# Tag Info

33

Physics cannot help giving physical significance to things. But, yes, the first item on your list should have been Lie Groups. Developed as a mathematical "sudoku" game generalizing rotations, in the late 19th century, by Sophus Lie, Felix Klein, Friedrich Engel, Henri Poincare, and, as associated applied mathematical structures by James Joseph ...

27

Not quite. Minkowski had the idea of representing special ralativity as geometry in 1907 under the direct influence of Einstein's 1905 paper, and he developed it in Raum und Zeit (1907) and Zwei Abhand lungen über die Grundgleichungen der Elektrodynamik (1909). See Minkowski on MacTutor. Before that only classical "spacetime" appeared, and only superficially....

21

fractal The so-called Cantor set was described by Georg Cantor, 1884 (or H. J. S. Smith, 1875?) Sets with "fractional" dimension were described by Felix Hausdorff, 1918 Investigated thoroughly by A. S. Besicovitch, 1930s - 1950s But these were only mathematical abstractions. Benoit B. Mandelbrot, 1960s and later claimed relevance of these sets in ...

18

This question was actually discussed on this site several times, for example here: When was the vector notation in physics and other sciences first introduced? It indeed looks strange to modern people that this simple idea came so late. Maxwell never uses vectors in his Treatise on electricity and magnetism, which makes his notation somewhat clumsy. In fact ...

17

I will focus on the history before the Yang-Mills paper. The first harbinger was the introduction of the scalar potential for the gravitational field by Lagrange in 1773. In 1864 Maxwell introduced vector potential for the magnetic field, which can be interpreted as a connection form, making magnetic theory the first gauge theory in hindsight. In the famous ...

17

I'll focus on the geometry of Yang-Mills theories specifically, but as Conifold's answer points out, gauge theories were studied geometrically long before the work of Yang and Mills. The foreward to volume 5 of Atiyah's collected works (on gauge theories) contains some historical comments on this from the mathematics side. You can read it here. This is ...

15

This seems a bit of a naive question to me. Einstein had been working on this problem for several years (starting as early as 1907), and had developed much of the physics by 1912. He greatly struggled to find the correct mathematical formulation of his theory, finally discovering the necessary tools from differential geometry during a collaboration with the ...

15

The following quote is from C. N. Yang, delivered at a 1979 symposium dedicated to the geometer Chern: "When I met Chern, I told him that I finally understood the beauty of the theory of fibre bundles and the elegant Chern​–Weil theorem. I was struck that gauge fields, in particular connections on fibre bundles, were studied by mathematicians without ...

11

I do not agree with the statement, that the lack of mathematical rigor is a major reason for not teaching the path integral formalism in quantum mechanics. The common physicist is normally not interested in complete mathematical rigor, as long as the concepts make sense from a physical point of view and produce the right results. A good example of this is ...

11

It is a strange idea that scientific laws can be only expressed with algebraic means. The Greek did discover several scientific laws. The oldest one is attributed to Pythagoras himself: it relates the length of the string to its pitch. This seems to be the oldest scientific law ever discovered. More laws were discovered in Hellenistic times: the law of ...

10

Potential theory (Green's formulas, Green's function etc.) was discovered by George Green who was doing physics. His work was called "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". Laplace equation was first written in a paper on Saturn rings. Eigenvalues, eigenvectors and adjoint operators were ...

9

You do not say what field of mathematics you are working in, and perhaps there are signs of separation there. Overall however, lively interaction between mathematics and physics is alive and well. John Baez has a blog This Week's Finds in Mathematical Physics, that is full of contemporary examples of it, so does Terence Tao. Nature, a leading journal in ...

9

First, to echo and elaborate on previous answers, Hilbert only appeared on the scene during the last few minutes of the last act, so to speak. Einstein made his first step towards the General Theory of Relativity (GR) in a 1907 paper "Relativitätsprinzip und die aus demselben gezogenen Folgerungen" (On the Relativity Principle and the Conclusions Drawn from ...

9

Group theory. A story I heard (perhaps enhanced over the years): In 1910, Princeton was about to begin a major in physics. The physicists sat down to decide what would be required for those majors to study. As part of the exercise, they went through a list of mathematics courses to determine which would be used in physics. They came to "Group Theory", ...

9

Ancient Greeks painstakingly avoided negative numbers, although they could have come handy in astronomical calculations and number theory, among other places. Brahmagupta in Correctly Established Doctrine of Brahma (c. 630 AD) uses the language of "fortunes" and "debts", which suggests the merchant origin of the negative number concept, but that remains a ...

9

This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations. Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

8

Minkowski space time was considered by mathematicians before Einstein and before Minkowski. Of course the name "space-time" was not used. The reasons were purely mathematical, not physical. The most important application was the Klein model of the hyperbolic geometry (non-Eucludean geometry of Bolyai and Lobachevski). More presicely, Klein considered the ...

8

Calculus was originally formulated in terms of infinitesimals. Hundreds of years later, a second formulation was found in terms of limits. There were originally some doubts about whether the version using infinitesimals was logically OK, but these doubts were cleared up by Robinson and others ca. 1961. The Leibniz notation $\int f(x) dx$ was invented in the ...

8

From where did the concept of operator in quantum mechanics came, historically? This was a gradual development started by Heisenberg's insight. He invented (infinite) matrices (without any prior knowledge of matrix multiplication). This was followed by by Born, Jordan and Dirac. Dirac's book Principles of quantum mechanics (1930) explains in great detail ...

8

Ampère did. Ampère's force law (not to be confused with one of Maxwell's equations, "Ampère"'s circuital law, which Ampère never wrote down, as Ampère didn't deal with the field concept), written in modern vector notation, gives the force that current elements $I_1 d\vec {\ell }_1$ and $I_2 d\vec {\ell }_2$ exert on one another to be: d^2\vec{F_{21}^A} = -...

8

Well, the history is not too simple, even though you have traced some correct events. It was Albert Einstein on November 4th, 1915 in his paper "Zur allgemeinen Relativitätstheorie" (Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1915)) in which he officially put "the gravitational field" (English translation) in connection (pun ...

8

I am afraid nobody noticed it, because nobody could have noticed it. A deviation is only a deviation when one has something that it is a deviation from. To "notice" the Magnus effect one has to operate from a theory that predicts a parabolic trajectory, and be able to separate the subtle deviations caused by spin from other deviations. Hall even wrote that ...

8

This probably refers to Galileo's "derivation" of Tartaglia's observation that cannon balls achieve maximal range when fired at 45°. Tartaglia's theory of projectile motion was wrong, he assumed that fired balls follow a line segment going up, then an arc of a circle to change direction, and finally fall vertically down, but the observation was ...

8

Abelian and non-abelian group theory -> quantum chromodynamics Noneuclidean geometry -> general relativity Sorry, I cannot write you any equations as examples.

8

I don't know if this can be counted for as physics, but to my knowledge the radon transformation was mostly something mathematicians thought about without any application. Now, it is widely used (and necessary) to transform images in tomography. https://en.wikipedia.org/wiki/Radon_transform

7

Essentially the same question was asked on Math Overflow: https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653 , and many examples were given. Such examples are really abundant. It is more difficult to name a major mathematical theory which did NOT eventually find an application in the real world. Philosophers who think ...

7

I would like to add a slightly more technical answer to supplement the other (although it is a bit late). It may be helpful, I think, and it touches a broader issue which interesting. It's a conflict of history of teaching versus history of the mathematical prerequisites for the material. First of all, supposing the topic material is taught early on, how ...

7

I would say that there is no good book which satisfies your description. The Book of Bottazzini and Grey mentioned in the comments is OK, but it certainly does not cover the role of complex analysis in physics. A better book is J. Dieudonne, Abrégé d'histoire des mathématiques 1700–1900, Hermann, Paris, 1978, in 2 volumes, both volumes have chapters on ...

7

Systematic study of turbulence originates with a series of experiments conducted by Osborne Reynolds starting in 1870s. His mathematical theory was developed in On the dynamical theory of incompressible viscous fluids and the determination of the criterion (1895). However, already in 1883 he described a classical experiment with a jet of dyed water at the ...

7

There is no point in giving examples on the second question, because for most truly groundbreaking results this time is really short. On the first question, one can mention almost all results of Archimedes and Apollonius, which had almost no development in antiquity, and during the middle age there were no people who understood them. (People read and ...

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