[calhoun137]

The key point is that energy, momentum, and angular momentum are additive constants of the motion, and this additivity is a very important property that ultimately derives from the geometry of the space-time in which the motion takes place.

> Is there any way to deduce which invariance gives which conservation?

Yes. See Landau vol 1 chapter 2 [1].

> I'm looking for the fundamental reason, as well as how to tell what will be paired with some invariance when looking at some other new invariance

I'm not sure there is such a "fundamental reason", since energy, momentum, and angular momentum are by definition the names we give to the conserved quantities associated with time, translation, and rotation.

You are asking "how to tell what will be paired with some invariance" but this is not at all obvious in the case of conservation of charge, which is related to the fact that the results of measurements do not change when all the wavefunctions are shifted by a global phase factor (which in general can depend on position).

I am not aware of any way to guess or understand which invariance is tied to which conserved quantity other than just calculating it out, at least not in a way that is intuitive to me.

[1] https://ia803206.us.archive.org/4/items/landau-and-lifshitz-...

[Aardwolf]

But momentum is also conserved over time, as far as I know 'conservation' of all of these things always means over time.

"In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant."

That means it's conserved over time, right? So why is energy the one associated with time and not momentum?

[rnhmjoj]

Conservation normally means things don't change over time just because in mechanics time is the go to external parameter to study the evolution of a system, but it's not the only one, nor the most convenient in some cases.

In Hamiltonian mechanics there is a 1:1 correspondence between any function of the phase space (coordinates and momenta) and one-parameter continous transformations (flows). If you give me a function f(q,p) I can construct some transformation φ_s(q,p) of the coordinates that conserves f, meaning d/ds f(φ_s(q, p)) = 0. (Keeping it very simple, the transformation consists in shifting the coordinates along the lines tangent to the gradient of f.)

If f(q,p) is the Hamiltonian H(q,p) itself, φ_s turns out to be the normal flow of time, meaning φ_s(q₀,p₀) = (q(s), p(s)), i.e. s is time and dH/dt = 0 says energy is conserved, but in general f(q,p) can be almost anything.

For example, take geometric optics (rays, refraction and such things): it's possible to write a Hamiltonian formulation of optics in which the equations of motion give the path taken by light rays (instead of particle trajectories). In this setting time is still a valid parameter but is most likely to be replaced by the optical path length or by the wave phase, because we are interested in steady conditions (say, laser turned on, beam has gone through some lenses and reached a screen). Conservation now means that quantities are constants along the ray, an example may be the frequency/color, which doesn't change even when changing between different media.

[calhoun137]

my understandinf is that conservation of momentum does not mean momentum is conserved as time passes. it means if you have a (closed) system in a certain configuration (not in an external field) and compute the total momentum, the result is independent of the configuration of the system.

[rnhmjoj]

It certainly means that momentum is conserved as time passes. The variation of the total momentum of a system is equal to the impulse, which is zero if there are no external fields.