Comment by KenoFischer | Hacker News Reader

Comment by KenoFischer | Hacker News Reader

I'm not sure the question is particularly well posed, but I think the most coherent answer you'd get from modern physics is symmetry. Restating Newton's law's slightly, you basically get two points:

1. Momentum is conserved

2. All interactions between objects are mediated by forces (i.e. interactions exchange momentum).

So you may ask, ok, but why do these two things happen, which is is where symmetry comes in. If you require that your theory is invariant under translations in space and time, your theory must conserve momentum and energy. Then you ask, ok, so what kind of long-range interactions can we have between particles in a theory where energy and momentum are conserved and the answer turns out to be, well those that exchange momentum. For example, in the hypothetical from the blog post where interactions are mediated by exchanges of acceleration (forces are proportional to jerks), you end up with a universe where absent interactions, acceleration is constant. Why is this inconsistent with symmetry? Well, in such a universe, the acceleration of particles would be constant in the absence of interaction, so their kinetic energy would keep growing and growing, violating conservation of energy.

https://en.wikipedia.org/wiki/Noether%27s_theorem

I think "symmetry" is actually not a great way to think about F=ma. The reason is that Newton's laws still work even if your system is non-symmetric with non-conserved dynamical quantities.

Consider releasing a rock and calculating how long it takes to hit the ground. In this system, the rock's momentum increases in time, and is therefore not conserved.

Consider a charged particle in an oscillating electric field. In this system, the particle oscillates, and so energy is not conserved.

What about Noether's theorem? In the first example the Lagrangian has a height dependence, and in the second one there's a time dependence. So the theorem does not apply.

Yet F=ma still works. So F=ma does not come from symmetry.

You might object that these quantities are conserved globally. But the point is that F=ma allows us to use model systems as affected by external forces; this is a very powerful simplifying technique.

Momentum is always conserved. You just have to draw the boundaries correctly.

Read about the mass defect.

m is derived as

m = P_S / c² * V_fohs

P_S and V_fohs do not exist in the standard model. The gravitational microcuviture of protons/neutrons cause changes in the P_S (static vacuum pressure) that therefor changes the mass. P_S is around 1.3 * 10^26 N/m².

This is from the Basic Structures of Matter - Supergravitation Unified Theory model, not the standard model.

BUT you don't have to. You CAN re-express whole systems as partial systems with external forces.

A ball falling to earth has a time-independent Lagrangian because you neglect the earth's acceleration. The electron orbiting the hydrogen nucleus moves in a fixed potential for the same reason.

"Draw the boundaries correctly" and you can't do physics.

Sure, but you cant turn around and say that momentum isn't conserved "because gravity".

This is great. Can you explain why the oscillating particle doesn't conserve energy?

Oscillating charges emit electromagnetic waves that carry away some energy.

This is the right answer. Noether's theorem can be used to derive newtons laws from symmetry properties of space alone. The article is wrong when it asserts that f=ma is taken as a given. The symmetry properties are the underlying assumptions.

Not when Newton discovered it they were not. The early versions of laws of thermodynamics were derived by just considering invariants (leading to conservation laws), not symmetries.

Such that things didn't just start moving on their own with no explanation. This is also why gravity was easier to approximate to them than heat. (Heat is symmetrical but hard to keep invariant, unlike motion.)

This is also why heat was considered an item (phlogiston / caloric theory) not a property of matter until quite late. (With Joule.) Or why the link between light and heat had to wait for so much longer.

Symmetries are actually much harder to pull off and verify. The big change there came with Maxwell or thereabouts.

This seems to lead to another question, which is why is the invariant of the system what it is? Why are momentum (mv) and energy (0.5mv2) conserved?

It directly follows from the assumptions that space is isotropic and homogeneous. I recall that there was a short and elegant proof of this link in Landau and Lifshits vol 1; you may want to look it up.

Indeed, it is at the start of the book. It also relies on the assumption that the Lagrangian is invariant up to a total time derivative under Galilean transformations, which leads to `dL/d(v^2)` being a constant (1/2 m). This is, I believe, the most fundamental way to look at it. Otherwise you could ask why the Lagrangian can't have other terms like v^4.

A sibling comment to yours mentions Noether's theorem that links symmetries (look up what symmetry means in this context if you're not sure) with conserved quantities. Here are some examples:

Energy is conserved because the laws of physics at two different times are the same.

Momentum is conserved because of translation invariance, as the laws of physics at two different points in space are the same.

There are others, like conservation of angular momentum because the laws of physics do not change under rotation.

Conservation of charge is because of gauge invariance of the electromagnetic field and might be the least understandable of the mentioned ones here. Check https://en.wikipedia.org/wiki/Charge_conservation#Connection... for some details, even though I'm aware that it might be way too technical.

It's interesting to note that, when you look at the universe as a whole, energy is not conserved. (It also isn't time- translation-symmetric.)

It's been such a successful concept that you'll even find people denying that that can be the case. I'm one of them, somewhat -- I suspect it implies that we're missing part of the system, hence the universe as we know it isn't closed.

Others have already given the modern answer, but the historical perspective and how it was discovered is also interesting. It's not obvious (some people thought that m|v| is conserved) but that those two are conserved can be deduced from intuitive facts about mechanics that you already know, plus the fact that gravitational acceleration is constant. Christiaan Huygens, who predates Newton, analyses the situation in this very interesting article, which Newton cites as an influence when he presents his laws: http://www.princeton.edu/~hos/Mahoney/texts/huygens/impact/h...

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