How do we know energy and momentum are conserved? Are these two facts taken as axioms or have they been proven by an experiment?
This question has been in part addressed here: Conservation of Momentum but I don't see how translational symmetry implies conservation of momentum. If the reasoning behind this could be explained that would be great.
Conservation of energy, like conservation of momentum, seems intuitive to me but similarly how do we know for certain that it is impossible to create or destroy energy? Is this taken as an axiom or has it been proved by an experiment?
I hope it is clear that I'm not trying to suggest that I don't trust these laws to be true but rather that I'd like to know how we know they are true.
Thanks for the help
Best Answer
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 can't really add any clarifying comment to that statement.