4 ms·

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?

dpq 7y ago
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.
- fferen 7y ago
  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.
_Microft 7y ago
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_to_gauge_invariance https://en.wikipedia.org/wiki/Charge_conservation#Connection... for some details, even though I'm aware that it might be way too technical.
- Filligree 7y ago
  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.
jules 7y ago
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/huyimpct.html http://www.princeton.edu/~hos/Mahoney/texts/huygens/impact/h...

dpq 7y ago

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.

fferen 7y ago
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.

_Microft 7y ago

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_to_gauge_invariance https://en.wikipedia.org/wiki/Charge_conservation#Connection... for some details, even though I'm aware that it might be way too technical.

Filligree 7y ago
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.

jules 7y ago

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/huyimpct.html http://www.princeton.edu/~hos/Mahoney/texts/huygens/impact/h...