Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

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11
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91 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\...
11
votes
0answers
253 views

What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
10
votes
0answers
250 views

Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, ...
9
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0answers
451 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
8
votes
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77 views

Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
8
votes
0answers
137 views

Closed form of $\int_0^1 \tan(\gamma\sqrt{1-x^2}) dx$

Some context: I'm studying the problem of nonperturbative pair creation from strong fields in quantum electrodynamics. For certain time dependent electric fields I can get some information about the ...
7
votes
0answers
138 views

Stokes' Theorem and Vector Fields with Jump Discontinuities

What are the continuity requirements on a vector field $\boldsymbol{A}$ such that Stokes' theorem, $$ \iint_S\nabla\times\left[(\boldsymbol{\hat{x}}\cdot\nabla\phi)\boldsymbol{A}\right]\cdot d\...
7
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400 views

Renormalization for mathematicians

Can someone explain to me the processes of renormalization and regularization used in quantum field theory and similar fields in a way that a pure mathematician might make sense of it? Is there a ...
7
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296 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: $$\int_{0}^{2\pi}\frac{1}{\sqrt{a^2-b^2\cos^{2}2\phi}}\exp{\left(-\frac{(x-c\cos\phi)^2}{a+b\cos2\phi}-\frac{(y-d\sin\phi)^2}{a-b\cos2\phi}\...
6
votes
0answers
88 views

Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
6
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201 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
6
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0answers
130 views

On the Visual Manifestation of Curves in Nature

All sorts of curves are useful in modelling and describing phenomena we observe. Trig functions, logarithms, exponentials, polynomials, hyperbolas, circles, and so forth are all very useful in this ...
6
votes
0answers
469 views

Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
6
votes
0answers
150 views

Solving numerically the equation of motion of D7 brane perturbation

I want to solve this equation $$ \partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0 $$ numerically. I know that ...
6
votes
0answers
110 views

Analytic caustics for 3D objects

Is it possible to efficiently calculate caustics for a given 3D object, like a torus, or a cube? To be more precise: let's assume that we have a 3d torus, resting on a 2d plane and a single light ...
5
votes
0answers
117 views

Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm (pragmatically)...
5
votes
0answers
79 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
5
votes
0answers
91 views

Show that the profile of the hill is a cycloid

I'm struggling with this problem: From George Simmons_ Differential Equations At sunset a man is standing at the base of a dome-shaped hill where if faces the setting sun. He throws a rock ...
5
votes
0answers
144 views

Earnshaw's theorem

Proposition Suppose $U\colon\Omega\to\mathbb R$ is a non-constant harmonic function, i.e. $U\in\mathcal C^\omega$, i.e. analytic, and $\Delta U=0$, where $\Omega\subseteq\mathbb R^n$ is a region. ...
5
votes
0answers
105 views

A light beam enters a closed room. What is the maximal number of reflections?

I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and the light beam enters the ...
5
votes
0answers
67 views

Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray diffraction....
5
votes
0answers
99 views

Second law of thermodynamics as a theorem about state space evolution

I once saw a mathematical explanation of the second law of thermodynamics. The statement was something like this: there is a mapping $f$ from the set of thermodynamic states $S$ to itself, and a ...
5
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0answers
951 views

Publication date for Michael Spivak - Physics for Mathematicians II?

I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
4
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0answers
38 views

The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
4
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58 views

Calculate the resistance between 2 adjacent nodes on a shape using graph theory

In shapes like regular octahedron or dodecahedron, how can Graph Theory be used to calculate the resistance between two adjacent vertices? All edges are assumed to have unit resistance. Is there ...
4
votes
0answers
40 views

A PDE problem on the Heat-Like differential equations

I came across the following questions in part of my work: Consider the Heat-Like equation of the form $\frac{\partial u}{\partial t}=\hat{H}u + f(x,t)u + g(x,t)$ where $\hat{H}$ is a Sturm-Liouville ...
4
votes
0answers
49 views

Integral of triple vector product

I'm following a Electrodynamics course and I'm currently stack in the following calculation: $\int{ \frac{ \vec{n} \times (\vec{n}-\vec{\beta}(t)) \times \vec{\beta'}(t) }{(1-\vec{n} \cdot \vec{\beta}...
4
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0answers
82 views

Fourier transform of $\frac{1}{r}$ (Coulomb potential)

When calculating the Fourier transform of a function of the form $f(\vec{r}) = \frac{1}{4 \pi \left|\vec{r}\right|}$, one encounters the problem that the resulting integral does not converge, i.e. $$\...
4
votes
0answers
378 views

Rolling a ball into a cone; what should the forces overall be?

Suppose there is a cone, with the apex pointing down, and the top of the cone at height $h$, apex half-angle $\psi$, ball of mass $m$, and the initial velocity into the cone (completely horizontal, ...
4
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0answers
67 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ~~~~...
4
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0answers
52 views

C. Neumann passage in Latin from *Annali di Matematica Pura ed Applicata*

Neumann, Carl. “Theoria nova phaenomenis electricis applicanda.” Annali di Matematica Pura ed Applicata 2, no. 1 (August 1868): 120–128. doi:10.1007/BF02419606. p. 121: Nova introducitur ...
4
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0answers
302 views

Developing solution for electrodynamics problem

Although it is a question related to physics, since the point it really matters is its mathematical aspect, I post this question on MSE. There's an additional exercise from Introduction to ...
4
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0answers
142 views

Where do I go from Linear algebra past Calc III to try to learn complex physics (relativity and quantum group theory)?

I'm mainly a programmer, but I have a love for Mathematics that's been, well, insatiable. I've had my eye on learning Quantum Groups and Relativity, but I want to stay in something I can do with ...
4
votes
0answers
124 views

Who established the word “ Degree of freedom ” in statistics?

I wonder who is the first one that established and applied the word : "degree of freedom" in statistics? Why he/she need degree of freedom in the calculation of many statistical values?
4
votes
0answers
178 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int w(R)\exp[-tw(...
4
votes
0answers
137 views

Gaussian Integral with non-polynomial exponent

I am currently trying to evaluate this Integral: $$\int\limits_{u_0}^{u_1} \exp\left[-\angle(H(u),N)^2\right]du$$ Where $H(u)=:\overrightarrow{a}+(\overrightarrow{b}-\overrightarrow{a})*u-\...
4
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0answers
87 views

Why are linear functions the natural analogue of exponential functions in a tropical semiring?

I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says It's not obvious how to interpret the ...
3
votes
0answers
107 views

generating functions for catastrophe theory

I am studying Thom's theorem in catastrophe theory and am having a hard time understanding what the "generating functions" actually do. How exactly are they used to classify generic caustics? The ...
3
votes
0answers
29 views

Dissipation term in wave equation

If we're given a string with mass density $\rho$ in units $\frac{M}{L^3}$ with constant cross-section $A$, tension $T$ in units $\frac{F}{L^2}$, and whose length is $L$; and then we assume that the ...
3
votes
0answers
46 views

Why don't more celestial bodies exhibit higher-order rotations?

It is well known that the Earth spins on its axis. It is also well known that the Earth's axis also precesses, i.e. spins around a secondary axis, much more slowly. Less well known is that we have ...
3
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0answers
21 views

Finding the work done, given a function.

A heavy metal 2 pound bucket initially is filled with 10 pounds of paint. Immediately after it is filled, it is pulled up at a steady rate to the top of a building 30 feet high. While being pulled, ...
3
votes
0answers
30 views

Which functions can be the resistance of a network?

Say we have $n$ resistors, with unknown resistances $r_1,\ldots,r_n$. We build a network using these, along with any finite number of other resistors of known resistance. We then measure the ...
3
votes
0answers
52 views

Davenport's Q-method (Finding an orientation matching a set of point samples)

I have an initial set of 3D positions that form a shape. After letting them move independently, my goal is to find the best rotation of the original configuration to try to match the current state. ...
3
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0answers
29 views

Arbitrary factors for the (modified) Mathieu equation

I am currently confronted with a physical equation that, after a fair amount of reworking, can be recast in the form of the modified Mathieu equation : \begin{equation} y(x)'' - (a - 2q \cosh(2x)) y(...
3
votes
0answers
38 views

The behavior of the 3D wave equation close to the origin

The general solution to the three dimensional wave equation is \begin{equation}u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r} \end{equation} where $F$ and $G$ are arbitrary functions. I want to ...
3
votes
0answers
103 views

Differentiation under the integral sign for an electrostatic field

Let $\rho\in C(\bar{D})$ be a continuous function on the compact set $\bar{D}$ and let us define $$\mathbf{E}(\mathbf{x}_0):=k\int_D\frac{\rho(\mathbf{x})}{\|\mathbf{x}_0-\mathbf{x}\|^3}(\mathbf{x}_0-\...
3
votes
0answers
107 views

Upper bound on the Lipschitz constant of entanglement entropy

I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full ...
3
votes
0answers
62 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page (...
3
votes
0answers
144 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) \...
3
votes
0answers
52 views

Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...