What is Energy? A Clear Definition and Explanation

[EnricoHendro]

TL;DR Summary

I have heard the word energy countless times but still haven't read or heard about what it actually is. Can someone explain it to me?

Hello there,

I have heard countless times about the word energy, but I still don't know what exactly an energy is. Like for example, we know the definition of a physical quantity called velocity which by definition is change in displacement (so we can easily describe what a velocity is). We can also easily describe what an apple is. But in case for energy, I have yet to find the exact definition of energy (other than the mathematical equation of, for example kinetic energy and potential energy). We know that work, electromagnetic radiation, electric transmission, heat, matter transfer are simply means to transfer energy to a system from the environment. But what is this energy? I mean what is this thing that work transfer? (like if we say a pass a basketball from our hands to other people, we know that it means we are transferring a basketball into other people, and we can define the "object" that we are transferring). I've read a thread in this forum asking this question, but I still haven't read a clear definite answer describing what an energy is.

 

[Dale]

I still haven't read a clear definite answer describing what an energy is.

I think the most fundamental definition is based on Noether’s theorem:

Energy is the conserved quantity associated with the time-translation symmetry of the Lagrangian.

That is probably the most clear and definite answer available, but unfortunately it takes a lot of background knowledge to appreciate and understand it. So the easiest “entry level” definition is:

Energy is the capacity to do work.

 

[EnricoHendro]

I think the most fundamental definition is based on Noether’s theorem:

Energy is the conserved quantity associated with the time-translation symmetry of the Lagrangian.

That is probably the most clear and definite answer available, but unfortunately it takes a lot of background knowledge to appreciate and understand it. So the easiest “entry level” definition is:

Energy is the capacity to do work.

ah, I see. So there is no "object" like definition of energy other than "capacity to do work" for entry level definition and "conserved quantity associated with the time-translation symmetry" defined by Noether?

 

[russ_watters]

But in case for energy, I have yet to find the exact definition of energy (other than the mathematical equation of, for example kinetic energy and potential energy).

ah, I see. So there is no "object" like definition of energy other than "capacity to do work" for entry level definition and "conserved quantity associated with the time-translation symmetry" defined by Noether?

Right; energy is not a physical thing it is a mathematical thing. The mathematical thing is all there is. It's just a mathematical thing that is found to be conserved in some situations, which sometimes makes it useful to pay attention to/calculate. This is a bit like asking: what color is fuel economy? What does entropy taste like? How much does temperature weigh?

 

[Dale]

So there is no "object" like definition of energy

Well, I am not sure what an “object like definition” is, but energy is not an object. It is a property of objects, one that is particularly useful primarily because it is conserved.

 

[EnricoHendro]

Well, I am not sure what an “object like definition” is, but energy is not an object. It is a property of objects, one that is particularly useful primarily because it is conserved.

Object like definition means like defining things like we define an object, like the definition of apple, or a definition of orange. i kind of invented that term to articulate my thoughts on what I want to understand. But thank you for the answer, I kind of understand the definition now. See I was having a conversation with my friend, he is a priest, and I raised an argument stating that we are nothing but a collection of atoms brought to life by energy. That’s when I am starting to wonder about what energy is, that’s why I asked this. But thanks to your

 

[A.T.]

I have heard countless times about the word energy, but I still don't know what exactly an energy is. Like for example, we know the definition of a physical quantity called velocity which by definition is change in displacement (so we can easily describe what a velocity is).

You have formulas for energy, just like for velocity. But since energy is a more generally applicable quantity, you have different formulas for different types of energy.

See also: https://www.feynmanlectures.caltech.edu/I_04.html

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 anything 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.

 

[A.T.]

Energy is the capacity to do work.

I don't like this one. Uniformly distributed heat energy doesn't have capacity to do work.

 

[DrClaude]

See also: https://www.feynmanlectures.caltech.edu/I_04.html

I really recommend reading the beginning of that.

 

[Dale]

I don't like this one. Uniformly distributed heat energy doesn't have capacity to do work.

Thermal energy has the capacity to do work. You just have to use a 0 K cold reservoir and a heat engine.

The capacity to do work is there, even if you cannot use it due to other circumstances. Your car’s gasoline likewise does not lose its energy when you misplace your keys.

I agree that it is not the greatest definition, but it isn’t wrong per se, and counter examples are not so easy. The problem to me is that “capacity” is too fuzzy. Once you know that you need a 0 K reservoir and a heat engine it is easy to see that thermal energy has the capacity to do work, but how are you supposed to know that from first principles?

The definition works, but requires a lot of case by case knowledge to apply.

 

[A.T.]

Thermal energy has the capacity to do work. ... The problem to me is that “capacity” is too fuzzy.

Yeah, for some definitions of "capacity" it has capacity to do work.

But more to the point : I don't think definitions are very helpful at the "entry level", even if formally correct and neat sounding. If the OP is not happy with the definitions, then maybe the OP is not really looking for a definition. I think the way Feynman introduces it, is a much better starting point.

 

[Dale]

I don't think definitions are very helpful at the "entry level", even if formally correct and neat sounding.

I have no disagreement with that. That is why I led with the Noether’s theorem definition.

 

[anorlunda]

Object like definition means like defining things like we define an object, like the definition of apple, or a definition of orange.

The key is where @Dale said that it is a property. Apples and oranges have the property of color, but color is not an object. Apples and oranges also have energy. Color and energy are not objects.

Apples and oranges also have heat energy. They can be warm or cold, but temperature is not an object.

But don't take the analogy too far. Energy and color are not the same thing.

 

[jbriggs444]

Object like definition means like defining things like we define an object, like the definition of apple, or a definition of orange. i kind of invented that term to articulate my thoughts on what I want to understand. But thank you for the answer, I kind of understand the definition now. See I was having a conversation with my friend, he is a priest, and I raised an argument stating that we are nothing but a collection of atoms brought to life by energy. That’s when I am starting to wonder about what energy is, that’s why I asked this. But thanks to your

You seem happy with the ideas of position and velocity. Are those "object-like definitions"?

 

[Dale]

You seem happy with the ideas of position and velocity. Are those "object-like definitions"?

I don’t think that @EnricoHendro (please correct me if I misjudge the issue) expects all definitions to be object like, but based on pop-sci discussions of energy he expected energy specifically to be object like. I.e. he expected that it was an object of some sort rather than a property of objects.

 

[Nugatory]

he expected that it was an object of some sort rather than a property of objects.

A quibble in this context, but it might be better to say that energy is a property of systems rather than of objects. That avoids the problems arising from trying to localize potential energy in multi-body systems.

 

[Dale]

A quibble in this context, but it might be better to say that energy is a property of systems rather than of objects. That avoids the problems arising from trying to localize potential energy in multi-body systems.

Excellent quibble! I agree completely

 

[EnricoHendro]

You seem happy with the ideas of position and velocity. Are those "object-like definitions"?

Yes, those are what I would call an object-like definition, because you see, we can describe it quite easily. Like we can have “simple” words to describe them. Like “what is velocity?” “It is simply a change in the object’s displacement over time”.

 

[EnricoHendro]

I don’t think that @EnricoHendro (please correct me if I misjudge the issue) expects all definitions to be object like, but based on pop-sci discussions of energy he expected energy specifically to be object like. I.e. he expected that it was an object of some sort rather than a property of objects.

um, yes, I expected energy specifically to be object like, so I was quite puzzled when I couldn’t find the “object like definition” of energy. I have no problem understanding the mathematical concept though, but I always prefer object like definitions. Because when people who have no knowledge in physics ask me what a concept is (in this case energy) I always look for a way to describe it to them without resorting to its mathematical definition. I still need to study more physics and math :D

 

[EnricoHendro]

The key is where @Dale said that it is a property. Apples and oranges have the property of color, but color is not an object. Apples and oranges also have energy. Color and energy are not objects.

Apples and oranges also have heat energy. They can be warm or cold, but temperature is not an object.

But don't take the analogy too far. Energy and color are not the same thing.

Ah I see...in a completely unrelated topic, talking about the “heat energy” part. In my textbook, it said that heat is not energy, it is simply a means of transferring energy. It also said when we expose a system (say a spoon) to a heat source (say a cup of hot coffee), the handle of the spoon gets heated because of the fast moving electrons/atoms inside the submerged part of the spoon bump into the relatively slow moving electrons/atoms of the handle, which causes the slow moving atoms in the handle to move faster, thus heating the handle. Is it correct if I say that the energy that the heat is transferring is simply the kinetic energy of the atoms inside the spoon?

 

[Dale]

Because when people who have no knowledge in physics ask me what a concept is (in this case energy) I always look for a way to describe it to them without resorting to its mathematical definition.

Yes, unfortunately some concepts are difficult to convey.

 

[EnricoHendro]

You have formulas for energy, just like for velocity. But since energy is a more generally applicable quantity, you have different formulas for different types of energy.

See also: https://www.feynmanlectures.caltech.edu/I_04.html

Ah, I see. Thank you for the quote man :D

 

[Dale]

Is it correct if I say that the energy that the heat is transferring is simply the kinetic energy of the atoms inside the spoon?

That is only true for a ideal gas. For more complicated substances there are many other internal degrees of freedom that can be energized. There are atomic and molecular orbitals, rotational modes, stretching, bending modes, torsion, and many longer range interactions.

 

[jbriggs444]

talking about the “heat energy” part. In my textbook, it said that heat is not energy, it is simply a means of transferring energy.

If one is a stickler for terminology, "heat" is to thermal energy what "work" is to mechanical energy. It is a transfer of thermal energy across an interface. Nonetheless, one will find people speaking loosely of "heat energy" as a synonym for "thermal energy".

It is not a mechanism of transfer (like conduction, radiation or convection). It is the amount transferred.

 

[jbriggs444]

Yes, those are what I would call an object-like definition, because you see, we can describe it quite easily. Like we can have “simple” words to describe them. Like “what is velocity?” “It is simply a change in the object’s displacement over time”.

Like @Dale, this is not what I would have expected a definition of "object-like" to entail. It seems that it is just a synonym for "simple".

If you had expected energy to be some sort of fluid-like substance that you could squeeze out of an object to measure how much it has then yes, that would be a wrong idea. It is not simple like that.

You might want to revisit the notions of position and velocity. Position is not an attribute of an object. It is an attribute of an object in the context of a reference system. You cannot directly measure position. You can only measure distances (or signal reception times, parallax, apparent magnitude, etc) and infer position relative to something else.

Similarly, velocity is not an attribute of an object. Since it is the first derivative of position with respect to time, velocity depends on the reference system. You cannot directly measure velocity. You can only measure velocity relative to something else.

Position, velocity and energy all depend on one's choice of reference system. Unlike attributes such as mass which do not depend on such a choice. Attributes that do not depend on the choice of reference system are called "invariant". Attributes that do depend on such a choice are called "relative".

 

[Dale]

If one is a stickler for terminology, "heat" is to thermal energy what "work" is to mechanical energy.

Well said. Stated like that it makes the first law of thermodynamics obvious.

 

[A.T.]

Like we can have “simple” words to describe them. Like “what is velocity?” “It is simply a change in the object’s displacement over time”.

I don't think starting a definition with "It is simply ..." makes it any simpler.

Because when people who have no knowledge in physics ask me what a concept is (in this case energy) I always look for a way to describe it to them without resorting to its mathematical definition.

Yes, as I suggested before: Definitions are not really useful to explain something to laymen. The description by Feynman is as best as it gets for an introduction.

 

[EnricoHendro]

I don't think starting a definition with "It is simply ..." makes it any simpler.
Ah, yes, you are right, my bad

Yes, as I suggested before: Definitions are not really useful to explain something to laymen. The description by Feynman is as best as it gets for an introduction.

Yeah, feynman’s description is easier to grasp for an introduction

 

[Infrared]

Energy is the conserved quantity associated with the time-translation symmetry of the Lagrangian.

I'm basically a layperson in physics, but can you say how you specify what your Lagrangian is without referencing energy? Or if the Lagrangian is your primitive notion, this seems a bit unsatisfying since once could equivalently ask what energy means in Hamiltonian dynamics, and then the Hamiltonian is just a primitive notion.

 

[Dale]

can you say how you specify what your Lagrangian is without referencing energy?

The Lagrangian is given by whatever formula matches observations. The Lagrangian is found through experiment.

 

[Gordianus]

If I guessed the scope of the OP correctly, I really prefer Feynman's approach.

 

[Infrared]

The Lagrangian is given by whatever formula matches observations. The Lagrangian is found through experiment.

Could one also answer "The Hamiltonian is given by whatever formula matches observations. The Hamiltonian is found through experiment"? And then just define energy to be whatever formula is found (and not reference Noether's theorem or anything)?

I understand that in some circumstances, the Hamiltonian is different from total energy, but I think those are the circumstances in which you wouldn't be able to define energy as Noether's theorem applied to time-translations anyway (e.g. a time-dependent Lagrangian), since (I think?) applying Noether like this is the same thing as taking the Legendre transform.

 

[Nugatory]

If I guessed the scope of the OP correctly, I really prefer Feynman's approach.

One is a definition, the other is an explanation. They're both needed.

The definition in terms of Noether's theorem becomes increasingly useful as we move further away from problems that play well with our intuiton, and with familiarity it becomes easier to appreciate its elegance... but in the contex of the original question and a B-level thread Feynman is likely more satisfying.

 

[Dale]

I understand that in some circumstances, the Hamiltonian is different from total energy, but I think those are the circumstances in which you wouldn't be able to define energy as Noether's theorem applied to time-translations anyway (e.g. a time-dependent Lagrangian), since (I think?) applying Noether like this is the same thing as taking the Legendre transform.

That is exactly the issue. When the Hamiltonian is not the energy how do you know it is not the energy?

In those cases using Noether’s theorem shows us that there is no conserved energy. That let's us know when the Hamiltonian is the energy and when it is not.

 

[EnricoHendro]

That is exactly the issue. When the Hamiltonian is not the energy how do you know it is not the energy?

In those cases using Noether’s theorem shows us that there is no conserved energy. That let's us know when the Hamiltonian is the energy and when it is not.

About the Noether’s theorem, will a standard textbook cover the details about the theorem? Or do I need to look elsewhere if I want to know more about it?