Classical non-relativistic newtonian mechanics can be derived from special relativity by letting the speed of light tend to infinity $c \rightarrow \infty $. But doing this for Einsteins famous formula $$ E = mc^2 $$ would yield $$ E = \infty $$ This would mean, that in classical mechanics the rest energy is infinite? Or how does one interpret this fact? Or should one rather look at $m =\frac{E}{c^2}$ and conclude that in classical mechanics the rest mass of a particle must be zero?
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In relativity, the absolute value of the energy of an object is important and measurable, because it contributes to the object's inertia by $E = mc^2$, as you wrote. But outside of relativity, energy and mass have no relation whatsoever. So it doesn't matter what the rest mass contribution to energy is, even if it's formally infinite -- we can just subtract it out, and nothing changes.
However, we do need to make sure that changes in energy agree. To do this, we need to use the full relativistic energy formula $$E = \gamma mc^2 = \frac{mc^2}{\sqrt{1-v^2/c^2}}$$ which applies to moving bodies as well as stationary ones. Performing a Taylor expansion in $v/c$, $$E = mc^2 + \frac12 mc^2 \frac{v^2}{c^2} + O(v^4/c^4)$$ which approaches $mc^2 + mv^2/2$ in the nonrelativistic limit. Subtracting out the constant rest energy contribution, we find $E = mv^2/2$, as we should.
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