### How was energy defined and discovered in the first place?

To explore this question, we'll consider the six most basic forms of energy in a little more detail. These are:

• Kinetic Energy
• Potential Energy
• Thermal (Heat) Energy
• Chemical Energy
• Nuclear Energy

In the process, we will hopefully shed a little more light on how energy is defined, and how these concepts were discovered by humans.

#### The Discovery of Mechanical Energy (the kinetic and potential energy of ordinary objects):

As described in the previous section on the various forms of energy, kinetic energy is the energy an object possesses by virtue of its motion. Anything that is moving or rotating possesses kinetic energy. The faster an object moves or rotates, the greater its kinetic energy. But how do we define kinetic energy mathematically, as something we can quantify? Well, for a simple particle of mass m (say, measured in kilograms) moving with some particular velocity v (say, in meters per second), the kinetic energy is defined as one-half of the particles mass times the square of its velocity (actually the magnitude of its velocity, its "speed", but we won't be pedantic here),

Ekinetic = (1/2) m v2.

To see first how this formula "behaves", note from the formula that if either the velocity or the mass is zero, then the kinetic energy must be zero, and that if neither are zero, the kinetic energy will be larger if either the mass or velocity is increased (assuming both are nonzero to begin with). Intuitively, you can think of kinetic energy as a measure of the work (or damage!) that something can do if it collides with something else; the larger the speed and/or the larger the mass, the larger the kinetic energy, and thus the greater the impact.

Below, after we define "potential energy", we'll discuss how the formula above was discoverd. But for the moment, notice just the following obvious thing again: Kinetic energy is defined by a specific formula. This formula was discovered by people who were trying to describe the behavior of the world with mathematical language; kinetic energy is not an intuitive, or vague, or mystical concept. This is true for all the forms of energy that we discuss here --- they have precise mathematical definitions and meanings. Energy can be quantified.

Potential energy, like kinetic energy, is also a measure of the work an object or system can exert on another object or system. Imagine a book falling off a table and crushing an egg. This is work being done to the egg by the book (pretty messy!) This potential work is a consequence of the position of the book relative to the floor. More specifically, it is the force of gravity that accelerates the book, giving it kinetic energy. So, as we noted in a previous section, because gravity, and hence the Earth, is a crucial component, the potential energy is really a condition of the book-Earth system. So how do we define  potential energy in this case? For the book sitting on the table, its potential energy is defined as the mass of the book times the acceleration of gravity g (which is about 10 meters per second squared at Earth's surface), and also times the height h of the table,

Epotential = m g h.

Again, as for kinetic energy, we see that there is a well defined mathematical formula that defines potential energy.

So, how then did people actually come up with these formula's for the kinetic and potential energies, and how did they prove the various special properties of energy? Amazingly, it took many people lots of hard work over at least a millennium to overcome various misconceptions and to discover the simple formulas above. First, some people, most notably Galileo Galilei and Isaac Newton (Newton's picture appears at right), gradually figured out how forces are related to acceleration --- this information is summed up by Isaac Newton's famous Laws of Motion, which we list here for completeness:

Newton's Laws of Motion:

1. An object at rest will remain at rest and an object in motion will remain in motion at constant velocity unless acted upon by a net force.
2. The net force on an object is equal to its mass times its acceleration (F=ma).

3. For every action there is an equal and opposite reaction.

Although most people are now familiar with these laws, they're really not all that obvious. The philosopher Aristotle, for example, wrote that all objects eventually come to a natural state of rest. From a practical point of view, he was correct, because most objects in our human experience do just that, they eventually stop, because they are subject to forces, such as friction with the air, and these forces generally bring objects in motion to rest with respect to the ground.

But this observation hid something deeper - that is, the crucial and not so obvious fact that objects not subject to interactions with other objects will simply keep moving unchanged. The world had to wait until Galileo, many centuries after Aristotle, to finally grasp this fact. Why was it so hard? Because its an abstract notion - in the real world, its impossible to completely turn off the interactions.

To analyze the consequences of these laws, Isaac Newton and Gottfried Liebniz both developed (independently) the body of mathematical techniques known as the calculus and applied it to analyze these laws. In the course of this analysis, they, and many people who followed them, found that it was extremely useful to formalize certain combinations of variables with special names which we now identify with the various different forms of energy.

Thus, to give a short answer, (mechanical) energy was "discovered" in the course of mathematically analyzing the equations derived from Newton's Laws.

More specifically, this was possible because it was found that Newton's Laws led, with the application of calculus, to formulas in which the parameter of time did not appear explicitly.

To see a concrete example, and how the particular  names for various forms of energy arose, consider again a simple mass m, such as book, which finds itself in Earth's gravitational field. We'll ignore air friction, to keep things really simple. Knowing ahead of time the definitions of kinetic and potential energy (which is really cheating!), we can add up the potential and kinetic energy as defined above, to get the total energy:

Total Energy = Potential Energy + Kinetic Energy

= m g h + 1/2 m v2.

(We read this as follows: "Energy equals mass times the acceleration of gravity times height, plus one-half the mass times velocity squared". Note that the multiplications are not indicated explicitly with an "x" - they are simply implied by the notation. Only the addition operation is noted explicitly: This convention makes the notation much simpler)

This is in fact a correct formula to calculate the total energy of the book at any moment. But what happened historically, before anybody knew how to define "energy", is that this equation was derived by "integrating" Newton's First Law (F=ma). "Integrating" is the fundamental process of calculus.

For those who want to see how this works in detail, we offer two options:

Surprisingly, as the derivations show, despite the fact that this quantity depends both on h and v, both of which change with time (say, as the book falls), it was found that this quantity equals a constant - i.e. it doesn't depend explicitly on time (that is, the variable time doesn't appear explicitly in the equation one gets from Newton's laws that contains the expression for the energy):

Great Discovery!  (m g h + 1/2 m v2 ) = constant

Now you might say to yourself "well, of course the energy doesn't contain the time variable, because we didn't include it when we wrote it down!". But this would be incomplete - Remember, we only wrote down the left hand side. How would we know to set this equal to a constant? To show this, we need to derive the complete expression from Newton's equations, and Newton's equations do involve time explicitly, so there is no a priori way to know that you would arrive at an expression that didn't!

But, you might say, how can this be? Don't h and v depend on time when the book is falling? And right you would be: What is meant that although the variables h and v both change with time as the book falls under the force of gravity, they both change in exactly the right way for the total energy, as given by the formula above, to stay always at the same, constant value!

In other words, h and v don't change in just any arbitrary way. They change exactly together in a way that keeps the energy expression constant.

Amazing you say, but how could this be exactly? Let's look at this more closely to see how it works. Before the book begins to fall, the speed v equals zero, so the kinetic energy is zero, and so the total (initial) energy just equals the initial potential energy:

Initial Energy = m g h, where h = table height.

Suppose that this energy is 5 Joules (the definition of a Joule, a basic unit for energy, is covered in a later section - just accept this term for now). As the book falls, it starts to pick up velocity, and therefore v, and its kinetic energy, begins to increase. But simultaneously, the potential energy of the book begins to decrease because the book's height h starts to decrease. The mathematical discovery that the total energy is constant tells us that the book falls in exactly such a way that the sum of the potential energy and kinetic energy remains exactly equal to 5 Joules. After the book has fallen (say, at the instant just before it hits the floor), its potential energy is now zero (because its height h above the floor equals zero), but the total energy is still 5 Joules, and the final kinetic energy is equal to this value:

Final Energy = Initial Energy = 1/2 m v2

Because the total energy doesn't change, we infer that the (initial) potential energy must have been completely converted into the (final) kinetic energy.

Note that the definitions of kinetic energy and potential energy were defined after the discovery that such a constant-in-time combination ( m g h + 1/2 m v2 ) existed. Because there is such a quantity, and only because there is such a quantity, does it make sense to break things down and call the combination ( m g h) "potential energy", and the other combination ( 1/2 m v2 ) "kinetic energy". If you couldn't add these things up into something that stayed constant in time, then these definitions wouldn't be useful! So the definition of energy expressions are "wholistic" in a sense.

Finally, people also analyzed these new physics equations to show that when objects interact, i.e. exert forces on each other, then the work exerted by one object on another, defined as

Work = Force x Distance,

is exactly equal to the loss in energy that the object experiences while doing that work. Likewise, this work is equal to the energy that the object being acted on gains. This discovery is called the "work-energy theorem" in physics texts, and is the fundamental connection between the concepts of energy and work. Moreover, its the reason that energy is conserved. Without this, the concept of energy might be interesting, but not very useful

In retrospect, it is really quite amazing that such a constant-in-time combination (m g h + 1/2 m v2 ) of the variables h and v even exists in the first place. Is this combination special to the particular case of a mass in Earth's gravitational field? Not all all! It turns out that there are such combinations for all physical phenomena known to us . There are very deep reasons for why this is so, and these are briefly discussed at the end of this section.

Discovery of Heat:

For more than a century after Newton, people didn't know that heat, which is now known to be the microscopic motion of molecules, was also a form of energy. They suspected instead that maybe it was some kind of substance not related to energy that was contained in things and could flow between things, and was released when things were burned or worn away by friction.

Some people called this supposed substance "caloric fluid". They started to suspect that there was more to the picture when somebody observed that when attempting to bore a cannon, one could grind and grind and make a lot of heat, but not grind away much of the cannon. Thus, it appeared that the "caloric fluid" was endless, and therefore it was hard to see how it could be coming through the material of cannon itself. Rather, it seemed to be produced somehow from the process of grinding the cannon.

Finally, an English physicist named James Prescott Joule 1818-1889, through very careful experiments, proved that heat is actually a form of energy by showing how it could come from conversion of other forms of energy, such as mechanical or chemical energy, and that when heat is considered in the calculation of the total energy, the total energy in processes involving heat is conserved.

#### Discovery of electromagnetic radiation and nuclear energy properties:

We now discuss how electromagnetic radiation (light) and nuclear energy (the so-called "rest-mass" energy of matter) came to be known.

Electromagnetic radiation and rest-mass energy may be thought of as representing two physical extremes of energy in nature. The phrase "rest-mass energy" refers to the intrinsic energy that an object has by virtue of its simply having mass, whereas light is a "pure energy state", and has zero "rest-mass". Ordinary objects that have both rest-mass and kinetic energy can be thought of as being in a state somewhere between these two extremes.

We use the phrase "rest-mass", because Einstein's Special Theory of Relativity tells us that the mass of an object is not actually constant, but actually increases with an object's velocity (a strange and wonderful implication of this theory). Here, we are specifically concerned with the relativistic energy that an object has when at rest, hence the term "rest-mass" energy. Einstein came up with his theory of special relativity when he attempted to explain certain inconsistencies between the theory of electromagnetic waves, which had been developed earlier in the nineteenth century by Faraday, Hemholtz, Maxwell, and others, with the mathematical properties of space and time as implied by Newton's Laws. Newton's Laws implied that all reference frames that only differ by a constant relative velocity should be equivalent, so that the laws of physics should look the same in all of these frames. But the equations for electromagnetic waves seemed to violate this idea.

These inconsistencies were particularly troubling because both Newton's Laws and the electromagnetic theory were by then well grounded in experiment. At the heart of the matter was the experimental finding that the speed of light was apparently independent of an observer's reference frame, which seemed consistent with the electromagnetic theory, but seemed at odds with Newtonian theory.  In the process of resolving this contradiction, Einstein deduced, much to everyone's great and continuing fascination, that  matter itself is a form of energy, the precise amount of this rest mass energy being given by his famous formula,

E=mc2,

where m is an objects mass, and c is the speed of light. As stated in the section on the various forms of energy, this formula tells us that a truly enormous amount of energy is bundled up inside ordinary matter. For example, it would cost over a million dollars to buy the amount of energy from a utility contained in the rest mass of a single penny!

Do we see any of this energy at use in the everyday world? Yes! A small fraction of that energy is released in nuclear reactions in nuclear reactors and nuclear weapons. More significantly, the energy given off by the Sun comes mostly from rest-mass converted into energy when hydrogen nuclei in the Sun fuse to form helium nuclei (fusion).

Einstein's theory should not be viewed as something different from the results deriving from Newton's Laws. Rather, Newton's Laws can be shown to be limiting case when velocities much less than the speed of light are considered. In other words, Einstein actually extended Newton's theory to large velocities, but in doing so, he changed our ideas about space and time forever.

Electromagnetic radiation, or light (although only some of it is visible to our eyes) may be thought of as pure form of energy. This includes visible light, the warmth you feel at a distance from a fire, and radio and television waves.

It is valid to think of light as consisting of packets of pure energy, called photons, that travel through space at about 186,282 miles per hour. Again, it is because of Einstein that we know that we can think of light as being the "pure energy state". This is because Einstein's theory also shows clearly that light, although made of discrete packets, has zero rest mass. Electromagnetic radiation is generated, for example, when the electrons in an atom jump to a lower energy level by emitting a photon, or when charged particles are accelerated back and forth in a radio transmitter's antenna.

Historically, the classical theory of light, upon which Einstein's work was largely based, was developed following a long period of research on electricity and magnetism. Initially, light was thought to be little "corpuscules" of energy, as suggested by Newton (for reasons which eventually proved erroneous). Then, in the nineteenth century, it was shown that light actually corresponds to electromagnetic waves, that is, coupled electric and magnetic fields which propagate in space via a kind of push-pull self-perpetuating manner. This discovery revealed how accelerating charged particles can generated light, and led to the invention of radio, and many other devices.

A little later on, around the turn of the century, however, it was found by Einstein and others that light also can be thought of as coming in discrete packets of energy (which we now call photons), as well as waves. The fact that light behaves both as particles and as waves is a strange and difficult to understand conceptual duality which underlies much of the theory of quantum mechanics in modern physics. This duality, in fact, lies at the heart of the deepest mysteries of present day particle physics.

The Reason there is Energy Conservation in our world

To conclude this section, let us take up the question of just why it is that the equations of physics should have led to the conserved quantity that we call energy in the first place? Is this just an accident? Nowadays, we have a deeper understanding of why there is such a quantity. It turns out that the true reason for such a quantity is the following innocent looking statement:

• The laws of physics do not change with time

From this very simple assumption, the principle of conservation of energy can be shown to hold.  The first person to fully appreciate this fact was the great mathematician, Emmy Noether, who first explained this fact in 1905, the same year that Einstein published his theory of special relativity.

The fact that the invariance, or symmetry of the laws of physics with respect to time could lead to something as concrete and useful as conservation of energy is really quite profound. As Noether showed, basic symmetries lead to many other laws of physics as well. Conservation of momentum, for example, another principle of physics, is a consequence of the fact that the laws of physics do not vary from place to place. Thus, symmetries allow us to derive very powerful "laws of nature" on very general grounds.

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