1. A short definition:

Energy is the conserved essence of structure and capacity for change/transformation.

To expand on the definition:

1. Conserved: This means a measurable quantity and it is neither created nor destroy: i.e., we have the conservation of energy.

Actually, the general relativity (GR) requires a generalization of the straightforward conservation of energy principle that applies to most physical systems of interest. We expand on this issue below, but the short version is that the generalization is the general-relativity energy-momentum conservation equation holds (see also Car-120).

2. Essense of structure: Structure of systems (e.g., nuclei, atoms, ... , galaxies, etc.) is given its most fundamental/abstract description by specifying the amounts of each form of energy they have. Structure, of course, has many other features besides energy, but energy is the key to simple understanding and analysis.

What maintains the structure? Forces.

3. Capacity for change/transformation: In any change in a system, energy is always added or subtracted and/or changes form (i.e., is transformed). Changes other than in energy happen too, but the changes in energy can be regarded as the most fundamental/abstract description of the overall change.

Note all forms of energy can be changed into all other forms energy. That is why they are all energy. The changes are NOT always easy.

What causes energy transformations? Forces.

4. The Units of energy: Without going into details, the connection to force leads to the unit of energy being the unit of force times the unit of distance. In metric units, the unit energy is the joule (J) = kg*m**2/s**2.

2. The only physics formula everyone knows is E=mc**2. What does it mean?

Three things:

1. First, E=mc**2 means that energy and mass are both manifestations of the same thing which we call mass-energy in Relativityspeak. In fact, nowadays people often just use energy and mass as synonyms for mass-energy. The choice being made often just by which aspect of mass-energy is being emphasized.

2. Second, E=mc**2 means all energy has an associated mass which is the thing that resists acceleration and that causes the gravitational force.

Q: But there are negative energies, but NO negative masses.

A: There are NO negative masses, but there are negative contributions to mass and that is what negative energies are. If you add up all the contributions to energy in a system, the result is always positive---or so we believe.

Q: Can you elucidate why we ordinarily do NOT notice changes in mass when we change energy: e.g., when we change the heat energy content of a system?

A: Most of those changes in energy create changes in mass that are too small to measure ordinarily. For example, the fiducial minimum food energy needed per day per human is

```                 7.5 MJ = 7.5*10**6 J = (1.7925 ... )*10**6 calories = (1.7925 ... )*10**3 food calories
= { (8.344875 ... )*10**(-11) kg ]*c**2
```
(see Wikipedia: Food energy: Recommended daily intake). You can see that this change in mass and all others like it are immensely small by everyday-life standards and even by ordinary laboratory standards. So it is NO surprise that the law of conservation of mass was thought to absolutely true, rather than just a usually valid approximation before the introduction of E=mc**2 with special relativity in 1905.

3. Third, E=mc**2 means that there is such a thing rest mass which is the mass-energy of existence NOT counting any kinetic energy (i.e., energy of motion) you do NOT want to count: usually this macroscopic kinetic energy. Logically, rest mass should be called rest mass-energy, but that is NOT a conventional term.

The rest mass of most objects includes heat energy and microscopic forms of binding energy.

Now some elementary particles possess apparently intrinsic rest mass. Examples include electrons and quarks. Note in modern understanding, the subatomic particles proton and neutron are NOT elementary particles. They are composite particles since they are made up of bound quarks. The subatomic particles and other composite particles (e.g., molecules) and all macroscopic objects have intrinsic rest mass from the elementary particles and other kinds of rest mass too (i.e., heat energy and binding energy).

Particles with rest mass are massive particles. Particles with NO rest mass are massless particles. The photon is the best known example of a massless particle. All massless particles are elementary particles (see Wikipedia: Massless particle).

3. What are the forms of energy?

The three fundamental forms seem to be the aforesaid intrinsic rest mass, kinetic energy (i.e., energy of motion) and force field energy (i.e., the energy of the fields that cause forces).

All other forms are, yours truly believes, sub-forms of some degree including forms that are sums of other forms. The forms can overlap: an amount of energy can belong to multiple forms. In fact, forms of energy are often defined just for particular contexts: e.g., sound wave energy for the sum of various other forms that add up to be the total energy of sound waves.

Below is a non-exhaustive list of main energy forms:

4. Energy, energy forms, and energy transformations provide a very useful way to understand/analyze systems and their evolution. In fact, as foreshadowed in the preamble, energy analysis often gives you partial information when complete information is hard to obtain.

For example, the food web in the image can be analyzed in terms of energy transfers. One aspect of the analysis is that biota need so much energy to live and can only get so much from their environment (i.e., the Sun and other biota). You can put limits on their behavior from energy analysis without knowing what ATP actually does. Of course, for full information you need to know about the details of the energy transfers.

A simpler example is from mechanics. Say you have a bead on a frictionless wire near the Earth's surface. Just from its initial conditions (i.e., initial speed and height) and the conservation of mechanical energy (a sub-form of the conservation of energy), you can predict the speed of the bead for any later height. To get the full time evolution of the bead's motion requires much more information and analysis.

5. Yours truly believes that the only direct measurement of energy is by measuring mass and using E=mc**2 to obtain the energy value. This direct method can always be done in principle, but in practice it is often impossible.

All other measurements are indirect: you measure several quantities that are characteristics of a body's state that are NOT energy and calculate energy from a formula that contains those quantities. For example, kinetic energy is the energy associated with the motion of body and is calculated in the classical limit from the formula KE = (1/2)mv**2, where m is the body mass and v is the body speed.

Another example is that there is an exact formula for the energy density of electromagnetic fields calculated from characteristics of the electromagnetic fields (see electric field energy and magnetic field energy). The formula is pretty simple actually:

### energy density = εE**2/2 + B**2/(2μ) ,

where E is the electric field magnitude at a point, B is the magnetic field magnitude at the point, and ε and μ are constants. The formula gives the energy density at the point.

6. Is conservation of energy absolutely valid?

Probably it is absolutely true in the limit of flat spacetime (AKA Minkowski space), but NOT absolutely in curved spacetime which is a spacetime predicted by general relativity (GR). This means the conservation of energy is probably as valid as we need in most local regions of the observable universe, but probably NOT absolutely in intense gravitational fields like very near black holes and NOT in application to the observable universe as a whole.

This failure of conservation of energy to be absolutely absolutely valid is disconcerting, but all is not lost. The energy-momentum conservation equation of general relativity is the generalization of conservation of energy law and it is absolutely valid in general relativity. However, the energy-momentum conservation equation is a lot harder to understand and work with than the conservation of energy law. See Carroll, 2004, p. 120.

The upshot is that conservation of energy is a valid emergent principle and we usually do NOT want to say more than it is absolutely valid which saves us from getting into long discussions of arcana of general relativity.

7. The history of energy can be very briefly summarized (see also Wikipedia: History of energy). In 17th century, Gottfried Wilhelm von Leibniz (1646--1716) (the codiscoverer of calculus) noted that the quantity Σ_i (m_i*v_i**2) was conserved in some contexts: i.e., obeyed a conservation law. He called this quantity vis viva (i.e., living force). In the 19th century, this quantity, now with correct coefficient (1/2) prepended, came to be called kinetic energy. Other forms of energy and the conservation of energy were then discovered. Albert Einstein's (1879--1955) discovery of E=mc**2 in 1905 in some respects completed the concept of energy---but in other respects the story goes on.

8. Outside of physics, there may have always been a vague idea of energy, the stuff that does things. The word power used in its non-physics sense corresponds to the vague idea of energy. Some cultures have a somewhat more mystical or spiritual vague idea of energy: e.g., life force, Chi (Qi), mana.