energy

(Definition)

Energy: Definition, Conservation, and Meaning Across Physics

Energy is one of the central organizing concepts of physics.

At the introductory level, it appears as kinetic energy, gravitational potential energy, elastic potential energy, thermal energy, and chemical energy.

At more advanced levels, energy becomes:

the conserved quantity associated with time translation symmetry;
the Hamiltonian that generates time evolution;
the time component of relativistic four-momentum;
an eigenvalue or expectation value of a quantum Hamiltonian;
a density and flux carried by classical fields;
a component of the stress-energy tensor in relativity.

These are not unrelated definitions. They are different manifestations of a common physical structure.

A useful introductory description is:

Energy is a scalar quantity used to account for physical change and for transfers between parts of a system.

The familiar phrase “capacity to do work” is useful at the beginning, but it is not a complete modern definition.

The deeper statement is:

When the laws of a system are invariant under translations in time, there is an associated conserved quantity called energy.

Figure 1. Energy can be stored in different physical forms and transferred between systems by work, heating, radiation, and matter flow.

1 Dimensions and units

Energy has dimensions

[E ] = ML2T −2.

(1)

The SI unit is the joule:

|--------------------------| 1 J = 1 kg m2 s−2 = 1 N m.| ----------------------------

(2)

Other useful units include

1 erg	= 10⁻⁷ J,	(3)
1 eV	= 1.602176634 × 10⁻¹⁹ J.	(4)

The electron volt is especially useful in atomic, nuclear, particle, and astrophysical physics.

2 Energy is a scalar

Energy is a scalar quantity.

It has magnitude but no spatial direction.

This distinguishes it from momentum.

Two particles moving in opposite directions can have equal kinetic energies even though their momenta have opposite signs.

For a particle of mass m and velocity v,

|-----1------| |K = --mv2. | ------2------|

(5)

Since

v2 = v ⋅ v,

(6)

the scalar nature is explicit.

3 Work as energy transfer

In undergraduate mechanics, energy is introduced most directly through work.

For an infinitesimal displacement dr under force F,

|-------------| dW--=--F-⋅ dr.|

(7)

For motion from point 1 to point 2,

|--------∫---------| | 2 | |W1→2 = F ⋅ dr. ----------1---------

(8)

Work is not a substance stored inside an object.

Work describes a transfer of energy.

4 The work-energy theorem

For a particle,

dv F = m ---. dt

(9)

Using

dr = v dt,

(10)

we find

dW	= F ⋅ dr	(11)
	= m ⋅ vdt	(12)
	= mv ⋅ dv.	(13)

Because

d(v2) = 2v ⋅ dv,

(14)

we obtain

( ) dW = d 1mv2 . 2

(15)

Integration gives

|W----=-ΔK.--| ---net---------

(16)

The kinetic-energy formula therefore has an operational meaning: net work changes kinetic energy.

Figure 2. Net work transfers energy into or out of translational kinetic energy. The dot product selects the force component along the displacement.

5 Potential energy

Some forces can be represented by a scalar potential-energy function.

For a conservative force,

|----------| F--=-−-∇U.--

(17)

In one dimension,

|------------| | dU- | |Fx = − dx .| -------------

(18)

The work done by the conservative force is

|--------------| -Wcons =-−-ΔU.--

(19)

Thus a conservative force doing positive work corresponds to a decrease in potential energy.

6 Potential energy is defined up to a constant

Only potential-energy differences affect the force.

U′(x) = U (x) + C,

(20)

then

dU-′ dU- − dx = − dx .

(21)

The zero of potential energy may therefore often be chosen for convenience.

This freedom does not make potential-energy differences arbitrary.

7 Familiar potential energies

Near Earth’s surface,

|----------| Ug-=--mgh.--

(22)

For a spring obeying Hooke’s law,

|------------| | 1- 2 | -Us-=--2kx-.-|

(23)

For Newtonian gravity between two point masses, choosing zero potential energy at infinite separation gives

|--------------| |U = − GM--m-.| ----------r-----

(24)

The negative sign is characteristic of a bound gravitational configuration under this choice of zero.

8 Mechanical energy

For a system with kinetic and potential energy,

|----------------| -Emech-=-K--+-U.-|

(25)

If only conservative forces do work and the potential has no explicit time dependence,

|------------------| -Emech-=-constant.-|

(26)

This is conservation of mechanical energy.

It is a special case of the broader conservation principle.

Figure 3. In one-dimensional conservative motion the total energy is fixed while kinetic and potential energy exchange. Turning points occur where kinetic energy vanishes.

9 Energy diagrams and turning points

For one-dimensional conservative motion,

K = E − U (x ).

(27)

Because

K ≥ 0,

(28)

the classically allowed region satisfies

|----------| E--≥-U-(x).-

(29)

Turning points occur where

E = U (x),

(30)

and therefore

K = 0.

(31)

Energy diagrams can reveal qualitative motion before the equation of motion is solved explicitly.

10 Energy in particle systems

For many particles,

∑ K = 1miv2i. i 2

(32)

The kinetic energy separates into center of mass motion and motion relative to the center of mass:

|--------------------| K = 1M V2 + Krel.| ------2----CM---------

(33)

This decomposition is important in collisions, molecular physics, celestial mechanics, and Thermodynamics.

11 Rotational kinetic energy

For a rigid body rotating about a fixed axis,

|-------1-----| Krot = -Iω2. | --------2------

(34)

For general rigid body motion, translational and rotational kinetic energies can both contribute to the total energy.

12 Power

Power is the rate of energy transfer:

|---------| | dE- | P = dt .| -----------

(35)

For mechanical work,

|----------| -P-=-F-⋅-v.-

(36)

For rotation about a fixed axis,

|--------| -P-=-τ-ω.-

(37)

The SI unit is the watt:

1 W = 1 Js−1.

(38)

13 Luminosity as an energy rate

In astrophysics, luminosity is radiated power:

|-----------| | dErad- | L-=----dt-.--

(39)

A source of nearly constant luminosity emits

ΔE ≈ LΔt

(40)

during a sufficiently short interval.

This is the direct connection between the energy concept and the PhysicsLibrary luminosity article.

14 Energy conservation is bookkeeping, not immobility

Energy conservation does not mean that the physical state remains unchanged.

A system can transform energy among different forms:

K	↔ U,	(41)
E_mechanical	→ E_internal,	(42)
E_chemical	→ E_thermal,	(43)
E_nuclear	→ E_radiation.	(44)

Conservation means that the total accounting remains consistent for an isolated system.

15 Thermodynamic internal energy

Thermodynamics introduces internal energy, usually written

|--| -U.-

(45)

It includes microscopic energy associated with degrees of freedom such as:

molecular translation;
rotation;
vibration;
intermolecular interactions;
electronic states;
chemical bonds;
nuclear degrees of freedom when relevant.

Internal energy is a state function.

Its change depends on the initial and final equilibrium states.

16 The first law of thermodynamics

Using the convention that W_by is work done by the system on its surroundings,

|----------------| |ΔU = Q − W .| --------------by--

(46)

Equivalently, with W_on = −W_by,

|----------------| -ΔU--=-Q--+-Won.-|

(47)

Here:

U is a state function;
Q is energy transferred because of a temperature difference;
W is energy transferred by generalized mechanical means.

Heat and work are modes of transfer, not stored properties.

Figure 4. In thermodynamics, internal energy is a state function while heat and work describe energy transferred across the system boundary.

17 Chemical, nuclear, and binding energy

Chemical energy is mainly associated with electromagnetic interactions and quantum states of electrons and nuclei.

Nuclear energy is associated with changes in nuclear binding and nuclear quantum states.

A reaction releases energy when the final state has lower total energy than the initial state.

For a bound system, one useful positive binding-energy convention is

|----------------------| |Ebind = Efree − Ebound.| ------------------------

(48)

The concept appears in gravitational systems, atoms, molecules, and nuclei.

18 Lagrangian mechanics

In analytical mechanics, dynamics can be described by a Lagrangian

L = L(q , ˙q ,t). i i

(49)

Define the generalized momentum

|---------| | ∂L- | pi = ∂ ˙q. | --------i--

(50)

The associated energy function is

|-----∑------------| |H = pi ˙qi − L. | i | -------------------

(51)

For a standard natural Lagrangian

L = T − U

(52)

with time-independent generalized coordinates and a velocity-independent potential,

|------------| -H--=-T-+-U.-|

(53)

In this common case, the Hamiltonian equals the familiar total mechanical energy.

19 Why time independence implies conservation

Differentiate

∑ H = p ˙q − L. i i i

(54)

Then

∑ _i

−

(55)

Expand

∑ ∑ dL- = ∂L-q˙i + ∂L-¨qi + ∂L-. dt i ∂qi i ∂ ˙qi ∂t

(56)

Using

∂L pi = --- ∂ ˙qi

(57)

and the Euler-Lagrange equation

p˙ = ∂L-, i ∂qi

(58)

the summed terms cancel:

|------------| |dH ∂L | |----= − ---.| -dt------∂t---

(59)

Therefore,

|--------------------------| |∂L- dH-- | | ∂t = 0 = ⇒ dt = 0. | ---------------------------

(60)

20 Noether’s theorem

Noether’s theorem gives the deeper interpretation.

Schematically,

|---------------------------------------------| continuous-symmetry---=⇒--conserved-quantity.--

(61)

Examples are:

spatial translation symmetry →linear momentum;
rotational symmetry →angular momentum;
time translation symmetry → energy.

Thus

Energy is the conserved quantity associated with time translation symmetry.

Figure 5. When the laws describing an isolated system are invariant under shifts of the time origin, Noether’s theorem associates that symmetry with a conserved energy.

21 A caution about the Hamiltonian

The Hamiltonian is often called the energy, but this should not be applied mechanically in every formulation.

Subtleties can arise with:

explicitly time-dependent coordinates;
electromagnetic gauge choices;
constraints;
nonstandard velocity dependence;
dynamical spacetime geometry.

For standard undergraduate systems, however, the Hamiltonian usually coincides with total energy.

22 Relativistic energy

In special relativity, energy and momentum are components of four-momentum:

|-----(-----)--| p μ = E-,p .| --------c-------

(62)

The invariant relation is

|------------------| |E2 = p2c2 + m2c4. | --------------------

(63)

For a massive particle at rest,

p = 0,

(64)

|----------| |E = mc2. | --0---------

(65)

Figure 6. Relativistic energy and momentum satisfy an energy-momentum relation with a nonzero rest-energy intercept for a massive particle.

23 Relativistic kinetic energy

For speed v,

|--------2-| E--=-γmc--,-

(66)

where

1 γ = ∘------2--2. 1 − v ∕c

(67)

The kinetic energy is

|----------------| K--=-(γ-−-1-)mc2.--

(68)

At low speed,

1-v2- γ ≈ 1 + 2 c2 + ⋅⋅⋅ ,

(69)

1 2 K ≈ --mv + ⋅⋅⋅ . 2

(70)

Newtonian kinetic energy is therefore the low-speed limit of the relativistic expression.

24 Photons and massless particles

For a massless particle,

m = 0,

(71)

|--------| -E-=--pc.|

(72)

For a photon,

|--------------| | hc | |E = hν = -λ-,| ---------------

(73)

and

|------| p = h.| -----λ--

(74)

This connects energy with wave frequency and wavelength.

25 Energy is frame dependent in relativity

Different inertial observers generally assign different energies to the same moving particle.

Energy is the time component of four-momentum, so it changes under Lorentz transformations.

The invariant relation

E2 − p2c2 = m2c4

(75)

is observer independent.

Thus relativistic energy is conserved in a chosen inertial frame for an isolated system, but its numerical value depends on the observer.

26 Quantum energy

In quantum mechanics, energy is represented by the Hamiltonian operator

|---| Hˆ. | ----

(76)

The Schrödinger equation is

|----------------------| | ∂ | |iℏ--|ψ (t)⟩ = Hˆ|ψ(t)⟩.| ---∂t------------------

(77)

The Hamiltonian therefore generates time evolution.

27 Energy eigenstates

An energy eigenstate satisfies

|-----------------| Hˆ|E ⟩ = E |E ⟩. | ----n------n--n----

(78)

For a time-independent Hamiltonian,

|ψ(t)⟩ = e−iEnt∕ℏ|E ⟩. n

(79)

The state changes only by an overall phase, so its observable probability distributions are stationary.

Figure 7. Quantum systems can possess discrete energy eigenvalues. Radiation exchanged in a transition carries the energy difference between levels.

28 Expectation value and energy uncertainty

A general quantum state need not have a definite energy.

Its expectation value is

|----------------| -⟨E-⟩ =-⟨ψ-| ˆH-|ψ-⟩.

(80)

The energy uncertainty is

|----------------------| |(ΔE )2 = ⟨ ˆH2 ⟩ − ⟨Hˆ⟩2. ------------------------

(81)

If the Hamiltonian is time independent, the expectation value of energy is conserved under Schrödinger evolution.

29 Quantum transitions

If a quantum system changes from energy E_i to energy E_f, conservation requires an energy exchange

|----------------| -ΔE--=--Ef-−-Ei.-|

(82)

For photon emission,

E > E , i f

(83)

and

---------------- |hν = E − E .| --------i----f--

(84)

This is the basis of discrete spectral-line energies.

30 Energy in classical fields

Fields can store and transport energy.

For the electromagnetic field in vacuum,

|------------------2-| u = 1𝜖 E2 + B--.| | EM 2 0 2μ0 | ----------------------

(85)

The electromagnetic energy flux is the Poynting vector:

|--------------| | 1 | |S = ---E × B. | -----μ0---------

(86)

The energy density has units of

−3 Jm ,

(87)

while the Poynting vector has units of

−2 W m .

(88)

31 Poynting’s theorem

Electromagnetic energy conservation is expressed locally as

|------------------------| |∂uEM--+ ∇ ⋅ S = − J ⋅ E.| ---∂t--------------------|

(89)

The terms describe:

change of energy stored in the field;
field energy flowing through space;
energy exchanged between the field and charged matter.

Figure 8. Electromagnetic fields store energy locally and carry it through space. The outward Poynting flux through a closed surface gives radiated power.

32 Luminosity from field energy flux

For radiation leaving a source,

|----∮---------| | | |L = S ⋅ dA.| ----------------

(90)

For isotropic radiation,

--L-- S = 4πr2 .

(91)

Thus luminosity is a rate of electromagnetic energy crossing an enclosing surface.

33 Energy in continuous matter

In fluids and continuous media, one often works with energy density u and energy flux F_E.

A generic local balance law has the form

|------------------| |∂u-+ ∇ ⋅ F = s ,| -∂t--------E-----E--

(92)

where s_E represents exchange with other subsystems.

For the complete isolated system, internal source and sink terms cancel in the total accounting.

34 Energy in general relativity

Matter and nongravitational fields are described by the stress-energy tensor

|----| |Tμν.| ------

(93)

In a local inertial frame, T⁰⁰ is associated with energy density, while mixed time-space components are associated with momentum density and energy flux.

The local covariant conservation law is

|------------| |∇ T μν = 0.| ---μ---------

(94)

This is the curved-spacetime form of local energy-momentum conservation for matter and nongravitational fields.

35 Why global energy is subtle in general relativity

A general dynamical curved spacetime need not possess a unique globally conserved scalar energy for the entire universe.

The reason again involves symmetry.

If spacetime has a suitable time translation symmetry, represented by a timelike Killing vector, a conserved energy can often be defined.

If the spacetime itself changes with time and lacks that symmetry, the corresponding global conserved energy need not exist.

This is not a failure of local conservation.

It reflects the role of spacetime geometry in gravitation.

36 Energy and cosmological redshift

A photon observed after cosmological expansion has lower frequency and therefore lower measured energy:

E = h ν.

(95)

In a spacetime without global time translation symmetry, asking where the photon’s “lost energy” went can be misleading.

The local relativistic conservation laws remain valid, but a simple Newtonian-style energy ledger for the entire expanding universe is not generally available.

37 Energy is not a material substance

Energy is extremely useful as a conserved accounting quantity, but it should not be imagined as a literal invisible fluid.

Several facts illustrate this:

the zero of Newtonian potential energy can often be shifted;
relativistic energy is observer dependent;
quantum states need not possess one definite energy;
classical fields store energy throughout space;
general relativity does not provide a unique local gravitational-energy density in arbitrary coordinates.

What remains fundamental is the mathematical structure of energy, transfer, symmetry, and conservation appropriate to the theory.

38 Energy conservation and entropy

Total energy can be conserved while its usefulness for producing macroscopic work decreases.

For example, organized mechanical motion can be converted by Friction into disordered internal energy.

The energy has not disappeared, but the second law of thermodynamics can prevent complete recovery of that energy as useful work in a cyclic process.

Therefore

Energy conservation and entropy increase are fully compatible.

39 A hierarchy of meanings

The concept can be summarized by physical level.

39.1 Introductory mechanics

1 K = --mv2, Emech = K + U. 2

(96)

39.2 Thermodynamics

ΔU = Q − Wby.

(97)

39.3 Analytical mechanics

∑ H = pi ˙qi − L. i

(98)

39.4 Special relativity

E2 = p2c2 + m2c4.

(99)

39.5 Quantum mechanics

iℏ ∂-|ψ⟩ = Hˆ|ψ⟩. ∂t

(100)

39.6 Field theory

Energy is represented locally by energy densities and energy fluxes.

39.7 General relativity

Local energy-momentum conservation survives, while global energy depends on spacetime symmetry and boundary conditions.

40 Worked conceptual examples

40.1 Example 1: kinetic energy

A 2.0 kg object moves at 3.0 ms⁻¹.

K	= (2.0)(3.0)²	(101)
	= 9.0 J.	(102)

If net work of 5.0 J is done on the object, its final kinetic energy is

|------| 14.0-J.-

(103)

40.2 Example 2: gravitational potential energy

Raise a 1.0 kg mass by 5.0 m near Earth’s surface.

Using

g = 9.81 m s−2,

(104)

ΔU	= mgh	(105)
	= (1.0)(9.81)(5.0)	(106)
	= 49.1 J.	(107)

40.3 Example 3: photon energy

For

λ = 500 nm,

(108)

E	=	(109)
	≈ 3.97 × 10⁻¹⁹ J	(110)
	≈ 2.48 eV .	(111)

40.4 Example 4: rest energy

For

m = 1.0 g,

(112)

E₀	= mc²	(113)
	≈ 8.99 × 10¹³ J.	(114)

This enormous scale explains why small mass differences can correspond to large nuclear energies.

41 Common mistakes

Treating energy as a vector.
Treating work as a substance stored in a body.
Treating heat as a state variable contained in an object.
Forgetting that potential-energy zero points can often be shifted by a constant.
Assuming mechanical energy is conserved when nonconservative transfers occur.
Confusing conservation of total energy with conservation of kinetic energy.
Treating negative gravitational energy as physically impossible.
Assuming the Hamiltonian equals T + U in every formulation.
Interpreting E = mc² as the complete energy formula for a moving particle.
Forgetting that relativistic energy is observer dependent.
Assuming every quantum state has one definite energy.
Forgetting that fields can store and transport energy.
Expecting a unique local gravitational-energy density in general relativity.
Assuming global energy conservation has the same simple form in every curved spacetime.
Confusing energy conservation with the claim that all energy can always be converted completely into useful work.