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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.

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.

However, there are formulas for calculating some numerical quantity, and we add it all together it gives “28” - always the same number.

Energy is the quantity conserved when the system's Lagrangian is invariant under time translation.

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.

would you for example say that order comes out from thoughts? Or that thoughts in themselves represent a sort of order?

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.

We better be careful here Ethos . Pete want us all to tell him what we think is 'energy', as I got it.

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',

I'm going to start from scratch. First let's said what I did to lightarrow's response where he saidQuoteEnergy 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.

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.

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 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.

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

Ok, can you make a summary of those specific conditions?

Further, it was proved in Section 2.7 that if the equations of transformation that define the generalized coordinates (1.38), r_{m} = r_{m}(q_{1}, …, q_{n}; 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.

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 q_{i}(t) and q_{i}'(t) which have to obey just minimum requirements (don't remember which ones).

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.

Note: I'm in the process of looking for an example of a Hamiltonian which is not the energy but is a constant of motion.

Quote from: lightarrowOk, can you make a summary of those specific conditions?From Classical Mechanics – Third Edition by Goldstein, Safko and Poole page 345QuoteFurther, it was proved in Section 2.7 that if the equations of transformation that define the generalized coordinates (1.38), r_{m} = r_{m}(q_{1}, …, q_{n}; 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: lightarrowBut 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: lightarrowI 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..

...In a sense, all energy is potential - the potential to do work. 'Potential energy' is just a generic convenience term for energy stored as spatial displacement, just as kinetic energy is the potential to do work due to an object's motion.

But shouldn't one definition be able to cover 'energy', if I assume that 'energy' to exist on its own Lightarrow?

"'Potential energy' is just a generic convenience term for energy stored as spatial displacement" ...

Can you explain thermal energy in that way? For example, you have an electric circuit made with a battery connected to a resistance and all is put inside an isolated box. The chemical energy of the battery is converted into electrical energy which heats the resistance wich heats the box and so it's converted into internal energy of the system. It's this internal energy stored in the sense you wrote?

That internal energy can be used to do work; for example, you could use it to turn water into steam to drive a motor or generate electricity.

Potential energy is just used to refer to potential for work associated with the position of an object in a system, usually involving restoring forces like gravity or tensile or elastic forces (e.g. a spring, or rubber band).

]I can give it to you: A rigid thin bar, in which a point mass can slide without friction, is kept in uniform rotation around one of its ends "O". The point mass is also connected to O with a spring.The system's energy is not constant but the hamiltonian is (i made the computations some times ago, if you want I go and try to find them).

Quote from: lightarrow]I can give it to you: A rigid thin bar, in which a point mass can slide without friction, is kept in uniform rotation around one of its ends "O". The point mass is also connected to O with a spring.The system's energy is not constant but the hamiltonian is (i made the computations some times ago, if you want I go and try to find them).Examples in which the energy is not constant but is the Hamiltonian is are trivial. Any case where the potential energy function is an explicity function of time is such an example. E.g. a charged particle moving in a time-varying EM field. But that's not what I was refering to. Please reread what I posted, i.e. Hamiltonian which is not the energy but is a constant of motion.

And so we throw away the second law of thermodynamics... []

Quote from: Pmb on 12/05/2013 22:01:44Quote from: lightarrow]I can give it to you: A rigid thin bar, in which a point mass can slide without friction, is kept in uniform rotation around one of its ends "O". The point mass is also connected to O with a spring.The system's energy is not constant but the hamiltonian is (i made the computations some times ago, if you want I go and try to find them).Examples in which the energy is not constant but is the Hamiltonian is are trivial. Any case where the potential energy function is an explicity function of time is such an example. E.g. a charged particle moving in a time-varying EM field. But that's not what I was refering to. Please reread what I posted, i.e. Hamiltonian which is not the energy but is a constant of motion.I sincerely don't understand what you mean. In the example I made, infact, the Hamiltonian is not the energy. I wrote:"energy is not constant but the hamiltonian is"Maybe my english is not correct?

In a sense, all energy is potential - the potential to do work. 'Potential energy' is just a generic convenience term for energy stored as spatial displacement, just as kinetic energy is the potential to do work due to an object's motion.

Quote from: dlordeThat internal energy can be used to do work; for example, you could use it to turn water into steam to drive a motor or generate electricity. And so we throw away the second law of thermodynamics... []

Quote from: lightarrow on 12/05/2013 19:13:42And so we throw away the second law of thermodynamics... []How so? You have a hot box that you can use as an energy source to do work. That's how steam engines and power stations work. How is that throwing away the 2LT? I'm not saying you'll get more out than you put in.

That internal energy can be used to do work; ...