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I recall in my basic electromagnetism and quantum mechanics lectures that choosing one coordinate system over another may greatly simplify the equations involved in solving a problem (think about computing the electric field of an infinite uniformly charged line in $\mathbb{R}^3$ where cylindrical coordinates are the best choice, for instance).

The "goodness" of the choice is often dictated by the symmetries the problem exhibits (cyclindric symmetry in the above mentioned example).

Question 1: is this analogous to some weak form of Noether's theorem?

In the running example, for instance, using cylindrical coordinates results in convenient expressions for the regions with constant magnitude of the electric field, namely surfaces of constant radius, as opposed to surfaces of constant value of $x^2+y^2$ when using cartesian coordinates (provided the charged line has been placed along the $z$ axis).

Thus, in the spirit of the question, I would argue in this case that the form of the surfaces of constant |electric field| is related with the symmetry of the problem.

Question 2: if the answer to Question 1 is affirmative, what is the precise mathematical formulation of the relation between symmetry and conserved quantities?

My motivation is that in my research I am faced with complicated polynomial equations that greatly simplify after choosing a clever coordinate system, otherwise rather ad hoc (e.g. a polynomial in 2 894 566 terms which becomes a polynomial in 13 249 terms in the new coordinates). They allow me to naturally define regions of constant "something", and I would like to know if there is a symmetry in my problem that dictates this simplification, and what symmetry it is.

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You seem to be confusing some things. Noether's theorem tells us that continuous symmetries of the Lagrangian of a system are in one-to-one correspondence with conserved quantities (you can find many references and explanations of this on the web, including on this site. The canonical example is $$\text{spacetime translation invariance}\Leftrightarrow \text{energy-momentum conservation} $$ However, this statement is independent of the coordinate system to choose to work in. It is clear that this must be the case, because the physics shouldn't depend on which coordinate system you choose to work in, so neither can the conservation of certain quantities.

That being said, it is true that working in a coordinate system that has the same symmetries as the problem under consideration may be very helpful. This is because it often allows you to reduce the number of relevant variables you're working with to the bare minimum (i.e. to the number of degrees of freedom of the problem).

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  • $\begingroup$ Thanks for your answer. Indeed those are different things, but please notice that I didn't intend to interpret my problem as a verbatim instance of Noether's theorem, but rather to view it in analogy to the philosophy "invariance under some group ~ some conserved functional" (where the ~ could be =>, or <=, or <=> as in Noether's theorem). $\endgroup$ – Camilo Sarmiento Sep 18 '14 at 11:03
  • $\begingroup$ The sentence starting with 'It is clear...' deals with this assertion. The physics can never depend on the coordinate system. $\endgroup$ – Danu Sep 18 '14 at 11:04
  • $\begingroup$ On the other hand, I recall vaguely from my mechanics lectures (over 6 years ago) that the number of (independent) degrees of freedom of a system was intimately related to the number of conserved quantities. So one could argue that knowing conserved functionals would help in setting up a convenient system of coordinates (correct me if I'm wrong, please). $\endgroup$ – Camilo Sarmiento Sep 18 '14 at 11:11
  • $\begingroup$ Sure, the symmetries can serve as a guide, showing you which degrees of freedom are really the relevant ones, so that we can pick our system conveniently. This is as basic as the examples you quote in your text, and there's nothing deep there. $\endgroup$ – Danu Sep 18 '14 at 11:13
  • $\begingroup$ I'm thinking about the Kepler problem, where the Laplace-Runge-Lenz vector is conserved, but where the nature of the related symmetry is not evident using spatial coordinates. However, there is a system of coordinates that renders the problem as that of a freely moving particle in a $4$-sphere, and in which it becomes apparent that conservation of the LRL vector relates to symmetry under some rotations of the $4$-sphere. $\endgroup$ – Camilo Sarmiento Sep 18 '14 at 11:16

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