What Is Energy: Physical Reality or Mathematical Construct?

[Phrak]

Energy is still conserved. For the reason that Cantab Morgan stated.

I would write that as, energy is conserved for the reason Cantab Morgan stated.

I shouldn't be so trit. Going to wikipedia, one can find a couple other conditions not stated. Also, on a curved manifold, are all the conditions met? I don't know.

I realize it's not easy to come up with all the conditions to ensure that some particular symmetry statement is true. As well, it may not even be fair to talk about curved spacetime in a general physics forum. A PF faq would be nice.

The thought experiment I rather like involves a laser beam, first split, and then crossed at some obtuse angle. It seems as though the energy should be conserved thoughout. The energy flux into the region where they cross is equal to the energy flux out. However, the energy flux density within the intersection can be made arbitrarily small as the angle approches 180 degrees. The conserved quantity in this case is mass_0 density, \rho_0, where rho is the density of the so called invariant mass.

 

[Cantab Morgan]

It is theories that are informed by observations and not the other way round observations being informed by theories.Any good theory, including Noethers, must conform to the observations it predicts and although Noethers,which is based on other theories, predicts conservation it does not explain why there should be conservation.

Your comment is intelligent, and it also reveals that I did not express myself clearly enough.

Noether's theorem is not a Physical theory. It is Mathematics. It is true.

Now, I suppose that what is up for grabs is whether the universe can really be modeled accurately by a Lagrangian formulation, because that is Physics. But there has been quite a lot of success doing this, so I hadn't questioned it in my original post. Strictly speaking, we say that such a Physical theory is useful rather than true. But if the universe can be modeled in this way, then Noether's theorem applies.

Noether's theorem tells us that if the equations of motion (basically F=ma) have no explicit dependence on time, then energy must be conserved. When we think of all the "F"s in the world, like Coulomb's Law for example, we don't see any time dependence. It is in this sense that we say that Noether's theorem tells us why energy must be conserved.

 

[Phrak]

Your comment is intelligent, and it also reveals that I did not express myself clearly enough.

Noether's theorem is not a Physical theory. It is Mathematics. It is true.

Now, I suppose that what is up for grabs is whether the universe can really be modeled accurately by a Lagrangian formulation, because that is Physics. But there has been quite a lot of success doing this, so I hadn't questioned it in my original post. Strictly speaking, we say that such a Physical theory is useful rather than true. But if the universe can be modeled in this way, then Noether's theorem applies.

Noether's theorem tells us that if the equations of motion (basically F=ma) have no explicit dependence on time, then energy must be conserved. When we think of all the "F"s in the world, like Coulomb's Law for example, we don't see any time dependence. It is in this sense that we say that Noether's theorem tells us why energy must be conserved.

I don't get energy conservation from a pulsed laser beam. The beam is split, then crossed. As the beams cross the energy is reduced during the time of crossing.

 

[zoobyshoe]

Yeah, we do - "the most fundamental level" is e=w=f*d. People who are looking for some deeper meaning to the word beyond the physics definition are attempting to attach a significance to it that simply doesn't exist and is completely unnecessary. IMO, this results from confusion about a possible link between the scientific and mystical definitions for the word "energy".

I think this is true, but the confusion also results from concepts of energy thrown around in science fiction movies, TV shows, and comic books. You end up with a notion that energy is a sort of self-contained, power-packed fluid transferable from one thing to another.

 

[Dale]

I don't get energy conservation from a pulsed laser beam. The beam is split, then crossed. As the beams cross the energy is reduced during the time of crossing.

That is incorrect. Maxwell's equations http://farside.ph.utexas.edu/teaching/em/lectures/node89.html" in general, the specific details of how you pulse your laser beam don't matter.

Please post a rigorous derivation if you believe otherwise.

 

[Phrak]

That is incorrect. Maxwell's equations http://farside.ph.utexas.edu/teaching/em/lectures/node89.html" in general, the specific details of how you pulse your laser beam don't matter.

Please post a rigorous derivation if you believe otherwise.

okay

 

[Phrak]

That is incorrect. Maxwell's equations http://farside.ph.utexas.edu/teaching/em/lectures/node89.html" in general, the specific details of how you pulse your laser beam don't matter.

Please post a rigorous derivation if you believe otherwise.

Yeah, you're right. The momentum of a particle tends to zero, the frequency and energy remain constant; the invariant mass changes.

 

[Dale]

Sorry, I don't follow. Are you talking about a massive particle as v->0 or about two photons interfering with each other or a Maxwell description of the laser or what?

 

[Phrak]

Sorry, I don't follow. Are you talking about a massive particle as v->0 or about two photons interfering with each other or a Maxwell description of the laser or what?

I don't know, Dale. The truth is, I haven't sorted this all out. Every consideration brings up something unexpected. A simple scenario is a one dimensional reflected light wave,

\Phi = \phi \: cos(kt - \omega t) - \phi \: cos (kx + \omega t)

The frequency is constant. The particle count doubles. I suppose the energy doubles, and so is conserved.

 

[ryuunoseika]

I've been throwing this term energy around for a while now, and thinking about it I have absolutely no idea what it is. Is it something that actually exists in the universe, or just a construct that we use to simplify problems?

Terms like kinetic energy, and even gravitational potential energy (from a Newtonian standpoint) are a little bit easier to understand because they are exist in everyday life, but when you get E&M (I've only studied classical E&M) you have this idea of a field, the field has energy and you need to calculate it? How does a field have energy?

Consider a conducting bar, moving on rails with a large resistor at one end bathed in a strong, uniform magnetic field. As the bar moves, the magnetic flux increases, inducing a current in the circuit and thus energy is lost in the resistor as heat, so the bar must be slowing down (it never actually stops even though it moves a finite distance, as I've calculated). I can ascertain what happens in this situation because I know about E&M, but it bothers me that I don't feel like I understand what is actually happening. If you showed me that and I didn't know physics, I would say the bar moves an infinite distance. How can some invisible field (with energy) stop a real, moving object? This is the whole idea of transfer of this energy which I don't know what it is?

Is there a good model of a physical interpretation of this phenomenon? Is it just a mathematical construct?

An electromagnetic field, like a gravitational or other field, only has energy when it acts upon something. You could think of energy as a mental/mathematical mechanism for understanding a series of interchangeable phenomenon in our universe, but consider this: according to E=MC2, Energy has mass (relativistic mass), thus it has gravitational pull. It even has momentum. Enough light concentrated in one direction can open a door. Energy has it measurable affects on reality, like mass, so in my mind it's real.