# Tag Info

26

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....

19

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 ...

16

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 ...

16

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 ...

13

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 ...

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 ...

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

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

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 ...

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

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: $$d^2\vec{F_{21}^A} = -... 7 First, to echo and elaborate on previous answers, Hilbert only appeared on the scene during the last few minutes of the last act, so to speak. Einstein made his first step towards the General Theory of Relativity (GR) in a 1907 paper "Relativitätsprinzip und die aus demselben gezogenen Folgerungen" (On the Relativity Principle and the Conclusions Drawn from ... 7 Systematic study of turbulence originates with a series of experiments conducted by Osborne Reynolds starting in 1870s. His mathematical theory was developed in On the dynamical theory of incompressible viscous fluids and the determination of the criterion (1895). However, already in 1883 he described a classical experiment with a jet of dyed water at the ... 7 Euclid wrote an Optica (300 BC) — surely “Visual rays proceed in a straight line indefinitely” ranks with the best physical laws. So did Ptolemy (160 AD), and Hero wrote a Catoptrica (50 AD). Aristotle knew the principle of virtual work. Jim Holt’s physics don’t seem to fare much better than his math. 6 According to Truesdell [1954]: (p. xliii:) As far as I can ascertain, it is Euler [1750, p. 196] which contains the first general statement of “Newton’s equations”. (p. xlii:) The axioms which Euler asserts “include all principles of mechanics” are$$ 2M\frac{d^2x}{dt^2}=P,\qquad 2M\frac{d^2y}{dt^2}=Q,\qquad 2M\frac{d^2z}{dt^2}=R.  (...) Anyone who ...

6

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", ...

6

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 ...

6

Discovery of non-Euclidean geometries in 19th century, it took about 40-50 years for them to get accepted.

6

I think because Richard Courant left Germany in 1933, earlier than many Jewish escapees. He did not lose his position due to being Jewish, as his previous service as a front-line soldier exempted him; however, his public membership in the social-democratic left was reason enough (for the Nazis) for dismissal. See the dates of the editions of Methoden ...

6

The equation in the picture is the Navier–Stokes momentum equation in convective form. The part inside the O looks like a cross-section of (the velocity field of) a flow between two rotating cylinders, mentioned in Ladyzhenskaya's Mathematical Analysis of Navier-Stokes Equations for Incompressible Liquids, 6.4, among other places, the so-called Taylor-...

6

On the non modern form of "vector"... from mathword The word VECTOR (which, like the word vehicle, derives ultimately from the Latin vehĕre to carry) was first a technical term in astronomical geometry. The OED’s earliest entry is from a technical dictionary of 1704: J. Harris Lexicon Technicum I. s.v., "A Line supposed to be drawn from any Planet moving ...

5

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 ...

5

One can probably say that the relevant parts of algebra were "known to experts", rather than "well-known", and the relevant parts of functional analysis did not exist at the time, see Moore's Axiomatization of Linear Algebra: 1875-1940. Even finite dimensional matrices were not exactly standard teaching item yet, although Cayley gave the definition of ...

5

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 ...

5

Lagrange (1736-1813) in 1777, followed by Laplace (1749-1827) in 1782, was the first to introduce the scalar gravitational potential.1 Lagrange's paper, Remarques générales sur lemouvement de plusieurs corps qui s'attirent mutuellement enraison inverse des carrés des distances, was read at the Academy of Berlin on 20 October 1777.2 So Weinberg is correct ...

5

The standard book about Newton's life is Never at Rest by Richard Westfall. On my opinion it is a very good book, it covers his life in great detail, and gives a general overview of his activities (not only in physics) but in astronomy, history, theology, alchemy, and as the Mint administrator. On physics, the latest English translation of Principia by Cohen ...

5

From an article by Freeman Dyson: In the year 1865, Maxwell published his paper 'a dynamical theory of the electromagnetic field' ... we, with the advantage of hindsight can clearly see that his paper was the most important event in the 19th C in the history of the physical sciences ... but the importance of Maxwells work was not obvious to his ...

5

Since I have no shame, I screen-shotted the reference that J.G. found:

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