] ### Mechanical equivalent of heat The [[Mechanical_equivalent_of_heat|mechanical equivalent of heat]] is the conversion factor between the unit in which heat was measured and the unit in which work was measured. Joule's paddle-wheel apparatus let a falling weight stir water in an insulated vessel and measured the rise in temperature; the work done by the weight and the heat gained by the water are the same quantity in different clothes.[^joule1850] The modern number is definitional rather than measured: the thermochemical calorie is defined as exactly 4.184 J, so 1 kcal = 4,184 J.[^murphy-units] The factor is what makes energy bookkeeping possible across domains. A diet of 2,000 kcal per day is 8.368 MJ spread over 86,400 s, which is 96.85 W — a human being runs at about the power of an old incandescent lamp — and heating 0.5 kg of water from 20 °C to 100 °C takes 40 kcal, or 167 kJ, which a 1,000 W kettle supplies in 167 s.[^murphy-units] One [[Kilowatt-hour|kilowatt-hour]] is 3.6 MJ, and the same ladder continues to the [[Units_of_energy|units]] used for national accounts: 1 quad ≈ 1.055×10¹⁸ J, and the United States runs at roughly 100 quads a year, about 3 TW, or 10,000 W per person.[^murphy-units] Every one of those conversions is the conservation law used as an accounting identity. ### Mass–energy equivalence In 1905 [[Albert_Einstein|Einstein]] showed that a body emitting radiation of energy E loses mass E/c², so that mass and energy are not two conserved quantities but one.[^einstein1905] The factor c² is enormous — the [[Speed_of_light|speed of light]] squared is 8.99×10¹⁶ m²/s² — which is why chemistry never noticed: burning a kilogram of hydrocarbon converts a few parts in ten billion of its mass, far below any nineteenth-century balance. Nuclear processes convert enough for the effect to be the whole point: [[Nuclear_fission|fission]] of uranium releases about 0.09 % of the fuel's rest mass, and [[Nuclear_fusion|fusion]] of hydrogen to helium about 0.7 %, the energy that powers [[Stellar_nucleosynthesis|stars]]. Murphy's table lists m·c² alongside m·g·h and ½·m·v² as one more entry in the same ledger, and that is the right way to read it.[^murphy-forms] Conservation of energy did not fail in 1905; it absorbed conservation of mass. ### Conservation of energy in beta decay The hardest test the law survived came from the [[Atomic_nucleus|nucleus]]. In [[Beta_decay|beta decay]] a nucleus emits an electron, and if the decay were a two-body process the electron would always carry the same energy. James Chadwick showed in 1914 that it does not: the emitted electrons come out with a continuous spectrum of energies, from nearly zero up to a sharp endpoint.[^chadwick1914] For two decades this looked like a genuine violation, and Niels Bohr was willing to consider that energy might be conserved only statistically in nuclear events. Wolfgang Pauli's alternative, floated in a letter of 4 December 1930 to a meeting of physicists at Tübingen, was to postulate an undetected neutral particle of very small mass emitted alongside the electron and carrying off the balance.[^pauli1930] Enrico Fermi named it the [[Neutrino|neutrino]] and built it into a quantitative theory of beta decay in 1934; direct detection took until 1956, when Clyde Cowan and Frederick Reines observed inverse beta decay beside a fission reactor.[^cowan1956] The episode is the standard example of how a conservation law is used in practice: the law was treated as the fixed point and the particle inventory as the thing to revise, and the bet paid. The same reasoning applied to [[Momentum|momentum]] and [[Angular_momentum|angular momentum]] fixed the neutrino's spin before anyone had seen one. ## First law of thermodynamics The [[First_law_of_thermodynamics|first law]] is conservation of energy written for a system that can exchange heat as well as work. For a closed system the change in [[Internal_energy|internal energy]] is ΔU = Q − W, the heat added minus the work done by the system, and every term is measurable: Q by [[Calorimetry|calorimetry]], W from pressure and volume, ΔU from property tables.[^yan-ch4] The law turns into engineering when it is written for a [[Control_volume|control volume]] — a region with fluid crossing its boundary — where the energy carried in and out by the stream appears as [[Enthalpy|enthalpy]] rather than internal energy, which is why turbine, compressor and nozzle calculations are enthalpy differences.[^yan-ch5] What the first law does *not* do is pick a direction. It permits a cup of coffee to heat up while the room cools, and permits a [[Heat_engine|heat engine]] to turn heat entirely into work; the [[Second_law_of_thermodynamics|second law]] forbids both. That division of labour is where popular accounts go wrong: "energy is conserved" and "energy runs out" are both true and are statements about different things. The energy in a litre of fuel is still present after it burns, as warm exhaust and a warm engine block. What is gone is its [[Exergy|exergy]] — its capacity to do work against the environment — and the [[Irreversible_process|irreversibilities]] of the burning are what removed it. ## Noether's theorem In 1918 Emmy Noether proved that every continuous symmetry of a physical system's action corresponds to a conserved quantity.[^noether1918] Invariance under translation in space gives [[Momentum|momentum]]; invariance under rotation gives [[Angular_momentum|angular momentum]]; invariance under translation *in time* gives energy. The theorem is the deepest answer available to "why is energy conserved?", and it sharpens the question into another: is the [[Lagrangian_mechanics|Lagrangian]] of this system independent of when you start the clock? In [[Hamiltonian_mechanics|Hamiltonian]] language the same statement reads dH/dt = ∂H/∂t, so the Hamiltonian is a constant of the motion exactly when it carries no explicit time.[^cline-symmetry] That is the switch the microsim throws. With ε = 0 the length is fixed, the Lagrangian contains no t, and the plotted energy is flat to the accuracy of the [[Runge–Kutta_methods|fourth-order Runge–Kutta]] integrator — any residual drift is numerical, not physical, which is itself worth seeing. With ε > 0 the length is a prescribed function of time, the system is no longer autonomous, and energy flows in and out through the agent doing the modulating. The Portal Book's canonical form writes the parametrically driven pendulum as θ″ + γ·θ′ + ω₀²·sin θ = −2·A·cos(ω·t)·sin θ, the drive entering multiplied by sin θ rather than added to it — the signature of a parametric rather than a directly forced [[Oscillation|oscillator]].[^comphys-param] Modulating the length and modulating the effective gravity enter the same way, which is why the sim's ε and the book's A play the same role. The interesting behaviour concentrates near Ω = 2·ω₀, where each half-cycle of the drive is timed to pump rather than to fight. With the book's settings — ω₀ = 1, ω = 2, [[Damping|damping]] γ = 0.2 — the pendulum hanging at rest stays there below A_c ≈ 0.18; from there to about 0.71 it settles into a closed orbit at twice the drive period; between 0.72 and 0.79 it rotates once per drive period; from 0.79 to about 1.033 the period doubles again; above that the motion is chaotic, with a periodic window near A ≈ 3.1, a further doubling cascade over 3.8–4.448, and chaos from 4.4489.[^comphys-regimes] None of this is a failure of conservation: add the modulating agent to the system and the total is constant again. What the sim shows is that the conserved quantity belongs to the symmetry, and when the symmetry goes, so does the constant. Both ends of the Ω slider are worth a look. As Ω/ω₀ → 0 the length changes so slowly that the pendulum's action stays constant — the adiabatic invariant — and the energy rides up and down with the length but returns. As Ω/ω₀ grows large the fast drive averages into an effective potential; for a pivot shaken vertically that is Kapitza's result, the inverted position becoming stable and the pendulum standing upside down with a small quiver.[^cline-kapitza] Between those limits sits the resonance, and the undriven [[Nonlinear_system|nonlinear]] pendulum underneath conserves ½·ω² − (g/L)·cos θ exactly, with a period departing from the small-angle 2π·√(L/g) by 3.8 % at 45° and 15 % at 90°.[^lebl-pendulum] ## Special relativity [[Special_relativity|Special relativity]] keeps conservation of energy but changes what is conserved and for whom. Energy and momentum are combined into a single four-component object whose parts mix under a change of reference frame the way space and time coordinates do, so observers in relative motion disagree about how much of a system's energy is rest energy and how much is kinetic, while agreeing that the total is conserved in every frame. The invariant built from the four-momentum is the rest mass, through E² = (p·c)² + (m·c²)², which reduces to E = m·c² for a body at rest and to E = p·c for a massless [[Photon|photon]].[^einstein1905] Because energy and momentum conservation are now components of one law, a process allowed by one and forbidden by the other cannot occur — the reason a free electron cannot absorb a photon, and the reason [[Positron_emission|positron emission]] and electron capture have the thresholds they do. ## General relativity In [[General_relativity|general relativity]] the clean global statement breaks down. The field equations imply a *local* conservation law, the vanishing of the covariant divergence of the stress–energy tensor: energy and momentum are conserved in every sufficiently small region.[^einstein1916] Integrating that into a single number for a whole [[Spacetime|spacetime]] requires a way to compare vectors at different points, and a curved, time-dependent geometry does not supply one. For asymptotically flat spacetimes — an isolated [[Black_hole|black hole]], a star, a binary — a total energy can still be defined at infinity and is conserved; gravitational-wave astronomy depends on it. For the [[Expansion_of_the_universe|expanding universe]] no such definition exists, which is why the energy of the cosmic microwave background falls as its photons redshift with no visible recipient. Noether's theorem explains the difference: there is no global time-translation symmetry to hang the conserved quantity on.[^noether1918] ## Quantum theory [[Quantum_mechanics|Quantum mechanics]] conserves energy in the sense that matters: for a system whose Hamiltonian does not depend explicitly on time, the expectation value of the energy is constant, and a state of definite energy stays a state of that energy. The [[Schrödinger_equation|Schrödinger equation]] makes the Hamiltonian the generator of time translation, so Noether's link between symmetry and conservation survives the change of framework intact. The energy–time uncertainty relation, ΔE·Δt ≳ ħ/2, is sometimes misread as a licence to borrow energy. What it constrains is how sharply a state's energy can be defined given how long the state lasts: an excited atom with lifetime τ emits a [[Spectral_line|line]] of natural width Γ = 1/τ in angular units, and with Δt = τ and ΔE = ħ·Γ the product is exactly ħ.[^raven-uncertainty] A 6.25 ns lifetime implies a 25.5 MHz linewidth (derived); a hundredfold longer life gives a hundredfold narrower line. Likharev's caution is worth repeating: time is a parameter and not an operator, so this relation is not on the same footing as the position–momentum one.[^raven-uncertainty] No experiment has shown energy failing to be conserved in an isolated quantum system. ## Status Conservation of energy is among the most stringently tested statements in physics, and in the [[Standard_Model|Standard Model]] it is exact: the Lagrangian has no explicit time dependence, so Noether's theorem applies without qualification. Every measured decay is a running check that the products carry the parent's energy, and the beta-decay episode stands as the template for how an apparent violation is handled. Two qualifications remain. The first is gravitational: a global energy for a non-stationary [[Spacetime|spacetime]] need not exist, so "the energy of the universe is constant" is not a well-formed claim in the way that "the energy of this laboratory is constant" is. The second is practical, and it is the one the rest of this flagship is about. Conservation makes energy a quantity that can be inventoried, but says nothing about usefulness: a joule in a warm lake and a joule in a tank of fuel are the same joule and are not the same resource. The [[Energy_transformation|conversions]] that matter — [[Photosynthesis|photosynthesis]], [[Combustion|combustion]], a [[Wind_turbine|turbine]], a [[Solar_cell|photovoltaic cell]], a [[Heat_engine|heat engine]] — all conserve energy perfectly and all destroy availability. Conservation is the constraint; the [[Second_law_of_thermodynamics|second law]] is the cost. ## See also - [[Noether's_theorem]] - [[Conservation_law]] - [[Perpetual_motion]] - [[First_law_of_thermodynamics]] - [[Mass–energy_equivalence]] - [[Mechanical_energy]] - [[Exergy]] - [[Irreversible_process]] ## References [^murphy-forms]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 5, "Energy and Fossil Fuels", p. 90 (Table 5.2, the forms of energy and where each reappears: m·g·h, ½·m·v², h·ν, H − T·S, c_p·m·ΔT, q·V, m·c²), and pp. 90–91 (the falling apple accounted at a constant total of 7 J). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-units]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5, pp. 88–98: W = F·d and 1 J = 1 N·m (pp. 88–89); power as energy per time (p. 91); 1 kWh = 3.6 MJ (pp. 92–93); 1 cal = 4.184 J and 1 kcal = 4,184 J (pp. 93–94); 2,000 kcal/day = 96.85 W and 0.5 kg of water from 20 to 100 °C = 40 kcal = 167 kJ, 167 s at 1,000 W (pp. 94–95); 1 Btu ≈ 1,055 J, 1 quad ≈ 1.055×10¹⁸ J, ≈100 quads/yr ≈ 3 TW ≈ 10,000 W per person (p. 95). Portal Book 097. [^yan-ch4]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 4, "The First Law of Thermodynamics for Closed Systems", pp. 127–186 (page to pin). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^yan-ch5]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 5, "The First Law of Thermodynamics for a Control Volume", pp. 187–238 (the enthalpy form for turbines, compressors and nozzles; page to pin). Portal Book 115. [^comphys-param]: Anagnostopoulos, Konstantinos (2016). *Computational Physics: A Practical Introduction to Computational Physics and Scientific Computing (using C++)*, 2nd ed. Chapter 6, "Motion of a Particle", p. 234 (the parametrically driven pendulum θ″ + γ·θ′ + ω₀²·sin θ = −2·A·cos(ω·t)·sin θ) and pp. 236–237 (RK4 integration with the angle wrapped into [−π, π]; defaults ω₀ = 1, ω = 2, γ = 0.2). Portal Book 061, https://open.umn.edu/opentextbooks/textbooks/computational-physics-a-practical-introduction-to-computational-physics-and-scientific-computing-using-c [^comphys-regimes]: Anagnostopoulos (2016), *Computational Physics*, Chapter 6, pp. 237–243 (the regime map: rest below A_c ≈ 0.18; a period-2T loop to ≈0.71; rotation at period T over 0.72–0.79; period doubling to ≈1.033; chaos beyond, with a periodic window near A ≈ 3.1, doubling over 3.8–4.448 and chaos from 4.4489; Poincaré sections sampled once per drive period). Portal Book 061. [^cline-kapitza]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised 2nd ed. Chapter 6, "Lagrangian dynamics", pp. 187–188 (the vertically driven pendulum b²·θ″ + b·Y″·sin θ + g·b·sin θ = 0 and its Mathieu-type small-angle form; the inverted state stabilised when the drive term dominates g) and p. 189 (non-autonomous systems have stable, unstable and chaotic regions). Portal Book 073, https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics [^cline-symmetry]: Cline (2018), *Variational Principles in Classical Mechanics*, Chapter 7, on symmetries and invariance, pp. 199–218 (page to pin), for the Lagrangian and Hamiltonian statements that a coordinate absent from the Lagrangian gives a conserved momentum and that a Hamiltonian without explicit time dependence is a constant of the motion. Portal Book 073. [^lebl-pendulum]: Lebl, Jiří (2014). *Notes on Diffy Qs: Differential Equations for Engineers*. Chapter 9, "Nonlinear systems", pp. 363–367 (θ″ + (g/L)·sin θ = 0 written as a planar system; centres at even and saddles at odd multiples of π; the conserved ½·ω² − (g/L)·cos θ; the period integral with no closed form, T_lin = 2π·√(L/g), and relative errors of 15 % at θ₀ = 90°, 3.8 % at 45° and 0.048 % at 5°). Portal Book 031, https://open.umn.edu/opentextbooks/textbooks/notes-on-diffy-qs-differential-equations-for-engineers [^raven-uncertainty]: Raven, Will (2025). *Atomic Physics for Everyone*, Chapter 3, pp. 58–60 (excited-state decay N(t)/N₀ = exp(−t/τ), Γ = 2π·γ, τ = 1/Γ) and Chapter 11, pp. 230–231 (the re-derivation from ΔE·Δt ≥ ħ/2: with Δt = τ and ΔE = ħ·Γ the product is exactly ħ). Likharev's warning that time is a parameter and not an operator, so the relation is less general than the position–momentum one, is at *Essential Graduate Physics, Part QM*, pp. 63–64 (Portal Book 047). Portal Book 046. The 25.5 MHz linewidth for a 6.25 ns lifetime is derived. [^mayer1842]: Mayer, J. R. (1842). "Bemerkungen über die Kräfte der unbelebten Natur." *Annalen der Chemie und Pharmacie* 42: 233–240. [^joule1850]: Joule, J. P. (1850). "On the Mechanical Equivalent of Heat." *Philosophical Transactions of the Royal Society of London* 140: 61–82. [^helmholtz1847]: Helmholtz, H. von (1847). *Über die Erhaltung der Kraft, eine physikalische Abhandlung*. Berlin: G. Reimer. [^clausius1865]: Clausius, R. (1865). "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie." *Annalen der Physik und Chemie* 125 (7): 353–400. [^einstein1905]: Einstein, A. (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" *Annalen der Physik* 323 (13): 639–641. [^einstein1916]: Einstein, A. (1916). "Die Grundlage der allgemeinen Relativitätstheorie." *Annalen der Physik* 354 (7): 769–822. [^noether1918]: Noether, E. (1918). "Invariante Variationsprobleme." *Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse* 1918: 235–257. [^chadwick1914]: Chadwick, J. (1914). "Intensitätsverteilung im magnetischen Spektrum der β-Strahlen von Radium B + C." *Verhandlungen der Deutschen Physikalischen Gesellschaft* 16: 383–391. [^pauli1930]: Pauli, W. Open letter to the participants of the Tübingen conference on radioactivity, 4 December 1930 ("Liebe Radioaktive Damen und Herren"), proposing a neutral, weakly interacting particle emitted with the beta electron. The letter was written against the background of Bohr's willingness to abandon exact energy conservation in individual nuclear events; it was not published at the time and circulated in manuscript. [^cowan1956]: Cowan, C. L.; Reines, F.; Harrison, F. B.; Kruse, H. W.; McGuire, A. D. (1956). "Detection of the Free Neutrino: a Confirmation." *Science* 124 (3212): 103–104. ## Bibliography The Portal Books and open texts this page draws on, grouped as the pair groups them. ### Modern accounts - Murphy, Thomas, *Energy and Human Ambitions on a Finite Planet* (2021), Chapter 5 "Energy and Fossil Fuels" — the unit ladder from the electronvolt to the quad, worked twenty ways. Portal Book 097. - Yan, Claire Yu, *Introduction to Engineering Thermodynamics* (2022), Chapters 4 and 5, the first law for closed systems and for control volumes. Portal Book 115. - Anagnostopoulos, Konstantinos, *Computational Physics*, 2nd ed. (2016), Chapter 6 "Motion of a Particle" — the parametrically driven pendulum and its route to chaos. Portal Book 061. - Cline, Douglas, *Variational Principles in Classical Mechanics*, revised 2nd ed. (2018), Chapters 6 and 7 — Lagrangian dynamics, the driven pendulum and the symmetry–invariance chapter behind Noether's theorem. Portal Book 073. - Lebl, Jiří, *Notes on Diffy Qs* (2014), Chapter 9 "Nonlinear systems" — the conserved energy of the plane pendulum and its period. Portal Book 031. ### History of ideas - Helmholtz, Hermann von, *Über die Erhaltung der Kraft* (1847) — the first general statement, argued from the impossibility of perpetual motion. - Joule, James Prescott, "On the Mechanical Equivalent of Heat" (1850) — the measurement that closed the gap between work and heat. - Noether, Emmy, "Invariante Variationsprobleme" (1918) — symmetry as the source of every conservation law. - Einstein, Albert, "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" (1905) — mass absorbed into the ledger. ## External links - [Energy and Human Ambitions on a Finite Planet](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet), Thomas Murphy, open textbook (Portal Book 097) - [Introduction to Engineering Thermodynamics](https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics), Claire Yu Yan, open textbook (Portal Book 115) - The Wikipedia pair's external links list further open resources, including the historical texts <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Conservation_of_energy.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Conservation of energy* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Conservation_of_energy.html" data-title="Conservation of energy"></div> *Built from `MICROSIM_GUIDE/specs/sims/Conservation_of_energy.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Conservation_of_energy) : [Wikitube](https://en.wikitube.io/wiki/Conservation_of_energy) · pinned revision [1370455457](https://en.wikipedia.org/w/index.php?oldid=1370455457) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E54 · sim pending (matter/Conservation_of_energy).*