How can energy be useful when it is 'abstract'?

Question

The topic which haunted me for two years until I gave up on it. But now I am doing engineering and this topic suddenly popped out of my textbook from nowhere. I seriously need to understand this topic because I wasn't able to do so in the past. I have read many book on 'energy' and it came to nothing. Maybe just because the many books I have read all said 'energy' is something which doesn't 'exist'! It is 'abstract'! It is just a number which represents the state or orientation of a system. But then I see so many examples that 'uses' energy to do 'work'. But the question here is that if anything doesn't exist in this universe, then how can the same thing be used to do something that exist?

My problem is that I didn't understood the topic 'energy' and all the other topics related to it (work, power etc.)

Answer 1

By Noether's theorem, there is a conserved quantity (a number) associated with every continuous symmetry of a physical system. Energy is by definition the conserved quantity associated with time translation invariance (i.e. that it doesn't matter whether we perform an experiment today or tomorrow, given that all circumstances are the same). In this sense, it exists.

Now, work is the idea that we can distinguish (arbitrarily) certain "kinds" of energy - for example, to every conservative force $F$ we can associate a potential $U$ with $\vec F = -\vec\nabla U$. We call the value $U(x)$ the potential energy at $x$. A particle can - and will - move to regions with a lower potential along a path $\gamma : [a,b] \to \mathbb{R}^3$, doing the work $W[\gamma] = \int_\gamma F(x)\mathrm{d}x = U(\gamma(a)) - U(\gamma(b))$ in the process (and now having $W[\gamma]$ less potential energy), but since energy is conserved, there must be another form of energy - in this case, the kinetic energy $\frac{1}{2}mv^2$ of the particle, which will be by $W[\gamma]$ greater than before.

It works always like this - we note that one kind of energy is taken away or added, and since we know that there is a way to get a conserved quantity out of this, we look for other forms of energy.

From the Noetherian viewpoint though, there are no different kinds of energy - to get energy, we simply use Noether's theorem for the Lagrangian (which is $L = T - V$, where $T$ is the kinetic energy term for all particles and $V$ is the sum of all potentials present) of our system to obtain a formula for the conserved quantity associated to time translation, and that's it. As with all conserved quantities, energy is useful in finding solutions to the equation of motion or in simplifying our system, or in predicting the result of interaction without solving them in detail - the classic example here is an elastic collision, where conservation of energy and momentum (which is that which is conserved through space translations) is all you need to know to predict the velocities after the collision.

Answer 2

To add to Dirk Bruere's answer. As I describe in detail in this answer here, the accountant's idea of a budget is a good analogy: the budget seems "abstract" but the law of conservation of energy is experimentally proven and in this way energy is very "real": a system fulfilling conservation of energy behaves in a way that is measurably very different from what would or could happen if energy were not conserved. Tt has been found in countless experiments over roughly two hundred years that systems behave as though they have a certain "budget" of work that they can do; it doesn't matter how you spend that budget, but if you tally up the work that can be done by the system in the right way (i.e. as $\int_\Gamma \vec{F}\cdot{\rm d}\vec{s}$, or $\int_0^T V(t)I(t){\rm d}t$ in an electrical circuit and so on), then the amount of work that can be done will always be the same.

You can also get "abstract" and foretell the existence of conserved quantities through Noether's Theorem: if there is a time shift symmetry, that is roughly, if a system's physics does not depend on where we put the time origin $t=0$, then there must be a conserved quantity. This is the quantity we call energy. It accompanies three other conserved quantites, one for each spatial shift symmetry - the notion that physics does not depend on where we put our co-ordinate axes - we can slide our origin around arbitrarily. These other three are the components of the linear momentum, and they are united with the energy in the momentum four-vector. This four dimensional object is conserved as a result of physics's invariance with respect to shifts in space and time, and its components are transformed with our co-ordinate systems by the Lorentz transformations. Special relativity brings a new reality to energy. If you confine "energy" in a box, such as in my thought experiment here, you raise its inertial mass. You now have to shove it harder than before to make it undergo a given acceleration.

General relativity brings yet another new "reality" to the notion of energy as a precise measure to how much "stuff" there is and where it is. The momentum four-vector is generalised into the so called stress energy tensor. This is the "source" term in the Einstein field equations. It therefore tells "spacetime how to bend" in J. A. Wheeler's famous one sentence summary of general relativity "Spacetime tells matter how to move; matter tells spacetime how to curve." So a generalised notion of energy directly influences the geometry of spacetime: in my mind you can't get more "real" than something which tells you precisely how much the geometry describing your reality deviates from the Euclid parallel postulate, which is pretty much what the "spacetime bending" half of the Einstein equations tells us. It's worth noting that in a general solution (such as an expanding Universe described by the so called FLRW metric) to the Einstein Field equations, global energy conservation - our exalted idea of a budget - no longer holds true. But a local version still holds: a generalised divergence of the stress energy tensor vanishes and this means that energy and momentum conservation must hold for a small enough region of spacetime. It is still physically impossible to "delete the Earth" or make a statue of Beethoven suddenly appear on to of your piano. These fanciful deeds would violate local energy conservation.

Answer 3

But the question here is that if anything doesn't exist in this universe, then how can the same thing be used to do something that exist?

Most people have no trouble understanding numbers, but they don't exist in the universe - we define them. Numbers are the first abstraction most people learn, and I'd be very surprised if you couldn't abstract a group of oranges (that exist) to a number. Learning abstractions is hard, but good abstractions make things simple once you understand them.

Energy is the abstraction common to all of the physical sciences. Energy is a scalar quantity that relates chemical reactions, optics, mechanics, materials, electronics and every other physical discipline. Mathematically, energy conservation results from the laws of physics being constant over time. The theorem that tells us that is even more abstract, so I apologize if that doesn't make sense yet. As an abstraction, energy has units of $m * l^2 / t^2 $ where $m$ is mass, $l$ is length and $t$ is time.

It is just a number which represents the state or orientation of a system. But then I see so many examples that 'uses' energy to do 'work'.

Energy is the most useful scalar in the universe. When we say that energy does work, we just mean addition and subtraction. Suppose I have 2 systems that interact. If system 1 does work on system 2, I add to system 2 the same amount of energy I subtract from system 1.

I have read many book on 'energy' and it came to nothing

Energy has meanings outside of physics that are related at best and profoundly contradictory at worst. I recommend reading the wikipedia pages on Kinetic energy and Work(physics). As a rule of thumb, if they aren't using algebra, calculus and numbers, be skeptical. By the way, I didn't use any algebra, calculus or numbers, so I strongly recommend that you keep your healthy skepticism and read some more sources that do.

Answer 4

It is Nature's bookkeeping to balance the accounts. The same with all conserved quantities.

Answer 5

The absolute quantity of energy is of little importance. What is useful knowledge to engineers and scientists is the change of energy, $\Delta{E}$.

Answer 6

Energy is conserved, and then you can treat it like a budget and say it is accounting. But then you just ask what is conserved and what is the budget making an accounting of and it gets rather circular.

So let's start with the basics. You have configurations. These are possible ways thing can be, for instance having two objects and one object here and another object there. If you had more than two objects that's a different configuration. If you had less than two objects that's a different configuration. If at least one was in a different location that's a different configuration. So now we have configurations and they are a basic thing.

Next we have time. This allows us to talk about having one configuration at one time and talk about having a different configuration at another time. This is essential for making predictions such as "this configuration can lead to that configuration" or "this configuration can not lead to that configuration" (and that latter is better because it can disproven).

Now it is possible to have a universe where knowing the configuration at one time means you know the configuration at later times.

We do not live in such a universe. This is empirically noticed because sometime we had the same configurations and yet things developed differently. For instance sometimes some pool balls are positioned a certain way and they stay there and don't move other times they might be in the same positions but one of them is moving and then the configuration changes (first it changes to a configuration where that moving one gets to a new location and then later as they bump together other ones move too).

This is a little bit of semantics because someone might say we should have included those initial velocities in our configuration, but I want to connect configurations and time since the whole point is to predict configurations over time so we should be explicit about how these are related.

So we have configurations, we have time, we have configurations at various times, and we have the rate at which configurations change.

Next we need a way to find out how configurations change over time. We empirically know that it depends on the initial configuration. And that it depends on how the configuration is initially changing in time. Does it depend on something else too?

That can only be answered empirically. But we do notice that for many system if you set them up with the same initial configuration and the same rate the configuration is initially changing in time that they evolve the same over time. And for the ones that don't we notice that it seems to be influenced by something else. So we say the something else should be part of the configuration (it affects the dynamics after all). There are two ways to have it effect the dynamics.

Let's look at the tides. You can just say that there something that makes the tides go up at some place and some times and down in some places and some times. If you did that, then energy would not exist and would not be conserved and things would seem a bit weird.

But let's look at what it means to have conserved quantities. It means that you can associate numbers with different combinations of configurations and rates if change of configurations. Then whatever number is associated with them initially will have to be associated with the entire evolution over time. So you start on some surface in some mathematical space and you are stuck on it, you can repeat but you can never leave the surface. So we can predict that you will never have a different value of that conserved quantity, this is falsifiable hence fantastic.

OK. But now look at the tides again. They seem to be falsifiable, we can see regularities. Why can't we make predictions that certain outrageously different configurations will never occur? We can. We can do so by noticing that the regions with highest tides tend to point towards the moon and so this externally put in variation in time can be replaced by including the location of the moon as part of the configuration.

Now we get a system where the things that tell us how the system evolves only depends on the configuration and the rate of change of the configuration there is no longer a part that depends on the time.

This leads to a conserved quantity, hence a surface that the total configuration stays on, hence the awesome ability to make a falsifiable prediction. This conserved quantity associated with thibgs not depending on the time parameter and only depending on the configuration and rate of change of the configuration is the thing we call energy.

It took a long time for people to accept it as real and that is partly because you have to include enough things for it to be real.

But that trust turns out to be well placed. Sometimes we look at a situation and it doesn't look like energy in conserved. But we can say now "hey that happens when we don't include all the stuff I think there is some new stuff" and every time that happens we act like detectives and search for missing stuff, and we always find it.

So now we expect energy to be conserved and can use it to learn new things, things that can be hard to notice.

But that isn't why it is trusted so much. Around 1900 we learned that time isn't a thing everyone agrees on. Those GPS satellites have clocks that tick differently than identically manufactured clocks tick down here on earth. And clocks tick differently at different heights and when moving at different speeds. These effects are small so you need accurate clocks but the effect is real. And it isn't just clocks, atoms vibrate differently, water falls differently, chemical reactions happen at different rates. Everything slows down together, which means a person standing next to a clock doesn't notice when the clock runs slower and the person thinks and ages and digests slower by the same factor.

In fact we predict that people deep in space think we are thinking, living, dying, and eating in slow motion and that our clocks tick in slow motion too (not much slower, but slower nonetheless).

So if time isn't a thing everyone agrees on then it doesn't make sense for your physics to depend on it. It can depend on clocks reading because that is a configuration of the parts that make the clock, and then whatever affects that can also affects the clocks too.

So it all works fine if we just say that everything is determined by the configurations and not by some mysterious parameter called time that we can't really measure anyway.

And it was after 1900 that we started to trust that things don't depend on time and then we learned that means that energy is conserved and by trusting in energy conservation we found evidence of whole new particles and then discovered everything about the whole new particles.

That's why there is trust in energy conservation. We learned not to trust clocks blindly, we learned not to trust time blindly, but we learned that energy conservation can be more trustworthy than any of the things from our everyday experience that seemed so trustworthy.

So it is partly that every other idea forsook as, abandoned us and tricked us, but energy conservation helped us.

Answer 7

I will try to give you a different perspective. Hopefully that will be of some help to you.

There is nothing abstract about an atomic bomb, the light produced by a light bulb, or your ability to move your arms. These examples are tangible proof that energy is real (it exists). The abstraction is in the minds of people, so that they can model its properties and behavior. If energy were not real, the universe would not exist!