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Noether's theorem states that if the functional $J$ is an extremal and invariant under infinitesimal transformation,

$$ t' = t+ \epsilon \tau + ...,\tag{1}$$ $$ q^{\mu'} = q^{\mu} + \epsilon \zeta^{\mu} +... .\tag{2}$$

Then the following conservation law holds:

$$p_\mu \zeta^{\mu} -H \tau - F = const.\tag{3}$$

What I am curious about is the forms of $\tau$ and $\zeta$. We can find strange conservation laws for any system so long as we find the right $\tau$ and $\zeta$. For instance, a free particle will admit to energy conservation when $ \tau = 1$ and $\zeta =0$. Also, a strange conservation for a damped oscillator can be found if $\tau =1$ and $\zeta = \frac{-bx}{2m}$.

My question is: What does the form taken by $\tau$ and $\zeta$ tell us about the system (or the laws of nature)? It seems to truly matter. Gallilean transformations ($\tau =0$, $ \zeta =t$) gives us regular old momentum conservation... Lorentz transformations do the same for relativity... But what do $\tau$ and $\zeta$ mean? What do they say? Why are they what they are?

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    $\begingroup$ Where did you see Noether's theorem stated in this strange looking way ? $\endgroup$
    – Antoine
    Commented Apr 6, 2017 at 18:30

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As a partial answer let us mention that (i) in a Hamiltonian formulation and (ii) for purely vertical infinitesimal quasi-symmetry transformations

$$\delta z^I~=~\epsilon \zeta^Iz^I,\tag{A}$$

meaning that OP's $\tau=0$ is assumed zero,

$$ \delta t~=~\epsilon\tau~=~0, \tag{B}$$

then the vertical generator

$$\zeta^I~=~\{z^I,Q\}\tag{C}$$

happens to be the Poisson bracket between the corresponding phase space coordinate $z^I$ and the conserved Noether charge $Q$. See e.g. this Phys.SE post for details.

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