It seems that Energy, Momentum and Mass (three conserved quantities) are important concepts. My question is, how were they originally developed? I have had plenty of exposure to the equations relating force, momentum and energy, and can manipulate the equations great. However, I feel I am shamefully ignorant of how each of these concepts were developed, and how the relation, say, between Force and Energy was initially determined.

It wasn’t just one discovery, it was many over a period of more than a century.

For a very quick overview, Wikipediahas a timeline of these events, including who first introduced these concepts.

For more detail, you can continue clicking on Wikipedia links, or your can read a history of physics. I particularly like the one by Isaac Asimov. Obviously it’s out of date for modern physics but perfectly fine for the period you are interested in.

I’m no historian, but I would guess that all three (as rigidly-defined mathematical entities, at least) date back to Newton. The idea of energy being absolutely conserved, meanwhile, probably goes back to Joule, who was the first to realize that heat was fundamentally a form of energy.

Yeah, **mass** (as distinguished from weight) was pretty much something Newton pulled out of his ass (hence the name:)). I am not sure if **momentum** is explicitly defined in Newton’s *Principia*, but even if not, the concept is certainly implicit in Newton’s mechanics (although in this case Galileo and Descartes deserve a lot of the credit for laying the conceptual groundwork).

I may be wrong, but I rather think **energy**, in anything like its modern sense, does not really arrive until well into the 19th century. As **Chronos** says, Joule was important in getting the idea of energy conservation going, but I think it was Helmholtz who really clinched it, and I have a feeling he may have been the first to have articulated a clear definition of energy.

My understanding is that Newton’s second law of motion was originally stated in terms of momentum, not mass.

**F = d/dt (mv) ** rather than **F = m dv/dt** ?

Seems to me I’ve read an article pointing this out before, that Newton was not so “reckless” as to assume mass is a constant under the influence of a force. And that therefore, in a small way, he anticipated Special Relativity by a couple centuries.

Nitpick: mass is not conserved.

Yes it is, so long as you don’t go and re-define the boundaries of your system in the middle of a problem. Would you care to provide a counterexample?

I was under the impression, from another post on GQ, that Mass and Energy were both conserved separately. Certainly E = mc^2 provides a relation between these two quantities, but mass and energy are conserved independent of each other.

Perhaps my understanding is flawed, though.

Yes. The momentum formulation is better, not just because it holds under Special Relativity, but also under conditions of an object changing mass, like a rocket.

It is if you define it properly.

Rest mass is *of course* not conserved. Take, for instance, the decay of the muon.

When the term ‘mass’ is thrown around, one tends to assume ‘rest mass’, not ‘invariant mass’. At least, that is my experience in the field of high energy physics.

If you want to know how the concept of *invariant mass* developed, that’s easy. It was introduced very shortly after Einstein published his papers on relativity in 1905. It is a property of a *system*, and it is conserved because it is a relativistic invariant constructed from actual conserved quantities (ala Noether) p and E.

Rest mass *is* invariant mass. A muon has a mass of 105.7 MeV/c[sup]2[/sup]. The decay products of a muon, when taken as a whole, also have a mass of 105.7 MeV/c[sup]2[/sup]. Note that this is *not* the same as the sum of the mass of an electron, a mu neutrino, and an electron antineutrino: That’s what I meant when I said that you can’t redefine the boundaries of your system in the middle of a problem.

You have your terminology wrong (or at best ‘loose’). The decay products of a muon, as a system, have an invariant mass, not a rest mass. You can call it that if you wish, but it’s a little sloppy. But in the end it’s just terminology. My nitpick is that historically speaking, in the context of Newton etc, and the first formal codifications of ‘modern’ concepts of mass and momentum, it’s obvious (to me at least) that the OP (like Newton would) is referring to the rest mass of individual particles, which is trivially not conserved. It was only after 1905 it could be understood that an abstract quantity called ‘invariant mass’ was conserved, even though the rest masses of all the individual particles in the system is NOT conserved.

I just want to do is make sure the OP, who seems to be asking basically about the development of classical Newtonian concepts, understands that the sum of the masses of the particles in a system is not conserved.

I’ll explain. The formula E = mc^2 applies only for a particle at rest. If the particle is moving, the formula is: E = sqrt( p^2c^2 + m^2c^4).

Now if the particle decays into 2 particles of energies E1 and E2, momenta vectors p1 and p2, and masses m1 and m2, then:

E_before = E1+E2

p_before = p1+p2

BUT

m_before does NOT equal m1+m2

There is, however, something that is conserved – it’s an abstract quantity called the ‘invariant mass of the system.’ But it’s not the same ‘m’ that you see in F=ma, for example.

You seem to be saying that invariant mass is a property of a system, but rest mass is a property only of individual particles. Except, what’s the difference between a particle and a system? What, for instance, is the rest mass of a proton? Is it just the sum of the rest masses of the three quarks? Because that’s a radically different value than what everyone quotes as “the mass of the proton”.

Do you mean to imply that Newton knew this, or that anyone before Einstein knew this? :dubious:

If not, and if the OP is indeed (as I also think is probable) asking about the origins of the classical Newtonian concepts, then you are certainly not clarifying anything by bringing up issues deriving from relativity theory. You are obfuscating.

But, in classical Newtonian mechanics, the sum of the masses of the particles *is* conserved, since having it not be conserved would require non-classical or non-Newtonian dynamics.

So, I agree with **njtt** that this issue is a distraction with respect to the historical question in the OP.

(Plus, we just had a big thread on these issues yesterday or so.)

Now you are confusing “invariant mass” with “invariant mass + binding energy”. The rest mass of the proton is simply its total energy divided by c^2 in its CM rest frame. The invariant mass of a system can be calculated in any frame, and is the sum of the squared energies minus the sum of the squared momenta in the system. The two concepts are different. If you calculate the invariant mass of the proton by looking at its constituents you will get a different mass from its rest mass. This is textbook particle physics.