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Can a law or a theory in physics be derived from solving and playing around with equations from other proven laws in physics?

That is if we take our current understanding of physics and try to combine mathematical equations, can the new equation that emerged be the basis of a new field or some new discovery?

Without prior observation and without experiment. That is an equation discovered by the laws of mathematics.

If it is possible how could a physicist achieve it? What should be his vision about an equation and where should be the starting point?

Can you give some examples of an equation that was built that way? i.e. what equations were derived mathematically?

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    $\begingroup$ Without observation or experimentation, how do your know that your newly-derived equations describe reality accurately? $\endgroup$
    – BMF
    Commented Sep 25, 2019 at 21:02
  • $\begingroup$ @BMF after the equation has been derived (which is build upon mathematical foundations that till this date had done a great job at describing reality by its logic) we can test the new equation by experiments. $\endgroup$
    – user144435
    Commented Sep 25, 2019 at 21:10
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    $\begingroup$ Einstein's theory of relativity predicted the existence of black holes before any were observed. Einstein originally considered that a "catastrophe", because black holes seemed too weird to exist, and worried the prediction might be evidence that his theory was wrong. Now, of course, we know they do exist as predicted by the theory. $\endgroup$
    – David Rose
    Commented Sep 25, 2019 at 22:08
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    $\begingroup$ No one, other than some contemporaries like Hilbert who were simultaneously racing toward a geometrical theory of gravity, had used Riemannian geometry in fundamental physics before, as far as I know. Einstein was not messing around with old equations. He was introducing fairly new mathematics into physics, based on deep physical intuitions, such as the equivalence principle. $\endgroup$
    – G. Smith
    Commented Sep 25, 2019 at 23:20
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    $\begingroup$ I encourage you not to imagine that playing with equations is likely to lead to a breakthrough. New physical ideas are more important. $\endgroup$
    – G. Smith
    Commented Sep 25, 2019 at 23:26

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Yes, it is possible to develop mathematical models that make predictions which have not been observed experimentally beforehand. There are many examples.

Perhaps one of the most famous resulted from the work of the English theoretical physicist P. A. M. Dirac, who developed the earlier work of Schrödinger and others to incorporate the principles of special relativity within quantum theory. The equations he developed had complementary pairs of solutions, each member of the pair having an opposite energy sign to the other. Subsequently, it was found that these corresponded to electrons and anti-electrons.

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I don't have the formalism but the conservation of energy was determined by showing, mathematically, that $\frac{\partial E}{\partial t} = 0$. It was checked more thoroughly before saying that E = constant.

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  • $\begingroup$ Why $\delta$ and not d or $\partial$? $\endgroup$
    – electronpusher
    Commented Sep 25, 2019 at 21:36
  • $\begingroup$ my bad, I wasn't thinking. I'll correct it now. Thank you for pointing that out. $\endgroup$
    – Natsfan
    Commented Sep 25, 2019 at 21:38

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