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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0answers
249 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
390 views

Nonabelian global symmetries, $SO(N)$ symmetric theory of $N$ scalar fields.

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda (\Phi^a ...
10
votes
0answers
245 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, ...
8
votes
0answers
405 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 $V(x)$ ...
7
votes
0answers
356 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
votes
0answers
286 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
6
votes
0answers
80 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
votes
0answers
192 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
votes
0answers
108 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 ...
6
votes
0answers
86 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
428 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
109 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
99 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 ...
5
votes
0answers
66 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
86 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
128 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
102 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
140 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 ...
5
votes
0answers
63 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 ...
5
votes
0answers
96 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
votes
0answers
872 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
votes
0answers
40 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
45 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 ...
4
votes
0answers
315 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
votes
0answers
66 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
votes
0answers
49 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
votes
0answers
273 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
votes
0answers
135 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
135 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 ...
4
votes
0answers
82 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
16 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)) ...
3
votes
0answers
36 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
96 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 ...
3
votes
0answers
104 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
51 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
113 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 ...
3
votes
0answers
73 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. ...
3
votes
0answers
91 views

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
3
votes
0answers
55 views

Mathematical interpretation of pressure (gradient)

I'm having some problem with the following. Usually, the pressure of some mass on an area A is defined by $$P=\frac{F}{A}$$ where $F$ is the force of the mass exerted due to gravity. However, here ...
3
votes
0answers
43 views

Expectation value in a Quantum derivation

I'm reading a physics paper (John Bell's 1964 paper on the EPR paradox if anyone is physics-curious) and I'm having an issue following his derivation. It's the probability distribution stuff -not ...
3
votes
0answers
107 views

What is $\int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \; $?

Given $F[u]$ and $G[v]$ are functionals of a real-valued function, what is $$ \int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \quad ? $$ I have encountered such expressions for ...
3
votes
0answers
100 views

Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$ \int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t) $$ for $(-\frac{1}{2}T< t < \frac{1}{2}T)$ is useful in photon ...
3
votes
0answers
543 views

Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
3
votes
0answers
121 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?
3
votes
0answers
175 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 ...
3
votes
0answers
100 views

Wilson lines, boundary condisions, surface defects of TQFTs

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
3
votes
0answers
456 views

Equation of motion for a wobbling disc

While looking at a frisbee the other day, I suddenly had a question. Suppose (in free space) you set a disc-shaped object spinning, and then you impart a sudden force perpendicular to the spinning ...
3
votes
0answers
152 views

Check my solution - Modelling of a spring with Differential Equation

I am doing some work with differential equations. I have solved the following problem but am uncertain if I'm doing it correctly. Could someone look over it for me and check if I'm doing something ...
3
votes
0answers
121 views

Stability of Orbits in Schwarzschild Spacetime

I'm looking at geodesics in the Schwarzschild geometry, and have come up against something I cannot prove. I've shown that for a particle moving on a geodesic with $r$ constant and $\theta=\pi/2$ we ...