Questions tagged [mathematical-physics]
For questions regarding the mathematical aspects of physics.
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Math concepts introduced by physicists and made rigorous later
I am looking for mathematical concepts (*) which have been introduced by physicists in a non rigorous way (e.g. without a formal definition, without rigorous proofs of the results, etc.) and used to ...
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What was the motivation for Minkowski spacetime before special relativity?
If I understand correctly, the concept of a Minkowski space/metric was already known before Einstein's paper on special relativity. Was there any physical motivation for studying this type of metric ...
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How did Schrödinger do quantum mechanics with wave functions?
On my way to learn about the very beginning of quantum mechanics and its different formulations, starting with Heisenberg infinite matrices and Schrödinger's wave functions, I can really not find till ...
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What is Heaviside's version of Maxwell's equations?
I have read, in many places, statements like this:
Heaviside was able to greatly simplify Maxwell's 20 equations in 20
variables, replacing them by four equations in two variables. Today we
...
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Did Euler or did d'Alembert incorporate initial conditions into the solution to the 1D wave equation?
My question is: Who is responsible for incorporating the initial conditions into the one dimensional wave equation solution? References or technical information would be appreciated, especially ...
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Surface Integrals History
Can't find any information about who and when first used surface/surface area integrals. What was the original motivation? In it's modern form it depends on some relatively modern notations like ...
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Did Rayleigh or Ritz prove the Rayleigh–Ritz theorem?
The maximum eigenvalue of a real symmetric (or complex Hermitian) matrix is given as the maximum of the associated the quadratic form:
$$
\lambda_{\rm max}(A) = \max_{\|x\| = 1} x^*Ax. \tag{1}
$$
This ...
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What was the real need of divergence and curl operators?
As I'm advancing my study in Electromagnetism I'm getting introduced to more mathematical operators which are exclusively used in Electromagnetism and Fluid Dynamics only.
Let me try to explain myself ...
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Did physics resulted in invention of maths or vice a versa [closed]
Did physics resulted in invention of maths or Maths resulted in invention of physics ??
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Why is $T_{\mu\nu}$ the Standard Notation for the Stress-Energy-Momentum Tensor
My question is simple: why do we use $T_{\mu\nu}$ to denote the stress energy momentum tensor, and when was the concept of the stress energy tensor first (or roughly the first) introduced (and by whom)...
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Has there been debate between relationship of philosophy of mathematics and physics?
Did there exist and does there still exist a debate over which school of mathematical thought (i.e. formalism, logicism, intuitionism, etc.) had the most affinity or application for physics? In ...
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The abstraction of mathematics from physics
When and how did mathematics come to be abstracted away from the physical world?
At first, mathematics would originate in its simplest form of counting and addition as to keep track of certain ...
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Did Sophie Germain find a flaw in Euler's equations for elastic vibrations?
I am a playwright working on a play about Sophie Germain. When Sophie was competing for the prix extraordinaire to find effective formulas to describe the vibrations of elastic surfaces, she believed ...
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What was the first physical theory to be mathematically expressed?
Newton's theory of gravitation is expressed as a mathematical relationship between distances and masses. It can be deduced from Kepler's equations (which are also mathematical).
Are there even older ...
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Physical theories and Mathematics [closed]
I study pure mathematics. In pure mathematics, we begin from some axioms and obtain theorems. I am also interested in studying physics. I have some questions about the relationship between physical ...
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Has physics ever given a physical significance to a mathematically abstract idea?
Consider a fundamental concept in maths that was created to 'solve' a problem that simply couldn't be solved by any other approach (or maybe for some other reason). Now let's assume that this concept ...
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Best history of Maxwell and his equations
I've done my B.S. in Electrical Engineering as well as mathematics but I'd like to get a proper, or complete history of Maxwell and the history of his derivation of the equations and the newness of ...
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Complex potential in E227?
I have a reason to believe that Euler introduced the complex potential in his Continuation des recherches sur la theorie du mouvement does fluides, published in 1757. However, I am having hard time ...
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Math development and under-appreciation of Maxwell's Equations
Freeman Dyson expresses the opinion in his 1972 essay titled "Missed Opportunities" that Maxwell's equations could have played a much bigger role, one that is comparable to classical ...
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Who first proposed the idea of "resolution of the identity"?
Who first proposed the idea of "resolution of the identity" as used in the functional calculus of self-adjoint operators? Was it von Neumann?
In Japanese, it translates as "resolution ...
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Who established the current standard demonstration of Euler-Lagrange equation in calculus of variations?
Who established the current standard(*) demonstration of Euler-Lagrange equations in calculus of variations, that is, $\displaystyle\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial ...
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Who first used complex analysis to account for singularities in field theory?
In 1925, Frenkel wrote a paper titled Zur Elektrodynamik punktförmiger Elektronen, which used complex analysis to treat an electron as a point, and its corresponding potential function as an isolated ...
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Example of abstract math theory that was later found to be applicable to physical world?
In this video about the Banach-Tarski paradox the host stipulates that history is full of examples of abstract mathematical theories that were later found to be applicable to the physical world. Is he ...
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In which work was Gibbs' Inequality introduced?
Gibbs' inequality
$$-\sum\limits_{i=1}^n p_{i} \cdot \log{p_{i}} \le -\sum\limits_{i=1}^n p_{i} \cdot \log{q_{i}}$$
is such a popular thing that I cannot find where it was introduced.
My findings
I ...
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Were Kepler's Laws of Planetary Motion the first formal definition of an ellipse?
It seems to me that Kepler's Laws necessitate some definition of an ellipse in terms of a coordinate system. I am wondering whether Kepler's Laws mathematically defined what an ellipse is, or if he ...
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Examples of Physical Discoveries with no Counterpart in Mathematics
Throughout the history of mathematics and physics, there has been many examples where mathematics was discovered first prior to its application in physics. Consider $i=\sqrt {-1}$ as an example, among ...
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Who discovered the wave equation?
https://link.springer.com/chapter/10.1007/978-1-4684-5772-8_2
says:
Using Newton's recently formulated laws of motion, Brook Taylor
(1685–1721) discovered the wave equation by means of physical ...
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Notation for Christoffel symbols
In Christoffel's 1869 paper in which he introduced the Christoffel symbols on the 3rd and 4th pages, they are written as $\left[\substack{ij \\ k}\right]$ and $\{\substack{ij \\ k}\}$. The notation $...
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Background on the Stone-von Neumann theorem
I'm a mathematician.
I'm required to give a lecture on the Stone-von Neumann theorem. I already have all the mathematical details figured out, but I wish to make the lecture more interesting by giving ...
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How did we arrive at the rule of addition of vectors?
I wanted to ask about how they arrived to the rule of addition of vectors. How did they know that if we add the X's and Y's of two vectors they would get a third vector which has exactly the same ...
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What is the origin in the discrepancy between engineers' and physicists' notation of waves?
my question is very simple. Physicists use this notation in order to write a (for example) plane wave:
$$
\xi(z) = \xi^+ \mathrm{e}^{+\mathrm{i}kz} + \xi^- \mathrm{e}^{-\mathrm{i}kz},
$$
where $\xi^+$ ...
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Finding sources for "computers will become so powerful that special functions will become obsolete" as a zeitgeist
In Why are special functions special [Physics Today 54, 11 (2007); eprint], Michael Berry makes the following observations:
This continuing and indeed increasing reliance on special functions is a ...
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Is Hermann Weyl's book “Space, Time, Matter” (1923) on General Relativity still relevant?
I really liked Hermann Weyl's mathematical books and would like to get accustomed to general relativity from his perspective, but wonder if it's still relevant after almost 100 (!) years?
Can this ...
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Were pictorial notations like Feynman diagrams for integrals used before Feynman?
In the book Mathews, Walker: Mathematical Methods of Physics, Addison-Wesley(1969),
there is a pictorial notation of the solution found by Fredholm about an integral equation.p.304, p.305This circle ...
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Who made the first derivation of the angle to maximise projectile range, which turned out to be wrong?
I remember hearing once that the first "proof" that the angle to maximise projectile range gave the correct answer, 45 degrees, but was later found that the proof was wrong. I can't remember ...
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Origin of operators in quantum mechanics
Historically, where did the concept of operators in quantum mechanics come from?
How did people first understand that momentum operator should be of the form of $i \hbar \frac{{\rm d}}{{\rm d}x}$?
...
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Why isn't Feynman's path integral taught more widely and earlier in today's academic physics curricula?
Anyone who has studied Feynman's path integral will know that it makes quantum mechanics more like classical mechanics. A student who has learned about the Lagrangian will easily understand the ...
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How has the definition of a tensor today changed compared to its original definition?
On page 71 of The Absolute Differential Calculus by Levi-Civita, a very clear definition of a tensor is given in terms of how the coefficients of a multi-linear form transform, such that the product ...
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Who was the first person to describe turbulence in mathematical terms?
Here I found that:
Sixty years later, Russian mathematician Andrey Kolmogorov furthered
our mathematical understanding of turbulence when he proposed that
energy in a turbulent fluid at length $R$ ...
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History of PDE's in the 19th Century (part 2)
This is a follow up to this question: History of PDE's in the 19th Century
The question I have been given to answer is:
The history of partial differential equations in the 19th Century belongs ...
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History of PDE's in the 19th Century
I've been asked to write an essay on whether the work on PDE's in the 19th century belonged to applied or pure mathematics. Does anyone know of any useful sources I could use?
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History of complex analysis
Does anyone know of a good book on the history of imaginary numbers and complex analysis and its role in physics?
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Origin of the Heaviside function?
I have tried to find the actual origin of the Heaviside unit step function and could not. I've searched and searched, read one complete biography of Oliver Heaviside, skimmed another, but nowhere can ...
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A peculiar quote from Oliver Heaviside
The best result of mathematics is to be able to do without it.
The above is a quote by Oliver Heaviside, an electrical engineer and mathematician. What does the quote really mean?
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Was Von Neumann and Birkhoff's original formulation of Quantum Logic related with projective geometry?
I was looking at how did von Neumann and Birkhoff formulate their Quantum Logic formalism back in 1936. To solve some questions, I contacted via email a philosopher who studied this topic.
I thought ...
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Who originally worked out the magnetic field produced by a solenoid and toroid?
Although, it seems very easy to find the magnetic field produced by a solenoid or a toroid, all we got to do is to make a suitable an Amperian Loop and take the $\mathbf B$ out of the integral and so ...
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Best books/papers on Newton and his mathematical physics
In your opinion, what are some of the best books/papers on Newton and his work that accurately cover the connections between his geometric proofs in the Principia and his development of the calculus ...
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Who pioneered the study of the sedenions?
I found lots of background information about the discovery of both imaginary and complex numbers, and enough information about the first two types of hypercomplex numbers; quaternions and octonions (...
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Who introduced the "dagger"symbol as conjugate transpose in quantum mechanics?
The $\dagger$ symbol is often used in quantum mechanics,and also often in general mathematics to represent the conjugate transpose operation.For Hermitian matrices we can write $$A^\dagger=A$$Who ...
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Is Vedic Science true? [closed]
I somewhere read that ancient Indian science was so advanced even in 2000 B.C. (or may be even before this) that Indian scientist and mathematicians calculated the value of distance between sun and ...
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