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Energy is the capacity to do work.

It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and we add it all together it gives “28” - always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

It's not that hard to define, it's basically force times distance, or anything that's convertible with that.

To further clarify matters, I really should give you a precise definition of energy. Unfortunately, I can't do this. Energy is the most fundamental dynamical concept in all of physics, and for this reason, I can't tell you what energy it is in terms of something more fundamental. I can however, list the various forms of energy - kinetic, electrostati, gravitational, chemical, nuclear - and add the statement that, while energy can often be converted from one form to another, the total amount of energy in the universe never changes.

A little calculus would show it.

Please show me a case where my definition of energy (i.e.the capacity to do work) is incorrect.

Abstract - The common definition is shown to be false. A modern definition must be based on the first and second laws of thermodynamics and in terms of a set of algebraic expressions written in such a way that their sum does not change when a system is isolated. (DF)

I could look up the definition of "Mammal"; it wouldn't include an expression of how to calculate anything.Definitions don't need to do that.

Work is done when a force F moves through a distance D. E=FdIf you add F=MAthen you can calculate the KE of a body by calculating the work done by bringing it to a halt.

Please answer my question, when does my definition fail?

The concept of energy is abstract and therefore not as easy to define as the concepts of mass and volume.

One definition of energy, as what Bored chemist said, is the capacity to do work.

Gravitational potential energy is negative.

Not unless you've found a source of Cavorite and are keeping it under your hat.

There's no such thing as a uniform gravitational field. It's often a useful approximation, but the actual potentials have divergence.

Why is it then that energy being the ability, or capacity to do work is recognised as a worldwide definition of energy?Since if it is a useless definition, why do most science textbooks include that as the basic definition?

No, in the real world you can't create perfectly uniform gravitational potentials extending over a finite space because of the way that matter consists of particles, and no number of particles arranged in any way can create a perfectly uniform potential.

QuoteI could look up the definition of "Mammal"; it wouldn't include an expression of how to calculate anything.Definitions don't need to do that.On the contrary. A definition is exactly what tells you what the value of a quantity is.QuoteWork is done when a force F moves through a distance D. E=FdIf you add F=MAthen you can calculate the KE of a body by calculating the work done by bringing it to a halt.You can't do that if you have not yet defined energy. In this case the energy is energy of motion. What you've done is to take another quantity, the value of the work done, and called it "kinetic energy". If all we've done so far is to define "energy" then its premature to define "kinetic energy". One can easily define the term "kinetic energy" and calculate it as th work done in bringing a body of mass m from speed 0 to speed v. However that does not tell you what energy is. I.e. we have no idea yet of whether energy = kinetic energy.If I were to define the term "energy" then I'd say that it's the sum of all the forms on energy such that the sum is an integral of motion (i.e. a constant of motion) for a closed system. Then we'd have to take care to find all those forms and then prove its such an integral. The problem is then reduced to finding all the forms of energy for a system. But to do that in general is not obvious and there is no general way to find all such quantities. I believe that's why Feynman said what he did.QuotePlease answer my question, when does my definition fail?I already did.

Could someone else please tell me what Pmb claims to have already explained.When is energy not the capacity to do work?

There is a confusion here between two things. firstly and simply the energy that we can make use of in physical machines which is easily defined as above and secondly the fundamental source of all energy in the universe. This second item is the really difficult bit.

Quote from: Bored chemist on 07/07/2010 06:57:15Could someone else please tell me what Pmb claims to have already explained.When is energy not the capacity to do work?I'm not sure this is what he was getting at, but to take the zero-point energy of a quantum system as an example. You can't extract that energy as work, but it exists and has physical meaning.

In QM the uncertainty relationship means that energy isn't strictly conserved. Perhaps it's better not to include that in a definition. Anyway conservation is a property of energy; not a definition.

Quote from: Bored chemist on 10/07/2010 18:48:45In QM the uncertainty relationship means that energy isn't strictly conserved. Perhaps it's better not to include that in a definition. Anyway conservation is a property of energy; not a definition.That is incorrect. For systems which have a time indepentant Hamiltonian the energy is conserved. If you're referring to the time-energy uncertainty relation, that's the incorrect interpretation.

Both of you are now talking about quantum mechanics. We were discussing quantum mechanics. If you wish to discuss this from a quantum mechanical point of view then "ability to do work" is meaningless since work cannot be defined in quantum mechanics.I mentioned this above but I think it was missed/ignored(?) Some physicists define energy as the generator of time translations in Hamiltonian mechanics.

And I was not thinking of a particle in a box. I had more of a mountain in mind. E.e. picuture a mountain which has a flat top and virtical sides. Then the potential in one dimension is like the one I gave.

Both of you are now talking about quantum mechanics.

It can do work in that case, but it can't spontaneously do work because it's not rolling off the edge already.

You are still wrong about the possibility of the system you talked about; it would violate the uncertainty principle.Do you accept that?

Well, you have to take the average or expectation value, ...since any single measurement could violate classical energy conservation...

Quote from: JP on 15/07/2010 01:44:24It can do work in that case, but it can't spontaneously do work because it's not rolling off the edge already.All that means is you have to move it which requires changing its velocity and to do that you have to do work on it. So what you're saying is that it will do work if you first do work on it.