What is energy ?

What is exactly Energy ? We all've heard about it and even have some intuition of what is all about.
Some will say it is the ability/capacity to do Work and they will be right, but, but... we want something more intrinsic and more general !
Let me first lay one line description and then we will explore a made up example, to make things clear.

Energy is simply a FUNCTION of the relations between Measurable properties of a Closed system.

Now this one was mouthful !! It seems more confusing that what we thought..but bear with me it will become clearer as we go.
Energy is a man made concept. The concept I think arose from thermodynamics. Read the definition above again... I hope you observe that energy is not a real thing but something to do with reality.
(For similar abstraction think of Information, there is no such thing as information but I hope nobody will dispute it has something to do with reality)
Just to start from somewhere lets postulate the first law of thermodynamics. Don't get scared you will get understanding of it over this article. OK, if didn't run out screaming by now here is it: (lets us signify Energy with the symbol U)
(1) Delta U = Q - W
Delta U - change of internal ENERGY in a closed system Q - heat in/out to/of the system W - work done onto/by the system
i.e. a closed system has "something" called Energy which could change if the system is influenced in some way.
[img in->box->out]
If there is no interaction with external system i.e. isolated system i.e. Q - W = 0, then :
Delta U = 0
"The internal energy of isolated system remain constant", this is how Energy conservation law is spelled most of the time.
In jargon it means : "You don't get anything for free" or "There is no free lunch".
OK, lets try to imagine this in some more down to earth way.
(Mind you in thermodynamics the energy is not parceled in pieces it is a smooth function, we are just using this as simplification for the purpose of our primer).

Say we have a room with window and 100 cubes sitting on the floor. We would need two volunteers Joe and Jane. Jane will come in from time to time and will have to figure how many cubes are still in the room, Joe will try to confuse her by hiding cubes or trowing them trough the window or other unforeseen ways. In the beginning it is easy for Jane, she has to just count the cubes.
Some formula like the one below could help her figure out what's left in the room :
(C)cubes on the floor - (O)cubes thrown over the window = 100
OR shorter :
C - O = 100
So far so good, but let's add a complication. Let give Joe a wooden box and allow him to hide cubes inside, Jane will be forbidden to look inside but will be able to weight the box. O'oooh and one more thing the box itself weight 2 kg and a cube weight 0.5 kg.
Could Jane invent a formula that will help her find the number of the cubes left in the room, just by counting what she can see and weighting the box, may be something like this may help (BW = box weight):
(C + (BW - 2)/0.5 ) - O = 100
That's good what if we carve a second window on some of the other walls in the room trough which we can throw cubes in, let's call this window I. After our mischief here is the updated formula to capture the new situation :
C + (BW-2)/0.5 - O + I = 100
So now Jane can handle situation with cubes being added, removed to/from the room or hidden in a box. She can make her life easier if she counts only the changes from one check to the next, instead of counting everything all the time. One additional benefit of using "change" instead of "hard counts" is that she can restart the experiment at any moment. She can just count the state of the system once (initial condition) and next time count the difference instead of the total.
So her little formula now becomes :
Delta C + (Delta BW)/0.5 - O + I = 0
here the Delta sign means "change in the quantity". And if you noticed this time the right side of the equation become zero, we can do this because we track changes not counts anymore.
Let's also combine all the "changes" in all the quantities under single symbol, namely U, we can interpret this pretending that all the cubes have been moved in the box, weighting is way faster than counting and simplifies things alot.
And then just for the sake of it Jane could use the symbols Q and W instead of I and O. And why not, one last thing let's move them to the right side of the equation. This way we have the things that change inside the room on the left side and all changes caused by the "interaction" with the outside world (cubes in/out the windows) on the right.
(Delta U)/0.5 = Q - W
FYI, it does not matter what arithmetic combination we select, namely -W-Q, +Q-W, +Q+W or -W+Q, we just have to settle on one of them and once settled use this combination all the time.
Jane is satisfied, but as often happen in life, something will screw things up. A friend of Joe comes to visit and brings him a bag of cubes with different weight.
The question is : Could Jane handle this new situation. Sure she can, she just needs one more cosmetic change to the formula :
(Delta U)/a = Q - W
where a is unknown (if you do a single measurement). But in our case Jane can find out what a is very easy if she do a consecutive measurements on all 3 parameters in the formula. Then she can substitute the measured values and calculate a. Remember Jane new game is to do at least two measurements in different instances of time.
She can find the changes, if we of course assume that whatever happens it comes in lumps, remember that was our simplification that we started the whole argument with.
Can we generalize the last formula even more ? For your disappointment, Yes we can !
If we are even more abstract the left side represent some sort of change, so we can disregard a altogether. We just want to know the gross change, we are no longer interested in counting things. In the real world the properties we measure (like pressure, volume, temperature) are mostly continuous. So after all the elaboration our final formula becomes :
Delta U = Q - W
Hmmm....where have I seen this ?!!
You can now clearly see how our initial formula after alot of deliberations and assumptions evolved from a way to count things to just measure the changes in the room (internal system) plus the impact of the surrounding. What this definition shows is that energy does not just pop-up out of nowhere it has to either be accounted of (part of the system) or has to come from outside. So unless energy is extracted or inserted out/in the system the energy content in the room has to stay the same.
What else did this whole discussion tells us. Remember the initial definition..

Energy is simply a FUNCTION of the relations between Measurable properties of a Closed system.

So we interpret it in the following way... we measure some parameters of a system (of course we will measure things that can be measured, f.e. volume, temperature, mass, pressure, magnetic field, position, etc.), then some time later we measure again. Whenever we find some relation that connect those properties in some meaningful way we call it ENERGY i.e. function of a measurable properties.
Different patterns of measurable things are different type of energy : kinetic, potential, chemical, nuclear, electrical ..etc. The sum of them is the TOTAL Energy of the system.
One inescapable but subtle conclusion from our reasoning is that we can never measure Energy "absolute" value per se, it is just a man-made Concept. This made up concept also requires the notion of state to make things comprehensible. We are interested in the change of things, knowing that change in pressure will cause change in temperature or volume (that is why we say it is a FUNCTION), help us to use this change in the real world to cause some other kind of change, like moving a piston. Or if we know that change in one type of energy can be converted in some other type of energy, then use this other type of energy to do some useful work.
See it does not matter what is energy as long as we can capture that change in mathematical equation which then allow us to calculate in advance what will the change be.
Remember Energy is function of measurable properties, if we can measure it we can calculate it, if we can calculate it we can predict it, if we can predict it we can use it to do some work for us.
Science is a quantitative discipline with the power to predict outcomes.
The concept of Energy and the law of conservation allow physicists to use double accounting.
Ok.. I poked the concept from so many angles it probably started to either make sense or become even more confusing for you ;)

Hope I did not, because next time we will tackle Entropy and boy that thing is confusing as hell...

Potential and Kinetic energy

We will use Gravitation to explore the idea of different forms of Energy in Classical mechanics.

To make useful the concept of Energy in Classical mechanics (Remember the idea of Energy comes from Thermodynamics), we would need at least two type of energies, the reason for this is that energy can only be converted from one form to another but it cannot be created or destroyed.
The scientists decided that those two candidates would be called Kinetic (the energy of motion) and Potential (the energy of position/state/"difference of position"). It make sense because Classical mechanics is primary concerned with the description and study of motion of objects.
On the right side in the image you would see all the various equations related to gravitation. I've decided to make this diagram, because most of formulas look very similar and at times it get's too confusing, use them as reference. And don't forget they are for circular orbits.
Now Kinetic energy has to measure the dynamic part of motion.

We know that the change in energy of a system is equal to the Work done on the system. But in Newtonian mechanics Work is equal to the Force applied over Distance. And we also know, force from Newton second law is F = m*a. "Mixing" them tighter will give us the following formula for Kinetic energy : (TODO: Add derivation) (1) E_k = 1/2 *m*v^2 m - mass of the object v - velocity of the object

Potential/Positional energy on the other hand does not have exact formula. In sense it is whatever we ascribe it to be. How's that ? Let's take for example the Gravitation field, which is a conserved field.
Which in our case means that the Total orbital/gravitational energy (E) of an object moving in this field is constant, instead what changes is the ratio between Kinetic and Potential energy i.e. E = E_k + E_p. Now we are going somewhere. To figure the exact formulation for Potential energy we would need to use the Universal gravitational law, because as we elaborated earlier change in Energy is the Work-done (and work depends on the gravitational Force).

Everybody knows the G-law : F = G * (m_1*m_2)/r^2 where : m_1, m_2 - mass of the two objects r - distance between the objects G - gravitational constant F - exerted force

We will look at the case when only 2 bodies are involved in which m_1 is much bigger than m_2 i.e. the influence of m_2 onto m_1 is negligible. (Ex: Earth orbiting the Sun, rocket orbiting Earth ....)
So we will say that m_2 is bound to orbit around m_1. And using gravitational law we can calculate what Force will be applied to m_2 at any point in space we wish. (In fact that is what the idea of gravitational field mean, if you were wondering i.e. we can calculate the Force acted upon any body who happens to be at any point near m_1.). Another simplification we will make is that m_2 orbits in circular orbit.
OK. Let's rehash. If we look at the G-law we will see that the Force acting on m_2 will be smaller the further the satellite is from the center of the more massive body. Our task is now to figure out the Work we have to do to move object from one orbit to another, this will give us the differences of the energy of the two orbits i.e. the potential energy.
See it was very easy...
Let see how our derivation will look like.
We start with the formula for Work : W = F * r Then we use the Work-energy theorem (which states that "change in Kinetic energy is equal to the Work done") : Delta E_k = W Delta E_k - W = 0 And according to Conservation of Energy if there is no outside influence : Delta E = 0 but : Delta E = Delta E_k + Delta E_p so finally : Delta E_p = - W = - F * r W - work done F - force applied r - distance over which the Force is applied

Because we need the difference between Work needed to move body from orbit at distance r_0 to orbit r_1, we would use little bit of Calculus. Don't run away, screaming :), let me try to explain it in simple terms.
If we were working with simple difference between two values we would have used simple algebra and subtracted the values calculated between the two points (Normally this is the so called Delta-delta, you probably have encountered often in high school physics). But as you may deduced the gravitational influence from point to point is changing gradually, because we can pick any point in between with any precision we like..
So if we calculate the Delta we will get just approximation not the real value.
OK you may say, what if we calculate differences every kilometer from one orbit to the other and then sum the list of results. Good point, we will get better approximation than the first case. But if we continue on this path we can go even further and do the calculation every meter, or every centimeter, or even every millimeter.
The smaller the interval the better approximation we will get, but you can see easily it will take us infinity to do all those calculations.
To save us from our misery sir Isaac Newton invented Calculus. What this mean for us is that instead of doing calculation for the infinite steps from orbit one to orbit two, we just apply neat simple formula, which the nice mathematicians cooked for us. And this formula is (definite integral, there is also indefinite integral which is even easier, but remember we are looking for a range between two orbits) :

int_(x_0)^(x_1) 1/x^2 dx = (- 1/x_1) - (- 1/x_0) = 1/x_0 - 1/x_1 = 1/(x_0 - x_1)
You can translate it to something like this : split the interval between ZERO and x_0 to infinite steps and calculate 1/x^2 for all those steps. After you are finished sum the results. Do the same for the interval from ZERO and x_1, sum again. Now subtract one sum from the other. Done.
Look at the integral sign it even looks like stylized SUM sign.
Mind you what really happens is not exactly summing, but it is close enough for our current understanding.
So let's apply this formula to our work equation and see what happens.
By the rules of calculus the constants can be moved outside the integration process. So we move G*m_1*m_2 out. W = int_(r_0)^(r_1) F\ dr = int_(r_0)^(r_1) (G * m_1* m_2) /r^2\ dr = (G * m_1* m_2) * int_(r_0)^(r_1) 1/r^2\ dr = (G*m_1*m_2) /(r_0 -r_1) and as we will see in a minute the ZERO point for Potential energy is chosen at infinity i.e. both distances are negative, so Delta r = (-r_0)-(-r_1) = r_1 - r_0, but I'm getting ahead of myself. Our solution for Potential energy then will be : Delta E_p = - W = - (G*m_1*m_2) / (Delta r) So this then, is our formula for gravitational potential energy : (1) E_p = - (G*m_1*m_2) / r
Let me go on a tangent here ... you can skip it if you want.. we already got our formula.
SPECIFIC "thing"
Normally in Physics where mass is involved there is this notion of 'SPECIFIC'-something, which is calculated by dividing the formula we are exploring by the mass. The trick is that "omitting" the mass in this way, we gain a formula for a unit-of-mass i.e. making it independed of the exact object mass. F.e. if we divide the above formula (1) by m_2. (And using the symbol M instead of m_1 for the big mass), we get : (2) E_p = - (G*M)/r You probably would say, so what's the difference ? The difference is that now in (2) we have formula were we do not care about the exact mass of the satellite. Or alternatively you can think of it like the formula for potential energy per 1 kg i.e. we have invariant way of doing it. So if we need to calculate what potential energy will be for 150 kg for example, we just multiply the result of our calculations by 150 at the end when we are finished. Ok to confuse us even more physicists do not call (2) "specific potential energy", but "gravitaional potential". And the earlier formula (1) is "gravitaional potential energy". Still confused... welcome to the club ;) Can you spot equation (2) in the graphic-diagram above ?

Pfuu... finally we have the formulas for both Kinetic and Potential energy in Gravitational field, but we are not done yet.

TODO