Deriviation for law of conservation of energy

[Amin2014]

I'm a bit confused about how the work-energy theorem for a single particle can be extended into the general law of conservation of energy for the macroscopic system, particularly the point where we divide the kinetic energy of the system into macroscopic kinetic energy and internal kinetic energy, and also the potential energy of the system into macroscopic and internal potential energy (and then define internal energy as the sum of internal potential and kinetic energies).

Can someone start from the familiar work-energy theorem, written for each individual particle of the system, and step by step derive the conventional form of the law of conservation of energy for the microscopic system?
And explain where ever you insert a new definition or experimental law (like heat and joule experiment). Thanks!

 

[Simon Bridge]

I'm a bit confused about how the work-energy theorem for a single particle can be extended into the general law of conservation of energy for the macroscopic system...

... I don't think you can. The work-energy theorem just says that work is the change of energy, and pretty much is a statement of the law of conservation of energy.

Though you realize that if the law of conservation of energy works for all the individual particles that make up the system, then it must hold for the system too?

How about going through the kind of derivation you are used to and showing us where you get confused?

 

[Andy Resnick]

I'm a bit confused about how the work-energy theorem for a single particle can be extended into the general law of conservation of energy for the macroscopic system,<snip>!

It can't- the work-energy theorem is actually a restatement of F = ma and *not* related to conservation of energy.

 

[anorlunda]

If a system comprised of many components does not conserve energy, then is it seems mandatory that one or more of the components must not conserve energy.

If you are satisfied that a single particle conserves energy, then try analyzing a two particle system, then three ... Any system with more than one particle is a macro system.

 

[jbriggs444]

If a system comprised of many components does not conserve energy, then is it seems mandatory that one or more of the components must not conserve energy.

If you are satisfied that a single particle conserves energy, then try analyzing a two particle system, then three ... Any system with more than one particle is a macro system.

There is a fallacy here.

One particle alone conserves kinetic energy. Another particle alone conserves kinetic energy. A pair of particles interacting gravitationally does not conserve kinetic energy. The whole is more than the sum of the parts.

 

[anorlunda]

There is a fallacy here.

One particle alone conserves kinetic energy. Another particle alone conserves kinetic energy. A pair of particles interacting gravitationally does not conserve kinetic energy. The whole is more than the sum of the parts.

No fallacy. Total energy is conserved. Particular forms of energy, kinetic, potential, heat, chemical, electrical, and so on are readily converted from one type to another. Indeed, most industries are based on energy conversion of some kind.

Only nuclear reactions and the cosmology of the universe do not conserve total energy. (And, even nuclear conserves total mass-energy.)

 

[jbriggs444]

But that conservation of total energy does not follow from particle-by-particle conservation. It follows, if anything, from pair by pair conservation.

 

[anorlunda]

But that conservation of total energy does not follow from particle-by-particle conservation. It follows, if anything, from pair by pair conservation.

I don't understand what you mean by pair by pair. In any case, what I said earlier stands. Energy is conserved particle by particle, in the interactions between particles, and in the macro system of particles.

If you really want to understand the derivation of conservation of energy, it follows from something called time translation invariance and Noethers Theorum. I don't think you can get there by considering the mechanics of particles. This wikipedia article on conservation of energy may also help.

Perhaps some teachers in the PF audience can suggest an easier way to prove conservation of energy, not just assert it.

 

[jbriggs444]

Energy is not conserved particle by particle. That is manifestly clear as the example of two particles under gravitational interaction demonstrates. However, if you consider those two particles as a pair and include the binding energy from their gravitational interaction then you can balance the energy books. That is what I have in mind when speaking about adding things up "pair by pair". You have to account for the gravitational interaction between every pair of particles in the system.

Of course, gravity is not the only interaction that may be present.

 

[anorlunda]

When we analyze the motions of particles, we presume that total energy is conserved. It is tremendously helpful, because every time it appears that energy is not conserved, it flags an error or omission in our math or faulty definitions in our heads. Then we can go back and correct our mistakes. That is what you need to do.

If on the other hand, you want to understand why energy must be conserved rather than presume it, then forget the particles and go back to Noether's Theorum. It takes some calculus and some effort to understand, but that is the true answer to the question posed by your title to this thread.

I recommend Professor Susskind's excellent video course on Classical Mechanics. He is a great teacher. Ater viewing his course, I guarantee that you will understand all this stuff including Noether's Theorum. But it will take some time and effort on your part. The course is available on youtube, here.