# Tag Info

28

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....

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 ...

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

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 ...

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

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 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 ... 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 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. 7 In a now-deleted comment, Consigliere ZARF listed a number of papers published in Zeitschrift für Physik in the late 1920's that used this notation. The earliest was Pascual Jordan's 1927 "Über eine neue Begründung der Quantenmechanik", using the notation on pp.816-817; with about 10 other papers published in the following few years, all in the ZfP, all ... 7 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 ... 6 Let me try to summarize what was said in the comments and add something. (I am a teacher of mathematics, not physics, but this is close). First of all not all students study Lagrangian mechanics at an early stage. It requires much more mathematical sophistication than most undergraduate students have. (I am talking of US here). Second, Feynman's integral ... 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 ...

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