# What Is Energy? Where did it come from?

The simplistic undergrad explanation aside, I've never really understood what energy really is. I've been told that it's something when converted from one kind of something to another kind, or does some "work", as defined by us, but what is that something?

Moreover, if the total amount of energy in the universe is finite and we cannot create energy. Then, where did it come from? I've learned from thermodynamics where it goes, but where does it come from?

I know this sounds like something trivially simple, but there is so much going on in the physical universe and I just can't grasp what it is. Maybe it is because I lack the mathematical understanding that I can't grasp the subtle things the universe is doing. Still, I want to understand what it is doing. How do I get to the point of understanding what it's doing?

(Note: What prompted me to ask this was this answer. I'm afraid that it just puzzled me further and I sat there staring at the screen for a good 10 minutes.)

• Energy conservation can be violated if time symmetry is violated. This happens in the vastness of space where dark energy makes space and time expand faster than ever. This violates energy conservation and is allowed as long as time symmetry is also violated. Commented May 31, 2020 at 14:41
• You were prompted by a rather sadistic one :) Commented Mar 11, 2021 at 11:54

Energy is any quantity - a number with the appropriate units (in the SI system, Joules) - that is conserved as the result of the fact that the laws of physics don't depend on the time when phenomena occur, i.e. as a consequence of the time-translational symmetry. This definition, linked to Emmy Noether's fundamental theorem, is the most universal among the accurate definitions of the concept of energy.

What is the "something"? One can say that it is a number with units, a dimensionful quantity. I can't tell you that energy is a potato or another material object because it is not (although, when stored in the gasoline or any "fixed" material, the amount of energy is proportional to the amount of the material). However, when I define something as a number, it is actually a much more accurate and rigorous definition than any definition that would include potatoes. Numbers are much more well-defined and rigorous than potatoes which is why all of physics is based on mathematics and not on cooking of potatoes.

Centuries ago, before people appreciated the fundamental role of maths in physics, they believed e.g. that the heat - a form of energy - was a material called the phlogiston. But, a long long time ago experiments were done to prove that such a picture was invalid. Einstein's $E=mc^2$ partly revived the idea - energy is equivalent to mass - but even the mass in this formula has to be viewed as a number rather than something that is made out of pieces that can be "touched".

Energy has many forms - terms contributing to the total energy - that are more "concrete" than the concept of energy itself. But the very strength of the concept of energy is that it is universal and not concrete: one may convert energy from one form to another. This multiplicity of forms doesn't make the concept of energy ill-defined in any sense.

Because of energy's relationship with time above, the abstract definition of energy - the Hamiltonian - is a concept that knows all about the evolution of the physical system in time (any physical system). This fact is particularly obvious in the case of quantum mechanics where the Hamiltonian enters the Schrödinger or Heisenberg equations of motion, being put equal to a time-derivative of the state (or operators).

The total energy is conserved but it is useful because despite the conservation of the total number, the energy can have many forms, depending on the context. Energy is useful and allows us to say something about the final state from the initial state even without solving the exact problem how the system looks at any moment in between.

Work is just a process in which energy is transformed from one form (e.g. energy stored in sugars and fats in muscles) to another form (furniture's potential energy when it's being brought to the 8th floor on the staircase). That's when "work" is meant as a qualitative concept. When it's a quantitative concept, it's the amount of energy that was transformed from one form to another; in practical applications, we usually mean that it was transformed from muscles or the electrical grid or a battery or another "storage" to a form of energy that is "useful" - but of course, these labels of being "useful" are not a part of physics, they are a part of the engineering or applications (our subjective appraisals).

• How can one say that laws of Nature DO NOT depend on time? Because It is well documented that prior to inflation rather during inflation the laws of nature were different. Isn’t that the whole concept of theory of cosmic inflation?
– A.M.
Commented Sep 23, 2021 at 21:19
• No, inflation, like all other phenomena, has variables that are time-dependent but the laws governing cosmic inflation are fixed. All experiments that have tried to find any change of the laws of physics have ended with negative or at most inconclusive results. Most of the possible variability would be a change of some parameters that would be upgraded to scalar fields. Commented Sep 25, 2021 at 7:15

I don't think the answer is trivially simple. I will try to give an explanation. In many problems of physics, what you are given is the initial and final states of the system. You don't know (or maybe no one does) what happens between these two states. Now there are quantities that you can measure before and after the system has undergone this change of state. The question is can you predict some of these quantities by knowing the others. Remember that we don't know the mechanism by which the system moves from these two states. But if you have something known as a conservation law, the problem becomes simple. (By saying that a quantity is conserved we mean that it doesn't change throughout some process). Suppose you have some magic function involving the quantities, which gives the same value no matter what the state of the system is, then you are done. The value of the function we call energy. And since its value doesn't change between these two states we say that its conserved.

This excerpt is from Feynman Lectures:

There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or any- thing concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. (Something like the bishop on a red square, and after a number of moves—details unknown—it is still on some red square. It is a law of this nature.)

• So, essentially, almost all of physics is based upon a quantity that no one has ever been able to truly define on it's own?
– Anna
Commented Jan 16, 2011 at 10:50
• @Anna: I would guess that the vast majority of physicists are able to define energy to their satisfaction. For some people the definition will be practical, in terms of the formulas you use to calculate it, while others will have some abstract idea of what energy is. It really doesn't matter. The fact is, all physicists, even if they are bothered by an apparent lack of definition, are able to use energy to develop theories and analyze experiments together, and as far as I'm concerned, that's all a physical concept needs to be useful. If you go beyond that, you're venturing into philosophy. Commented Jan 16, 2011 at 11:11
• Energy is never a fundamental entity of a theory as positions or wavefunctions are. Energy just shows up because our theories have some symmetries and it would always show up regardless of the kind of theory, as long as it contains the appropriate symmetries. Commented Jan 16, 2011 at 12:32
• I would like to endorse Raskolnikov's important comment, too. Energy is not something one starts with while constructing a theory; energy is a "cherry on pie" that one can find out to be conserved. It is given by a fixed formula that can be found and when this formula is evaluated at any time, the number is always the same. It didn't have to exist at all. However, energy is really special because the information about the formula for the energy is equivalent to the information about the way how the system evolves in time - so it's a bit more fundamental than Raskolnikov suggests. Commented Jan 16, 2011 at 18:01
• The great thing about Feynman is that when he didn't know something, he knew he didn't know something. Commented Sep 19, 2011 at 20:31

To understand what energy is, it is necessary to understand the concept of work.

Work is defined as the action of a force over a path.

$$W=\vec{F}\cdot\vec{d}$$

What does this means? It describes how "exerting" or "draining" a particular action is. For example, imagine lifting a shopping bag of mass $$10\ \mathrm{kg}$$ vertically by $$1\ \mathrm m$$. This takes work, and exactly the following amount, given by the weight of the bag times the distance.

$$W= \vec{F}\cdot\vec{d} = Fd\cos{0}=mgd=10\ \mathrm{kg}\times9.8\ \mathrm{m\ s^{-2}}\times1\ \mathrm m=98\ \mathrm J$$

Energy is classically defined as the capacity of a physical system to do work, or in other words: as you perform work, you exchange energy for some physical effect by doing work. Or in other terms again, by exerting a force over a distance you convert energy into work.

In our example, you need to use some form of energy to lift the shopping bag. The quantity you need is exactly the amount of work we calculated.

What happens to this work? It's converted to energy again – to gravitational potential energy:

$$U_\text{final} = U_\text{initial} + W$$

or

$$\Delta U = U_\text{final} - U_\text{initial} = W = mgd$$

which is the classical definition of gravitational potential energy.

So in practice – we never see or measure energy directly. When energy changes form, it is called work, which we can measure. So work, in a way, is a "transport" concept for energy. Energy, on the other hand, is like a "reservoir" of work in potential.

Why is energy a useful quantity? After all, work seems to be a more "fundamental" quantity from an experimental point of view.

The answer to this lies in the conservation law of energy. Work in itself describes a change in energy, so it's not a conserved quantity in itself unless you embed it in the more general concept of energy, which is conserved.

In fact, we can derive large swaths of classical mechanics using conservation of energy as a prime principle, together with the principle of least action.

Caveats

In more advanced theories, conservation of energy is a much more complicated matter and does not apply as simply as in the classical sense. For example in SR, energy can be converted to apparent mass and vice versa.

There are also very interesting mathematical properties of potential energy and its relation to forces and especially fields of forces. These explanations, though are way more abstract and mathematical – I assume you want an intuitive, instinctual explanation of what energy is.

If you are looking for the former please see this question.

• Yep, I've studied that, but the thing you're overlooking is that what is this quantity? Think about it. Where does this capacity comes from? What allows us to exert a force in the first place? What is that unknown x?
– Anna
Commented Jan 16, 2011 at 10:48
• @Anna: I don't understand what is the x you are looking for. Energy is what allows us to exert a force. With "zero" energy you can't exert a force. Commented Jan 16, 2011 at 10:56
• As LM put it I was searching for potatoes only there are none to find.
– Anna
Commented Jan 16, 2011 at 13:44
• This is the Physics 101 version, and is a good answer to have on tap for beginners. Commented Jul 15, 2011 at 14:55
• Energy is NOT the capacity to do work. Textbooks published in the last few decades are usually careful not to define it that way. See R.L. Lehrman, "Energy is not the ability to do work," The Physics Teacher 11 (1973),15-18, and E. Hecht, "Energy and Change," The Physics Teacher 45 (2007), 88-92.
– pwf
Commented Nov 11, 2020 at 3:57

Wow, this is going to be a risky answer. However, I think you may be looking for an answer that is more conceptual than mathematical or philosophical, so here goes:

Energy is change. That is, energy is present if we observe relationships between objects and fields changing in some way from moment to moment.

Heat energy is just a very fine-grained version of change, expressed in the motion of many molecules whose average motion is negligible.

Potential energy is the possibility of future change. It requires the additional idea that change can be absorbed by some sort of spring-like capability, stored for a length of time, then released again in the future as explicit change.

This spring-like storage effect always seems to boil down to some form of stretching or compressing fields in ways they don't want to go. Thus winding an old style grandfather clock with a key captures explicit change (winding) in the form of interesting stresses on the bonds that hold metal atoms together. For nuclear energy the fields are different, but the concept of stretching or compressing them in interesting ways remains pretty much the same.

Finally, within the idea of potential energy lies an important hint at the relationship between energy and mass. Mass is in a quite real sense the ultimate form of potential energy. In matter, the energy of the past is so well safeguarded from release that it takes an extraordinary key -- specifically an equal quantity and type of antimatter -- to unwind it fully and release all of its energy. For matter, it is the various unbreakable rules of conservation, such as charge conservation, that keep this energy bound up and unavailable. But if some lock-canceling antimatter does happen to show up, watch out!

Photons, the constantly-moving quanta of changing electromagnetic fields, come close to being the purest form of energy possible, with some quibbles I won't bring up here. Not surprisingly then, photons are the majority of what is released when matter and antimatter cancel each other's locks.

With that, I should emphasize again that this is not intended to be a mathematical or philosophical answer. All I'm trying to convey is that energy is all about change. It can be ongoing change, as when objects move in large unidirectional ways (kinetic) or microscopic multidirectional ways (heat), or it can be potential change. The latter is change that was captured and stashed away sometime in the past by stressing fields. The most extreme form of potential energy, one in which the release of the energy is safeguarded by profound conservation laws, is what we call matter.

• So it has to do something with entropy. A low entropy state can change to a high entropy state, while two "equally" high entropy states usually won't change to each other. Commented Jan 8, 2019 at 7:24
• "Energy is change"? I think one could probably make a stronger argument that energy is "what doesn't change", given that it is conserved. Commented Jun 7, 2019 at 0:23
• Photons usually aren't the majority of what is released when baryons and anti-baryons annihilate. Commented Aug 13, 2020 at 0:18

Energy is a convenient way to account for a system's ability to do useful work. There are certain modern qualifications we attach to energy, mostly that the total energy of a closed system is always conserved (barring cosmological effects), which is now explained by the use of symmetry and Noether's theorem (as explained by other's comments).

To try to get to a more satisfying everyday notion of energy, it is best to resort to the concept of useful work and the accounting of it. We understand that we exert effort (force over a distance) by lifting an object from the ground to the top of a table. For practical accounting, we need to understand how much effort was expended. It was these sorts of accounting problems that led early engineers to the concept of energy.

Energy is useful to us only if it has the ability to change its current form into another form. One way energy can change is to start with potential energy and convert it to kinetic energy. For example, consider the static energy stored in the bonds of carbon and hydrogen in a gallon of gas. That bonding energy can be released and converted into useful kinetic energy, such as causing the relative motion of a car. Energy can also change from kinetic energy into potential energy and do useful work. Consider a ball rolling on a table; it has kinetic energy. If the ball collides with a spring, compresses it, and a latch catches it, the ball will lose its kinetic energy, and the spring will gain potential energy.

If the energy is in a form that is useless to us, then we measure it in terms of entropy. In a closed system, there is a maximum possible value of entropy associated with it. If the entropy of a system is lower than its maximum entropy, then that system is "not at equilibrium" and still has usable energy internal to it. This means that work can be performed within the system to increase the multiplicity of states (which is commonly interpreted as disorder), by converting that potential energy into kinetic energy internal to itself.

In everyday terms, we only think about energy it terms of the useful work that can be derived from it. So when we talk of selling energy in an energy market, what is being traded is a commodity that can be used to do work. There are different ways that energy can be stored, but when we buy a certain amount of energy, we expect it to allow us to accomplish certain tasks in a predictable way.

This is a somewhat simplified discussion. There is a lot more that can be added and several clarifications are needed. I don't know your level of understanding, so I have abbreviated my explanation.

Energy is just a numerical quantity that never changes when nature changes its courses. It being an abstract idea can be illustrated by an analogy. And who can make you enjoy this other than Mr. Feynman. In his lectures, Feynman gave an extraordinary analogy to this:

Imagine a child Dennis who has blocks which are absolutely indestructible, and cannot be divided into pieces. Each is the same as the other. Let us suppose that he has 28 blocks. His mother puts him with his 28 blocks into a room at the beginning of the day. At the end of the day, being curious, she counts the blocks very carefully, and discovers a phenomenal law— no matter what he does with the blocks, there are always 28 remaining! This continues for a number of days, until one day there are only 27 blocks, but a little investigating shows that there is one under the rug—she must look everywhere to be sure that the number of blocks has not changed. One day, however, the number appears to change—there are only 26 blocks. Careful investigation indicates that the window was open, and upon looking outside, the other two blocks are found. Another day, careful count indicates that there are 30 blocks! This causes considerable consternation, until it is realized that Bruce came to visit, bringing his blocks with him, and he left a few at Dennis' house. After she has disposed of the extra blocks, she closes the window, does not let Bruce in, and then everything is going along all right, until one time she counts and finds only 25 blocks. However, there is a box in the room, a toy box, and the mother goes to open the toy box, but the boy says "No, do not open my toy box," and screams. Mother is not allowed to open the toy box. Being extremely curious, and somewhat ingenious, she invents a scheme! She knows that a block weighs three ounces, so she weighs the box at a time when she sees 28 blocks, and it weighs 16 ounces. The next time she wishes to check, she weighs the box again, subtracts sixteen ounces and divides by three. She discovers the following:

In the gradual increase in the complexity of her world, she finds a whole series of terms representing ways of calculating how many blocks are in places where she is not allowed to look. As a result, she finds a complex formula, a quantity which has to be computed, which always stays the same in her situation.

Owing to the above analogy, it is being abstracted that energy is such a manifestation of a number which has a large number of different forms but will never change except for going in and out...

Any concept in physics- energy, mass, entropy- is explicitly defined by the set of mathematical relationships for the concept. Any linguistic definition of a concept is an attempt to provide a physical understanding of the concept. For example, for force, one linguistic definition is "A description of an interaction that causes a change in an object's motion". For me, that definition is not useful unless I understand the mathematical definition defined as F = ma and how it is used in applications.

For a broad concept like energy, the linguistic definition must necessarily be rather vague, and to understand such a definition you need to understand the mathematical relationships for energy and their use in applications.

For a basic understanding of energy, I like the simple definition of energy stated in an old engineering thermodynamics textbook. "Energy is the capacity, either latent or apparent, to exert a force through a distance." Obert and Young, Elements of Thermodynamics and Heat Transfer. For a defined system in basic thermodynamics, we consider the internal energy, the energy in/out of the system due to work and/or heat, and the energy in/out of the system due to mass transfer. [The internal energy is sometimes called the "heat", but this is technically incorrect from a thermodynamics viewpoint. Heat is energy that crosses a system boundary- without mass transfer- solely due to a temperature difference. Work is energy that crosses a system boundary- without mass transfer- due to any intensive property difference other than temperature.]

Over time, the concept of energy has been extended to include rest mass energy, field energy, and so on to preserve the concept of the conservation of energy. So again the linguistic concept has to be very broad/vague to accommodate such considerations.

Hope this helps.

The following answer converges on the same ideas as in the answer by contributor maenju, submitted sept. 23, 2023.

The derivation that I present here is a reworked version of a PSE answer by me to a question about the work-energy theorem, submitted on july 17, 2021.

I will argue the following:

• the concept of energy arises naturally in classical mechanics
• the concept of energy that arises in classical mechanics carries over to all other fields of physics.

In classical mechanics the second most important expression, second only to $$F=ma$$, is the work-energy theorem. If $$F=ma$$ is granted as axiom then work-energy follows as theorem.

The starting point:

$$F = ma \tag{1}$$

The next step is to evaluate an integral: integrate both sides with respect to the position coordinate, integrating from starting point $$s_0$$ to final point $$s$$

$$\int_{s_0}^s F \ ds = \int_{s_0}^s ma \ ds \tag{2}$$

The work-energy theorem hinges on the fact that the two factors in the right hand side of (2), acceleration $$a$$ and position coordinate $$s$$, are not independent of each other; acceleration is the second derivative of position.

We proceed to develop the right hand side of (2).

In the steps starting with (5) the integrand $$a$$ is converted to velocity and the differential $$ds$$ is converted to $$dv$$, using the relations (3) and (4)

$$v = \frac{ds}{dt} \ \Leftrightarrow \ ds = v \ dt \tag{3}$$

$$a = \frac{dv}{dt} \ \Leftrightarrow \ dv = a \ dt \tag{4}$$

(5) corresponds to the right hand side of (2)

$$\int_{s_0}^s a \ ds \tag{5}$$ $$\int_{t_0}^t a \ v \ dt \tag{6}$$ $$\int_{t_0}^t v \ a \ dt \tag{7}$$ $$\int_{v_0}^v v \ dv \tag{8}$$

First (3) was used to change the differential from $$ds$$ to $$dt$$, with corresponding change of limits. Next (4) was used - with change of limits - to arrive at (8).

So we have the following mathematical relation:

$$\int_{s_0}^s a \ ds = \tfrac{1}{2}v^2 - \tfrac{1}{2}v_0^2 \tag{9}$$

Notice especially:
The definitions (3) and (4), in combined form, are sufficient to imply (9).
Stated differently: (9) expresses: whenever you have a quantity $$s$$, its first derivative, and its second derivative, an expression with the form of (9) is valid.

We use (9) to go from (2) to the work-energy theorem:

$$\int_{s_0}^s F \ ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2 \tag{10}$$

The concepts of potential energy and kinetic energy exist by virtue of the existence of the work-energy theorem, in the sense that when (10) does not obtain there is no reason to take an energy concept into consideration.

Mathematically the origin of the concept of energy is the act of evaluating an integral. That is why the expression for kinetic energy has that factor $$\tfrac{1}{2}$$, it stems from evaluation of an integral.

Multiple degrees of freedom

I take it as obvious that the above derivation generalizes to multiple degrees of freedom, be it cartesian coordinates or generalized coordinates.

Kinetic energy: proportional to square of velocity.
Pythagoras' theorem: the square of the resultant is the sum of the squares of the components.
Therefore: with multiple degrees of freedom it is sufficient to use an expression for the resultant kinetic energy. We have that either with one degree of freedom or with multiple degrees of freedom: it is sufficient to express the kinetic energy (at any instant) as a single scalar value.

In contrast with that: when there are multiple degrees of freedom it is necessary that the expressions for the potential cover all degrees of freedom. The force is recovered by evaluating the gradient of the potential.

We always know in which direction an acceleration will be: down the gradient of the potential. That is sufficient.

Sum of potential energy and kinetic energy

The outcome of an integration is inherently an incremental value. As we know: the outcome of an integration does not have an intrinsic zero point; for any potential energy we choose a zero point for it.

The left hand side of (10) is the expression for work done. We have that potential energy is defined as the negative of work done.

Hence:

$$- \Delta E_p = \Delta E_k \tag{11}$$

Which of course rearranges to:

$$\Delta E_k + \Delta E_p = 0 \tag{12}$$

This demonstrates:
In and of itself the work-energy theorem is already sufficient to imply that the sum of potential energy and kinetic energy is a constant.

Of course, that should not be blindly generalized to a blanket statement of conservation of any energy.

An example:

Inductance is a phenomenon where a form of kinetic energy is recognized that is distinct from mechanical kinetic energy.

Let there be a coil with significant self-inductance. For simplicity we make the setup superconducting.

We take the position of charge as our state, in such a way that the first derivative of that state is the electric current.

In the absence of an electromotive force the current will continue indefinitely. To change the current strength an electromotive force is required. A change in current strength results in an change of magnetic field that in turn produces an electromotive force that acts in opposition to the change of current strength.

Because of that opposition: the second derivative of the state is proportional to the applied electromotive force.

We have that the relation between second derivative of the state, electromotive force and inductance has the same form as $$F=ma$$, thus giving rise to a electromagnetic counterpart of the work-energy theorem.

As we know: the recurrent pattern is: in physics taking place energy is conserved.

That observation suggests that all interactions are of a nature such that the second derivative of the state is proportional to the gradient of the potential energy.

Well energy is only a part of something else that is much more important, and that is called "Action". There is a bumper sticker that reads "Physics is where the Action is". One of the most important quantities in the universe is Planck's constant, and it has the units of action. (Joule sec). The universe is designed in such as way that the no matter how things move about or change their structure, the action changes represents the efficiency of that change. Or to put it better, the probabilities of things happening or existing can be found out by accounting for this quantity called "Action". From the principle of least action we can derive conservation laws for quantities recognisable as energy, momentum, and angular momentum, it is a consequence of the symmetries involved in the action principle. (Lubos said it better).

Now we know that of these various kinds of conserved quantities, the energy one relates to the application of force (dear old Sir Isaac Newton figured that one out) and so the nice thing about energy is that it can be stored within structures by arranging a force to be stored. And so we have food and fuel. And chemistry. And evolution.

Chemists don't often use energy directly in calculations though, they also use a kind of minimisation principle that involves both energy and another useful quantity called entropy which is a measure of the amount of freedom of choice that we allow energy to have - this measure is called the "Free energy" and this is what allows you to calculate exactly what chemical reactions will occur and to what extent. And so it goes. This free energy is not conserved, the universe is winding down like a big clockwork spring.

The big bang (if you believe in that) is simply an earlier state when the energy density was very high. It doesn't necessarily mean that the universe was a single black hole of finite size. Quantum mechanics also tells us that there is a ground state to just about everything including spacetime, so if there is a vacuum there is a ground state energy. It is generally not worth trying to create a perpetual motion machine from the vacuum though, in spite of legendary pages of Youtube videos.

One thing that energy is not, is a kind of cosmic fluid. It is just a perspective on how change can occur - Einstein relativity theory teaches us that any kind of cosmic fluid including spiritual enlightenment fluids are impossible.

Cause effect means that there is something asymmetric that has happened. Asymmetry is closely associated with the idea of information, the problem of transferring information requires that there is a net displacement in space and time. The strictures on information transfer are the same as those on energy transfer, and we find that the movement of energy suddenly becomes about the movement of bits! So knowledge, energy and time and space must be considered in the same picture.

Firstly the connection betwen energy and time is very profound. We do not understand time fully but we know that our sense of real time requires there to be a meaningful succession of different states, take away clocks and time literally loses its meaning in such a context. For pure energy, every day is groundhog day - there is an intrinsic period associated with energy states but no sense of succession.

How does "real" time enter the picture? We know that there is an opportunity in spacetime for "timelike" intervals between events, in this zone a succession of events can be established that can maintain a cause-effect relationship in all reference frames. But that doesn't give us the clock itself. Systems also entangle when creating a time ordering, but this is all unclear.

The upshot is that just as there is a energy cost to doing things in the world, there is also a "cost" for systems to even exist in the world we know - aspects of that world must be unknowable. The converse is also true, if we encounter a system that is unstable and can be stabilised by releasing specific energy, then the specificity of that energy means that the time at which the event will occur is unknowable. We simply cannot consider a concept such as energy in isolation - without understanding the nature of concepts such as knowledge and time. Its a package deal.

Finally open systems through which energy flows are better able to maintain clocks and establish a time order, so life is a phenomenon associated with unstable energy flows.

The simplest explanation that I know as to why time runs in one direction, is that events in the reverse direction are "unobservable". I know that sounds like a tautology but if you show why they are unobservable then you have a better explanation. Likewise the positive observable energy could have in its negative counterpart a reason why it is unobservable, but now I really am not qualified to comment, I have already badly exceeded my limits.

ENERGY is a mathematical concept!!!

Many people often find the concept of energy quite confusing as they attempt to visualize or intuitively understand it. However, in reality, energy is indeed intuitive, but not in the way one might initially think. It's crucial to recognize that energy is not a tangible substance; instead, it is a conceptual and mathematical tool developed based on the fundamental principles of kinematics and kinetics, as laid out by Newton's Laws.

Sure, let's start with kinematics. We'll define ($$r$$) as the position vector, ($$v$$) as the velocity of the position vector, and ($$a$$) as the acceleration of the position vector. Now, let's express the kinematics equations in their differential form:

1. Position to Velocity (First Derivative):

$$\frac{\mathrm{d} r}{\mathrm{d} t} = v$$

1. Velocity to Acceleration (First Derivative):

$$\frac{\mathrm{d} v}{\mathrm{d} t} = a$$

Certainly, we can express these quantities as functions of each other without explicitly involving time by eliminating ($$\mathrm{d} t$$). Let's proceed:

$$\mathrm{d} t = \frac{\mathrm{d} r}{v}$$ $$\mathrm{d} t = \frac{\mathrm{d} v}{a}$$

Certainly, let's proceed by equating the two equations:

$$a \mathrm{d}r = v \mathrm{d}v$$

Now, let's integrate this expression to find as function of ($$r$$) and ($$v$$), then we have:

$$\int_{r_1}^{r_2}a \mathrm{d}r = \int_{v_1}^{v_2}v \mathrm{d}v$$

$$\int_{r_1}^{r_2}a \mathrm{d}r = \frac{1}{2}v_2^2-\frac{1}{2}v_1^2$$

Certainly, assuming a constant acceleration makes the integration process more intuitive. Let's go ahead and integrate the left side of the equation:

$$a\Delta r= \frac{1}{2}v_2^2-\frac{1}{2}v_1^2$$

Understanding this equation is very important. If we displace a distance $$\Delta r$$ with a constant acceleration $$a$$, then we should observe a variation in the quantity $$\frac{1}{2}v^2$$. This implies that with constant acceleration, if you cover a greater distance ($$\Delta r$$), you'll naturally achieve a higher velocity ($$v$$). This relationship is reasonably intuitive.

Certainly, to account for inertia and incorporate Newton's second law ($$F=ma$$), we can introduce mass ($$m$$) into the equation. Here's how we can do it:

$$\int_{r_1}^{r_2}F \mathrm{d}r = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2$$

And just like that, we arrive at this fascinating mathematical relationship. In principle, we could have assigned any names to these quantities, but for some compelling reasons, they've been given specific names in the realm of physics. The quantity ($$\frac{1}{2}mv^2$$) is referred to as kinetic energy, often denoted as ($$K$$), while the integral of ($$F \mathrm{d} r$$) is known as work, symbolized as ($$W$$).

$$W=K_2-K_1$$

Let's not lose sight of the essence of this relationship. It fundamentally answers the question of how much our velocity changes when we experience a specific acceleration over a given distance. When we bring mass ($$m$$) into the picture, it allows us to gauge how much our kinetic energy ($$K$$) changes when a force is applied over a distance, or what we call work done ($$W$$).

What is energy?
Energy is the capacity of a system to do work.

Where does it come from?
It generally comes from another source of energy, as in energy gets converted from one form to another.

Where does it ultimately come from?
That my friend is a question for MetaPhysics.stackexchange.com, which sadly doesn't exist as of now. You might want to hop over to Area51 with a proposal.

• -1 Energy is the capacity of a system to do work. What is Work? Change in Energy. Isn't this cyclical? Commented Jul 30, 2011 at 22:18
• @Bernhard en.wikipedia.org/wiki/Energy
– abel
Commented Aug 8, 2011 at 16:28
• Agree, that (ultimately) this is a metaphysical question.
– jim
Commented Mar 10, 2018 at 18:36
• ::sigh:: It is not a metaphysical question and even in the classical version the cycle is broken because there is a definition of work that does not depend on energy: $\mathrm{d}W = \vec{F} \cdot \mathrm{d}\vec{s}$. The fact that it requires deep insight to notice that this quantity is important doesn't change the fact that it does not depend on a definition of energy. You use this definition of work to bootstrap the work-is-energy-transfer-and-energy-is-ability-to-do-work structure from the underlying mechanics. And that's all before we introduce the Noetherian definition. Commented Jan 1, 2019 at 21:03