I'm trying to understand Noether's theorem from an intuitive perspective. I know that time-translational symmetry implies the conservation of energy. Is it possible to convince oneself that time-translational symmetry implies that energy is the only conserved quantity, and not linear momentum for instance, without going through the mathematical derivation?
Classical Mechanics – Why Does Time-Translational Symmetry Imply Energy Conservation?
classical-mechanicsenergy-conservationnoethers-theoremsymmetry
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We know through experimental observation. That is the beginning and end of the subject of physics, at least the part of it the tells it apart from, say mathematics. Conservation of momentum is simply an inductively reasoned hypothesis to summarize certain patterns in experimental data.
You are alluding to the conservation of momentum's being "explained" through Noether's Theorem. As I discuss in my answer to the Physics SE question "What is Momentum, Really?" here, whenever the Lagrangian of a physical system is invariant with respect to co-ordinate translation, there is a vector conserved quantity. That fact is wholly mathematical result, that continuous symmetries of a Lagrangian always imply quantities conserved by system state evolution described by that Lagrangian, one for each "generator" of continuous symmetry (i.e. basis vector of the Lie algebra of the Lie group of the Lagrangian's symmetries).
Note carefully, however, that Noether's theorem is an "if" theorem: a one-way implication. It's far from being the only way that a conservation might arise. Experimentally, it has been found to be fruitful to act on the hunch that it is the explanation, in the following way. Since the conserved quantity in a Lagrangian formulation of Newtonian mechanics implied by co-ordinate translation invariance is Newtonian momentum, we hypothesize that the result is more general and therefore deliberately construct Lagrangians for other theories to have this symmetry so that they too will have conserved momentums (i.e. spatial co-ordinate translational invariance). When we make predictions with these theories, they turn out, again determined experimentally, to be sound.
We say that the symmetry "explains" conservation of momentum, but what we really mean that is that we have found a compelling translation of the conservation law: it translates conservation into symmetry terms.
It is nonetheless an important translation; in my opinion it makes physics much more "visceral". The statement of conservation laws as givens seems abstract and, from a 21st century standpoint, arbitrary and open to question. In stark contrast, a symmetry description is much more accessible to us: even tiny children begin to understand that the World's behavior is independent of the way we choose to describe it. Why should the laws of physics change simply because I decide to shift my co-ordinate origin to another place, or rotate my co-ordinate system (rotational invariance of a Lagrangian gives rise to conservation of angular momentum)? Unless, of course, there is a clear, outside, experimentally measurable agent breaking this independence (e.g. grain structure in a crystal making laws depend on their orientation relative to the grain).
User knzhou adds:
... I would just add that we are now so confident in energy/momentum conservation that it can be used "in reverse" to your method in paragraph 3: if we saw events at the LHC with missing energy, this would be taken as evidence for dark matter, not evidence against conservation of energy! We would change our Lagrangian, nothing more.
I can't really add any clarifying comment to that statement.
Here is an easier way to think about this.
Imagine that the laws of physics could be time dependent, and you arrange for the law of gravity to be turned off in the shank of an otherwise dead Thursday afternoon.
In anticipation of this useful event, you have a machine set up which uses falling weights to perform useful work, and a pile of weights ready to place inside a container mounted high above the machine, from which they are fed downwards into it.
Then comes Thursday, and with gravity shut off it is easy for you to lift all those weights up and into the container. Later that afternoon, when gravity comes back on again, you turn the machine on and it produces useful work, which cost you nothing because gravity was off while you loaded the machine with weights.
In this way, turning gravity on and off allows you to violate energy conservation.
Best Answer
The main point is that if an action has a given infinitesimal quasi-symmetry with a single infinitesimal parameter then there is a single non-ambiguous & unique on-shell continuity equation; not 2 or more as OP seems to suggest.
Importantly, a translation quasi-symmetry in one variable implies that the canonically conjugate variable is conserved. This is easier to see in the Hamiltonian formalism, cf. e.g. my Phys.SE answer here.
Concerning time-translation quasi-symmetry and energy conservation, see e.g. this related Phys.SE post.