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Offline Pmb

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What is energy?
« on: 06/05/2013 20:32:18 »
I'd like to discuss the concept of energy.  Energy, like several concepts in physics (like mass, time and space) is very hard, if not impossible, to define. H.A. Kramers put it this way. From a statment at the Princeton Bicentennial Conference on the Future of Nuclear Energy, 1946, in K.K. Darrow, edd., Physical Science and Human Values (Princeton: Princeton University Press, 1947), page 196.
Quote
My own pet notion is that in the world of human thought generally, and in physical science particularly, the most fruitful concepts are those to which it is impossible to attach a well-defined meaning.
In my experience I have found that to be quite true.

Before I start please consider the arguments that I've presented to date. They are on my web site at
http://home.comcast.net/~peter.m.brown/mech/what_is_energy.htm

My viewpoint is about the same as Richard Feynman and A.P. French. Feynman wrote the following in his Lectures
Quote
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.
A.P. French said something quite similar in his text Newtonian Physics, MIT Introductory Course in Physics.

Please note one important point that both Feynman and French make, i.e.
Quote
However, there are formulas for calculating some numerical quantity, and we add it all together it gives “28” -  always the same number.
The underline is mine. So while we do know what some forms of various types of energy we do not know what energy itself is. At least not to date.

Many times throughout history we make  calculations and find out that the amount of energy that we've calculated was not conserved and conservation of energy, if and when it's defined, will be one of the defining requirements of it. In each case so far (besides gravity in GR) we've been able to define a quantity made up of other things in the system such that when taken into account lead to a conserved quantity. EM energy is one example.

Recently I got into a debate about the definition of energy and of course they disagreed with Feynman and French (I guess they think that they're smarter than Feynman and French, but not I) claiming that my argument is wrong because energy is a defined quantity. Their argument is that we know the forms of energy and the sum we call total energy and therefore its defined. The problem with that sort of definition is that its failed in the past and will likely fail in the future. The reason we know the form of EM energy is because we demanded that energy is a conserved quantity. But what is energy that it is conserved? How do we define EM energy so that when added to the rest there will be a conserved quantity? Perhaps there's a definition out there we haven't figured out and EM energy is really different by a constant factor, let's call it k, than we now define it. I believe that will still lead to a conserved quantity but now we have a different value.

The suggestion made was that energy is that which is defined through Lagrangian mechanics (there's a formula for it). I'll use the term Jacobi's integral to refer to it and give it the letter h.

Note: When h is expressed in terms of generalized coordinates and canonical momentum it's called the Hamiltonian

This is one of it's names. You can find this in Classical Mechanics - Third Edition by Goldstein, Safko and Poole. Another name for it is the energy function. I'm not using that term for a good raason which I'll now describe. Goldstein et al describe the conditions under which Jacobi's integral is constant (also known as an integral of motion). In those cases h is not the energy. So the attempt to define it as energy leads to an error in general. Also there are forms of energy and situations which can't be described by h. For example; you can't use it to show that energy is constant in particle interations and decays.

It's for the reaons I mention above that there is a law of conservation of energy. If we knew how to define it and then were able to prove it was constant then we'd call it the energy conservation theorem, not the law of energy conservation.

If you have Feynman and/or French's text then I recommend reading them so that you can understand the reasons behind their conclusions rather than simply stating that their conclusions are wrong.

Often you'll see texts define energy as the ability to do work. That's too vauge and can't be used to write an expression for the energy of a system or it's form. E.g. a moving particle can do work, right? How do we define the quantity of energy from that statement in order to find the energy, whatever that is? How about we use the quantity m^2 v^2? What about the quantity mv? Each expresses a moving body and the mass of the body, since more of each can do more work. Suppose for the sake of arguement that we don't know or have not yet defined momentum. Some authors say that it's potential energy that is what has the capacity to do work. Do you think we should define the term "energy" to be synonymous with potential energy?
« Last Edit: 06/05/2013 22:27:58 by evan_au »


 

Offline yor_on

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Re: What us energy?
« Reply #1 on: 06/05/2013 21:57:51 »
No, energy and potential energy can't be the same. If I assume a bubble with a frictionless gas inside, that I chose to call energy, then I have defined a set amount constrained inside that bubble. If I now let there be objects in relative motion that now and then collide, releasing energy, assuming them to be made up of the same 'stuff' as the frictionless gas, then the release they create in their collision is 'energy' transformed. But as long as they are in relative motion you can chose any object you like, or several of them simultaneously, to define several 'potential energies' to a same object, simultaneously. So to me they can't be the same, and the frictionless gas and 'bubble' I'm using for it is just a naive description of the way we define it, defining it as a 'closed universe'. It's all a question of what you think makes a universe to me. you can choose to define it such as what makes it a universe to us is force and information carriers (light). Those tell us about distances and dimensions, clocks and rulers, and inside those we find 'energy'.
 

Offline yor_on

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Re: What us energy?
« Reply #2 on: 06/05/2013 22:12:32 »
There is also a possibility of defining a 'potential energy' from different uniform motions, depending on your choice of referent for a speed, also on your choice of 'who is moving' relative who. I'm not happy about potential energy, from that point of view, as it only will make sense from considering a defined system in its whole. You could say it represent a sort of symmetry perhaps? But 'energy' as in releasing new radiation in a collision exist, and is definably there in a experiment, whereas its potential energy only exist theoretically, until colliding.

Then again, what about 'bending space' as you get up to a relativistic speed, is that a result of 'energy', or is it a result of a geometry? And if light has the possibility of distorting the space it propagates in, would it be possible to measure?
=

Although, using matter solely, ignoring a speed, you can define a energy to it, but that's not a potential one, it's the energy released from its rest mass. Or you can turn that one around stating that everything we measure on is in some way in 'motion', even quantum mechanically, as the only way you can define a position is by measuring. Before that you have a probability of position, but no final guarantee, although I'm slightly unsure on that one as I can imagine a experiment repeated using 'identical setups' giving us a same result, position wise. As lights speed in a vacuum for example. It's a tricky universe.
==

There is a third point confounding it to me, and that is frames of reference, making all SpaceTime positions a locally defined property, although able to (Lorentz) transform into each other. So what are the 'energy' defined for a patch of space positionally? The space may be contracted from one observer accelerating (for simplicity) relative one uniformly moving. Both descriptions are locally true, according to how I read relativity, and experimentally defined. And the only way you can experiment is 'locally' as I see it. All other descriptions leaves what we measure directly, instead introducing theoretical frameworks transforming one SpaceTime position to another, but, directly measured, A and B will measure on two different 'patches', joined through a Lorentz transformation.
==

Assuming that Casimir effect is a result of the 'vacuum energy' it should then give us a different result depending on your acceleration/'speed', as you could be considered to be 'shrinking' a set amount of 'energy', and so volume (or area) of 'space' locally measured. If we define it as existing a set amount of energy, existing inside a 'closed universe'. That one I think might be testable? But I've also made some presumptions that are crucial to such a test. That there indeed is a 'vacuum energy' existing, that the Casimir effect is a result of vacuum energy, expressed in waves fitting, or not fitting, between plates creating a 'pressure'. I've also seen it defined as a result of matter. And yeah, 'compressing a vacuum' is a tricky one :) not making much sense, does it? And as you and the plates are of restmass you might get a result consistent with the assumption I make, without that vacuum locally having changed any 'properties' what so ever.

but not really, if it is a result of matter, and assuming that it could differ, then you will see a difference between defining that 'equivalent speed', relative a uniform motion, or getting to it under a acceleration, all as I think for the moment :) But if it indeed is possible to compress a 'vacuum' energy-wise, then, and only possibly (still a acceleration and restmass involved), it might make no difference for the energy difference?

Or you won't see a thing differing it? Which then, to me, and assuming a defined volume of space to contain a defined amount of (vacuum)energy, means that A and B:s local definitions of a area/volume doesn't matter for that definition. which becomes a hard nut to crack from a idea of a seamlessly same common (closed) space in where we all co-exist, to me at least. Because how do you define a energy to a volume, if it is observer dependent?
=

We can indeed simplify it, by defining Lorentz contractions to exist relative different uniform speeds too. Accepting this, one will find it simpler to define. And can one really expect the Casimir effect to differ, relative 'different' relative (uniform) motions? I don't expect it too.

So?
« Last Edit: 06/05/2013 23:54:04 by yor_on »
 

Offline lightarrow

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Re: What is energy?
« Reply #3 on: 07/05/2013 15:18:34 »
Energy is the quantity conserved when the system's Lagrangian is invariant under time translation.
 

Offline yor_on

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Re: What is energy?
« Reply #4 on: 07/05/2013 19:29:21 »
Would that define it Lightarrow?
Why then not say, energy is 'transforming' :)

Because that is the only time you will see it in action. The other definitions do not touch what it is, only try to describe it as a 'quantity'. but it's not a quantity, it's not bosons, well, I don't think it is although those seems closest to it somehow. If space is energy, then you can assume a length contraction to locally give you more, in a direct local experiment. That one is worth testing. If it doesn't, where exactly is that 'energy'?
 

Offline yor_on

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Re: What is energy?
« Reply #5 on: 07/05/2013 19:43:32 »
assume that a object in uniform motion does not contain any 'potential energy' by itself. Actually they don't, as locally measured. and it won't matter what speed you define it too. Now use two such objects, defining them as in 'relative motion' relative each other. They still won't get any extra 'potential energy' as measured locally. The collision will present us with a kinetic energy, but that one is not a result of any preexisting 'extra energy' created through me defining them as a system. There was no extra energy to be found in any of the object local experiments. What we might have is a definition of energy as something needing a symmetry, if we by that mean that one object by itself is not enough, you need something more than one object, by itself, to get a transformation as it seems to me.
=

The point I'm making is that it is only in a 'system' you can define that (potential) energy created through relative motion. There is no such thing as far as I know, locally measured, only relative some other object. Or otherwise expressed, would you expect that different uniform motions, creating 'uniform speeds' as measured relative some other reference frame, will be measurable in Earths matter? As its atoms jiggling more? Defining it as more 'energy' in a object, one should be able to expect that object to express it, as locally measured, too.
==

And that one keeps taking me to if a frame of reference can be defined by scaling. Because using that, you might have some 'amount/magnitude' to define, with being 'at rest' as matter could be seen as, consisting of 'frames of reference', still being correct. As long as one accept that different uniform motions are locally equivalent.
=

That as from such a point of view a uniform motion, and a piece of matter, must be equivalent relative any real local 'motion'. It doesn't exist from that point of view, and uniform 'speeds' becomes a description from frames of reference in relative motion. but from a point of your 'local experiment' both that matter, and your motion, gives you a same local description of a energy. The one related to restmass.
=

One more point, as I see it this will be true for 'energy', even if we had a 'absolute reference frame' from where would get 'real uniform speeds', differing relative it. It doesn't matter, what matter is your experiments, and if they don't give different uniform motions a added, locally definable, energy, then they don't.
« Last Edit: 07/05/2013 20:58:49 by yor_on »
 

Offline Pmb

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Re: What is energy?
« Reply #6 on: 07/05/2013 19:54:46 »
Energy is the quantity conserved when the system's Lagrangian is invariant under time translation.
You didn't read the first post carefully and/or completely, did you?
 

Offline yor_on

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Re: What is energy?
« Reply #7 on: 07/05/2013 20:39:59 »
We find uniform motions everywhere, don't we? What about something quantum mechanical. A atom?
Does it consist of uniform motion or accelerations?

As the definition of uniform motion I use, expect all uniform motion to be equivalently 'still', then there's no problem, as I see it, defining it from a probability instead of calling it a motion. Like if we had accelerations, describing a locally defined change, uniform motion locally defined as being 'still'.
=

Using 'energy' for it I would say that all accelerations expend 'energy'. Does a uniform motion do the same? Not if you define a uniform motion my way. Resistance and friction is relative rest mass, or a momentum, involving accelerations, decelerations, and annihilation.
« Last Edit: 07/05/2013 21:21:37 by yor_on »
 

Offline Pmb

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Re: What is energy?
« Reply #8 on: 07/05/2013 21:48:03 »
Quote from: yor_on
Using 'energy' for it I would say that all accelerations expend 'energy'. Does a uniform motion do the same? Not if you define a uniform motion my way. Resistance and friction is relative rest mass, or a momentum, involving accelerations, decelerations, and annihilation.
I don't understand what you're talking about. What does uniform motion have to do with the definition of energy?
 

Offline yor_on

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Re: What is energy?
« Reply #9 on: 08/05/2013 08:57:36 »
Uniform motion to me, as a description of something moving in a vacuum expends no energy Pete, as I know. Uniform motion on a planet does though. And to me that has to do with different types of friction and resistance. It must have to do with how the universe has split it into two (ideal) descriptions, and it is a question of what you wonder about too, energy. Lately I've started to think that all live, be it plants or men, accelerates. Every time something change it must involve a acceleration somehow, I think. It's like you can minimize the description of what differs, and also knit it to expending energy locally defined. And that's what is my main interest, to find what I can define locally.

Accelerations is a local definition to me expending energy.
Uniform motion does not expend energy, defined in a vacuum.

It's also a question of the way one define it. If I would define me uniformly moving on a airless planet, you could say that I expend energy relative a gravity, but now we involve a system as I think. Me relative that planet, and that becomes frames of reference in my mind, and a 'system'. It's two principles to me, uniform motion relative a acceleration, in where a acceleration always expends energy.
 

Offline yor_on

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Re: What is energy?
« Reply #10 on: 08/05/2013 09:17:12 »
One can also think of it this way, assume that gravity is a preferred direction. Then everything has to follow that direction, as long as nothing stops you from doing so you won't expend energy, moving in a geodesic. That geodesic is also a uniform motion, no matter if one define it as gravitationally accelerating from some inertial observation point, as I see it. And you can knit it to expending energy. I'm as fascinated as you by the idea of 'energy', and still don't know what it is, unless we define it as transformations, which doesn't tell us anything more than that is a point in where we can measure it. Although there is one thing, accelerations-decelrations must be present in any change.
« Last Edit: 08/05/2013 09:19:26 by yor_on »
 

Offline yor_on

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Re: What is energy?
« Reply #11 on: 08/05/2013 09:32:55 »
Using that, can a 'space', or a vacuum, move 'faster than light' :), or expand and inflate. Well, if one believe so, then why not be able to compress it too? And if a (local) Lorentz contraction shrink that universe, and the universe also is defined as of being of 'one magnitude' energy-wise, also implying it to be closed. Then you should be able to use Casimir force in different uniform motions, to find it differ.

No, I do not believe a vacuum to compress, neither do I expect it to do ftl. There has to be some other way to describe a expansion and inflation, not involving accelerations at all.

So no.
=

Or expressed this way, can a vacuum transform (change)? Not as I know, using my way of looking at it, then it can't accelerate either. And if you use 'uniform motion' as something equally 'still', expending no energy, then you can connect that to a locally defined arrow, if we define that as something observably changing. And so a atom can exist 'for ever', as long as it does not expend (or gain) 'energy'. Which, to me, then also should relate to a question of accelerations-decelerations. I differ between what I call a arrow, relating that to 'c', and a locally invariant clock, and then refer to 'time', which I don't really know what it is. It reminds me of the question of a background dependence, or not. I want something from where a arrow can exist, and saying that matter transforming is 'change', is 'time', does not catch it to me. I use 'c' for defining that arrow (as a local constant), and if I only could understand why 'c' is, I think I would find 'time'.
« Last Edit: 08/05/2013 10:23:30 by yor_on »
 

Offline evan_au

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Re: What is energy?
« Reply #12 on: 08/05/2013 12:55:56 »
I sometimes use the concept of "Work" to describe "Energy" because it has an intuitive meaning to those without a solid physics background - "If I were to exert this Energy, I would be working hard, and I'd get tired".

That doesn't require any mathematics, but for a more advanced audience you can certainly describe equations for the kinetic energy, various forms of potential energy (gravitational, electrostatic, chemical, etc) and also matter energy (usually not very accessible unless you have a Star Trek drive handy).

And then you can describe how one form of Energy can be transformed into another.

Certainly the current paradigm in physics is that the sum of all these forms of energy is conserved; that if some energy is missing from an interaction, then you have either miscalculated, or you have discovered a new, slippery particle (like a Neutrino or the hypothesised Dark Matter particles).
 

Offline yor_on

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Re: What is energy?
« Reply #13 on: 08/05/2013 13:24:43 »
As far as I get it a energy conservation of a energy-momentum 4-vector as defined under SR is what I would call a 'local definition'. It defines it such as the restmass of something is presumed to be invariant under a coordinate change. And if I now got that one right :) then that is what I think too. Local constants being invariant, then introducing 'frames of reference' between them, comparing ones local constants (clock ruler) relative what one define as others. But if joining whatever frame you found to differ from yours, time and distance, you will find it to be the exact same as yours.

As for work?
I don't know, once I thought of that as 'straining', not really related to what reaction you got. Then I found the main stream definition. Nowadays I would prefer to call it accelerations I think, locally defined.
 

Offline Ethos_

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Re: What is energy?
« Reply #14 on: 08/05/2013 14:46:18 »
The answer to this question is truly much more complex than the one I've chosen to offer, nevertheless, in simple terms the following is the best I can do.

The energy in a system is equivalent to the amount of order it possesses. And the law of entropy states that; as the order is replaced by disorder, the amount of usable energy is decreased.

One might say that energy and entropy are in a death struggle. Of course, entropy will eventually win out.

Another question is: What gave rise to the initial order in the first place? If our universe is singular and without neighbors, unlike the multiverse concept , and if there were no collisions of so-called branes, what caused the Big Bang?

For it is the Big Bang that gave rise to any order we are left with and any usable energy that may be extracted from our present circumstances.

The energy we observe is like seeing the time clock of entropy unwind.



« Last Edit: 08/05/2013 15:29:14 by Ethos_ »
 

Offline yor_on

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Re: What is energy?
« Reply #15 on: 08/05/2013 16:06:46 »
Then life is fighting entropy Ethos :) As its complexity (order) grows. You also have theoretical frameworks to consider, would you for example say that order comes out from thoughts? Or that thoughts in themselves represent a sort of order? If you do you now have left a purely physical definition for a slightly meta physical, although quite correct I suspect? Because it is thoughts, and logic, that defines the mathematics and physics we define. Or one might assume that the universe at its very 'bottom' is a purely mathematical construct, leading to complexity, and that entropy is a thing related to life, not to the construct in itself. And linear time is what creates the processes we see, as I think. We would define even timelessness from that, if we ever could do a real experiment on it. That as you are observing whatever you define from causality.
 

Offline Ethos_

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Re: What is energy?
« Reply #16 on: 08/05/2013 16:24:17 »
would you for example say that order comes out from thoughts? Or that thoughts in themselves represent a sort of order?
Yes, but these assemblies of order are only temporary on the universal scale. As the time clock of entropy winds down, there will be fewer and fewer opportunities for order to exist. In the end, all order will disappear. We can only redirect the course which entropy takes, we can never create additional order without diminishing the  surrounding order in the process. For a closed system, the net result  is a decrease in total order.
 

Offline Ethos_

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Re: What is energy?
« Reply #17 on: 08/05/2013 16:37:19 »

 We can only redirect the course which entropy takes, we can never create additional order without diminishing the  surrounding order in the process. For a closed system, the net result  is a decrease in total order.
However, on the other hand, if our universe is not somehow closed, there remains the possibility for further increases in order. This brings us around to the question; What caused the Big Bang? And if one is a believer, then; Who may have caused the Big Bang?
 

Offline yor_on

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Re: What is energy?
« Reply #18 on: 08/05/2013 16:45:00 »
We better be careful here Ethos :). Pete want us all to tell him what we think is 'energy', as I got it. Which indeed is a question very close to philosophy, and you and me both like that, I think. But if we now also lift in religious beliefs, and 'wonderings', in it, I suspect this thread to never end.

Let me ask another, rather cool question, instead. One can also imagine a 'superposition' of energy, in where it isn't defined, until it becomes a outcome. What type of 'energy' would one call that? A singular one, somehow represented by all its aspects, outside of causality? Or a 'composite', a multitude co-existing before a outcome? Or is it so that the universe use relations defining it as a outcome, those becoming its 'hidden parameters'? How should one define that from causality, and linear time?

http://www.researchgate.net/post/Superposition_does_not_conserve_energy
 

Offline Ethos_

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Re: What is energy?
« Reply #19 on: 08/05/2013 19:14:04 »
We better be careful here Ethos :). Pete want us all to tell him what we think is 'energy', as I got it.
Yes yor_on, I was only making a point about the difference between a closed or open system. While I would like everyone to have a say in this discussion, I agree that inviting religious views would probably be counterproductive.
Quote from: yor_on
Let me ask another, rather cool question, instead. One can also imagine a 'superposition' of energy, in where it isn't defined, until it becomes a outcome. What type of 'energy' would one call that? A singular one, somehow represented by all its aspects, outside of causality? Or a 'composite',
My own take on this would be, discounting the complexities, that energy is solely the unwinding of time with respect to system order. The result of the equation represents only a different direction taken and does not, in my opinion, change the identity or character of energy.

Think of it this way: However the original order was established, the unwinding of the order gives us the notion of energy release. Because time is involved, motion, force, magnitude, frequency, along with numerous other identities detail the observed changes to local conditions. And all happening at different rates and directions all in accordance with the law of entropy. But energy remains the same quality, and that quality remains the change in order that time has on reality.

Time = arrow of change
Energy release = change in system order
Entropy = order to disorder
 



« Last Edit: 09/05/2013 17:35:30 by Ethos_ »
 

Offline simplified

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Re: What is energy?
« Reply #20 on: 09/05/2013 16:54:18 »
Kinetic energy is a lost spatial communication.
 

Offline Pmb

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Re: What is energy?
« Reply #21 on: 09/05/2013 23:11:39 »
I'm going to start from scratch.

First let's said what I did to lightarrow's response where he said
Quote
Energy is the quantity conserved when the system's Lagrangian is invariant under time translation.
My response to him was You didn't read the first post carefully and/or completely, did you?  The reason for that response is that in the opening post I explained the following.

1) The quantity he is referring to is known as Jaconi's integral, given the letter h, because its an inetagral of motion under certain circumstances, i.e. when the Lagrangian is not an explicit function of time. It's also known as the energy function because under specific conditions the value equals the total mechanical energy of the system. This value is called the Hamiltonian when expressed in generalized coordinates and canonical momenta.

2) In general h does not equal the energy of the system even when its conserved. Therefore it is illogical to call make any attempt to define it as energy.

3) lightarrow is thinking of Noether's theorem which states that when the Lagrangian is independant of time h is constant. However even when h= T + V = total energy the Lagrangian must first be defined and it's defined in that instance as L = T - V. That is to say that we must know two forms of energy before we can know what the total energy is. In that case it's simipler to define energy as T + V since that definition will hold even when h is not constant, i.e. when lightarrow's condition does not hold.

3) h only cotains mechanical energy when it's the lagrangian involved. When there's a Lagrangian density involved EM energy and certain other forms for continuous matter can be defined. In general there is no lagrangian for all possible systems.

4) Noether's theorem is only intended to give a theorem relating a propery if of energy. It cannot, for the reasons stated above, actually define energy.

All this was stated in the first post. Too bad those reasons were ignored.

I'd like to know that both kinetic energy and potential energy and hence their sum is a well defined quantity. But that's merely two forms of energy and forms of energy are quite well defined.
« Last Edit: 09/05/2013 23:40:39 by Pmb »
 

Offline yor_on

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Re: What is energy?
« Reply #22 on: 10/05/2013 07:57:30 »
Then you're not discussing what 'energy' might be, instead questioning what mathematical definitions, and constraints, are most valid, if I get you right Pete? To do that you better first define how you see it. Energy as in a vacuum, does it exist? Can you prove it other as an assumption from mathematics? And if there is a 'energy' in it, do you think that explains the definition of a super-position? Seems to me that Noether's theorem is a description of principles, that we expect the universe to step out from, but not discussing what energy is?
=

I'm not sure of the Casimir effect, but you have the Higg 'field' though? You might be able to state that the LHC have proven something more than what we measure directly to exist. But to prove it as a field it needs to explain observer dependencies. Does it do that?
« Last Edit: 10/05/2013 08:28:04 by yor_on »
 

Offline lightarrow

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Re: What is energy?
« Reply #23 on: 10/05/2013 16:00:16 »
I'm going to start from scratch.

First let's said what I did to lightarrow's response where he said
Quote
Energy is the quantity conserved when the system's Lagrangian is invariant under time translation.
My response to him was You didn't read the first post carefully and/or completely, did you?  The reason for that response is that in the opening post I explained the following.

1) The quantity he is referring to is known as Jacobi's integral, given the letter h, because its an integral of motion under certain circumstances, i.e. when the Lagrangian is not an explicit function of time. It's also known as the energy function because under specific conditions the value equals the total mechanical energy of the system.
Ok, can you make a summary of those specific conditions? If you only want to refer to a mechanical system, for the moment, it's ok.
Quote
This value is called the Hamiltonian when expressed in generalized coordinates and canonical momenta.

2) In general h does not equal the energy of the system even when its conserved. Therefore it is illogical to make any attempt to define it as energy.
Right.
Quote
3) lightarrow is thinking of Noether's theorem which states that when the Lagrangian is independant of time h is constant. However even when h = T + V = total energy the Lagrangian must first be defined and it's defined in that instance as L = T - V. That is to say that we must know two forms of energy before we can know what the total energy is.
But only if you already know how to write the energy, that is, if you already know what IS energy for that system, so in this case the problem is already solved  :)
What I intended is a generic system with a lagrangian defined as a generic function of qi(t) and qi'(t) which have to obey just minimum requirements (don't remember which ones).
Quote
In that case it's simpler to define energy as T + V since that definition will hold even when h is not constant, i.e. when lightarrow's condition does not hold.

3) h only contains mechanical energy when it's the lagrangian involved. When there's a Lagrangian density involved EM energy and certain other forms for continuous matter can be defined. In general there is no lagrangian for all possible systems.

4) Noether's theorem is only intended to give a theorem relating a property of energy. It cannot, for the reasons stated above, actually define energy.
I don't agree with this, in the sense that, if I can describe with precision (even if, e.g., for mechanical systems only) for which conditions a certain function has the desired properties, then what I'm looking for is exactly that function with those conditions. If, instead, you are looking for a definition of energy which is more general and that holds even for more complex systems, then it's another story and then I agree with you.
Quote
All this was stated in the first post. Too bad those reasons were ignored.
Ok, let's say that we needed to understand better what you wrote, and, at the same time, to review our knowledge  :)

--
lightarrow
« Last Edit: 10/05/2013 16:06:16 by lightarrow »
 

Offline Pmb

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Re: What is energy?
« Reply #24 on: 10/05/2013 18:13:55 »
Quote from: lightarrow
Ok, can you make a summary of those specific conditions?
From Classical Mechanics – Third Edition by Goldstein, Safko and Poole page 345
Quote
Further, it was proved in Section 2.7 that if the equations of transformation that define the generalized coordinates (1.38),

rm = rm(q1, …, qn; t)

do not depend explicitly upon time, and it the potential is velocity independent, then H is the total energy T + V. The identification of H as a constant of the motion and as total energy are two separate matters, and the conditions sufficient for one are not enough for the other. It can happen that Eqs. (1.38) do involve time explicitly but that H does not. In this case H is a constant of motion but is not the total energy/ etc.

Quote from: lightarrow
But only if you already know how to write the energy, that is, if you already know what IS energy for that system, so in this case the problem is already solved  :)
I stated above that if you know the Lagrangian, L = T - V then you know the energy E = T + V so long as V is not velocity dependant. In that case you don’t need to bother righting the energy. But that’s mechanical energy, i.e. one of the forms of energy, and not energy itself. As I stated the forms of energy are well known and defined. Its energy itself which remains without a proper definition.

Quote from: lightarrow
What I intended is a generic system with a lagrangian defined as a generic function of qi(t) and qi'(t) which have to obey just minimum requirements (don't remember which ones).
There’s no such think as a generic Lagrangian. There are instances when a Lagrangian cannot be defined such as systems which include friction, thermal energy, zero point energy, the energy etc.

Quote from: lightarrow
I don't agree with this, in the sense that, if I can describe with precision (even if, e.g., for mechanical systems only) for which conditions a certain function has the desired properties, then what I'm looking for is exactly that function with those conditions.
That doesn’t make sense to me. It’s phrased in a confusing way. What properties are you referring to? Please rephrase more clearly..
 

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Re: What is energy?
« Reply #24 on: 10/05/2013 18:13:55 »

 

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