What is Energy? A Clear Definition and Explanation

[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?

 

[EnricoHendro]

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".

Ah, i see, thank you for pointing this. I need to revise my notes about this :D

 

[EnricoHendro]

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.

I see. So, for complicated substances, the heat (in terms of work) would affect not only the kinetic energy inside the system, but also could affect those degrees of freedom?

 

[EnricoHendro]

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.

I see, so there is thermal energy which is associated with heat. Just to be sure, is thermal energy the same as internal energy (which arises due to nonconservative force such as friction)? Or they are completely different?

 

[Dale]

I see. So, for complicated substances, the heat (in terms of work) would affect not only the kinetic energy inside the system, but also could affect those degrees of freedom?

Yes, generally all of them that can hold sufficient energy will be occupied

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?

I am sure it is covered in most textbooks that deal with the Lagrangian approach. It is quite straightforward.

If the Lagrangian has some differential symmetry then there will always be a conserved quantity associated with it. Symmetry under spatial translations leads to conservation of momentum. Symmetry under rotations leads to conservation of angular momentum. Symmetry under gauge transformations leads to conservation of charge. Etc.

 

[Shane Kennedy]

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

uniformly distributed, in what ? You need matter in order to have heat. You do need to transform that energy into a different form in order for it to do work, but it is the same energy. If you place one side of a Peltier device in contact with the original matter, and the other side in contact with some lower-heat-energy matter, some of the higher energy will convert into electricity, some of it will be transferred to the cold side.

 

[A.T.]

uniformly distributed, in what ?

In the Universe:
https://en.wikipedia.org/wiki/Heat_death_of_the_universe

 

[FactChecker]

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

It certainly does have the capacity to do work. It just needs to be in the right situation, which is next to something colder. Your statement is similar to saying that the air at the center of a tank full of compressed air does not have the capacity to do work because it is surrounded by air at the same pressure. That is wrong.

 

[A.T.]

It certainly does have the capacity to do work. It just needs to be in the right situation, ...

Is it sensible to define the general quantity energy via the quantity work, if it also relies on having "the right situation"? I think the definition via work applies to thermodynamic free energy, but not to energy in general:
https://en.wikipedia.org/wiki/Thermodynamic_free_energy

 

[FactChecker]

Is it sensible to define the general quantity energy via the quantity work, if it also relies on having "the right situation"?

Yes. It is your scenario of "Uniformly distributed heat energy" that requires a very specific and contrived (impossible?) situation to propose that an object with heat does not have the potential to do work. In any other, much more common situation, it clearly has the potential to do work.

I contend that a compressed spring does not lose its energy simply because it is not in a situation where it can not expand. Also, the compressed air in the center of a bottle of compressed air does not lose its energy simply because it is surrounded by air at the same pressure. A heated object does not lose its energy simply because it is in an area of "uniformly distributed heat energy".

The reason that I am defending the definition of "potential to do work" is that I think it has some great advantages. Energy appears in so many forms and can so easily change from one form to another (potential gravitational, kinetic, heat, compressed spring, etc.) that I have always admired that "definition", vague as it is.