"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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42 views

Generalized Gell-Mann Matrices

The Generalized Hermitian Gell-Mann Matrices (in dimension $d$ ) consist of the $h_k^d$, where $1\leq k \leq d$, and the $f_{k,j}^d$, where $1\leq k, j\leq d$. There should be $2^d -1$ matrices in ...
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19 views

Representing an operator in different bases

Say I have a random operator $\hat {A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ represented in the basis $\mathbf {e} = \left \{ \hat {e}_1, \hat {e}_2\right \}$ How should ...
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1answer
57 views

Is there a solution to this unidirectional wave equation, with initial value $v=f(x)$ and $x=t^2$

unidirectional wace equation: $$\frac{du}{dt}+c\frac{du}{dx}=0$$ The initial value $u=f(x)$ is given on the parabola $x=t^2$. Is there a solution to this problem, discuss why the solution is unique ...
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1answer
33 views

Coherent states - operator algebra problem with physics motivation

Motivation: I have a mathematical problem motivated by quantum field theory in physics. It should be quite easy to prove, but for some reason I can't do it. Intro: Let there be operators $\hat{a_i}$ ...
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60 views

How To Prove The following equation?

The equation arised in the paper:Exact and asympototic representations of the sound field in a stratified ocean.That is the equation(3.12) for solving the problem $$\Delta ...
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71 views

Atiyah-Segal axioms for TQFT [closed]

Could someone explain the importance of the Atiyah-Segal axioms for TQFT? Why is this studied by mathematicians, why is it interesting or useful?
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60 views

Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...
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1answer
8 views

Moment of Inertia of Rectangular Prism about one of its edges

Question: What is the moment of inertia of a rectangular prism with dimenions $l\times w\times h$ represented by $a\times b\times c$ about one of its edges? Are my bounds correct, and what is $r$? ...
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2k views

What are some math concepts which were originally inspired by physics?

There are a number of concepts which were first introduced in the physics literature (usually in an ad-hoc manner) to solve or simplify a particular problem, but later proven rigorously and adopted as ...
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1answer
48 views

Solve equation on mathematical physics

Show that $$\gamma_+ - \gamma_-=\frac{2\beta_0\beta}{\sqrt{(1-\beta_0^2)(1-\beta^2)}}$$ where $$\gamma_+=(1-\beta^2_+)^{-\frac{1}{2}} \ \mbox{and} \ \beta_+=\frac{\beta_0+\beta}{1+\beta_0\beta}$$ ...
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0answers
24 views

How to normalize the order of an expression?

Suppose that for $p, p'>0$, $i,j>0$, \begin{align} [y_{i,p}, y_{-j, p'}] = -\frac{1}{p}(1-q_1^{p})(1-q_2^p)\tilde{c}_{i,j}^{[-p]} \delta_{p,p'}, \end{align} where $\tilde{c}_{ij}^{[-p]}$ is some ...
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1answer
68 views

What does Wolframalpha's definition of “contravariant vector” mean?

http://mathworld.wolfram.com/ContravariantVector.html Wolframalpha offered a one line definition to contravariant vector which is a bit confusing to me Contravariant Vector: The usual type of ...
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1answer
30 views

Mathematical physics - Expand the a series of binomial [closed]

Expand the a series of binomial $\left(1-\frac{v^2}{c^2}\right)^{-\frac{1}{2}}$. Enter the first three terms. What is the ratio of third term to second if $\frac{v}{c}=0,1$?
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1answer
45 views

What is a “moment” in mathematics, and what does it mean?

This is a general question. I would like a better conceptual understanding of what a moment is, it's meaning, and it's applications (not just in probability). I already looked at Wikipedia, but I ...
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1answer
48 views

Getting started on Celestial Mechanics

I am searching for a math-accurate book on this subject, in particular for this topics: $n$-body problem, getting more detailed when $n=2$. Efeméride calculation. Orbit determination. Perturbation ...
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0answers
47 views

Gauss´s law proof “details”

I know that this question has already been asked multiple times but I´m still not getting on the mathematical details behind the answers... So I hope that this question doesn´t get closed; also I ...
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1answer
27 views

Regarding gauss law differential form

I have a big issue regarding the equality of integrands in gauss law. Given the integral form we have that $$\oint_{\partial\Omega}\vec{E}\cdot\vec{dS}=\int_{\Omega}\nabla\cdot \vec{E}dV={1\over ...
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1answer
51 views

Is the Wave function a “Smooth” function of the Potential?

Consider the Schroedinger equation in a spherically symmetric system. In the unit system under which energy is measured in Hartree and length in Bohr radius $a_0$, the schroedinger equation can be ...
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2answers
27 views

Algebra with proportionalities?

Do the rules of algebra apply when you’re working with proportionalities? For example, I know that $P \propto \rho$, where $P$ is pressure and $\rho$ is density, and $\rho \propto m$, where $m$ is ...
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2answers
32 views

Finding center of mass for tetrahedron

I am given a tetrahedron with the following points: $$\begin{align} P_1 &= (2,0,1)\\ P_2 &= (-1,1,1)\\ P_3 &= (1,0,2)\\ P_4 &= (3,1,4) \end{align}$$ and I am tasked with finding its ...
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1answer
50 views

Unit and dimension of angles

usually in physics (at least in the SI) angles are regarded as dimensionless. Is it possible to give a dimension (and a unit) also to angles and still have a system of units of measure as coherent as ...
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3answers
35 views

How to understand the equality about $(\mathbf{q}\cdot \mathrm{grad})\mathbf{q} $?

For the relation, $$(\mathbf{q}\cdot \mathrm{grad})\mathbf{q} = (\mathrm{curl}\mathbf{q})\times\mathbf{q}+\frac{1}{2}\mathrm{grad}|\mathbf{q}|^2,$$ is there any physics, geometry, or basic intuitive ...
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2answers
68 views

Otimization on a city with infinite many traffic lights.

Province Ave has infinitely many traffic lights, equally spaced and synchronized. The distance between any two consecutive ones is $1500m$. The traffic light stay green for 1.5 minutes, red for 1 ...
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30 views

Question on inclined plane with velocity of projection $u$ both up and down plane.

Show that for a given velocity of projection the maximum range down an inclined plane of inclination $\alpha$ is greater than up the plane in the ratio $$\frac{1+\sin(\alpha)}{1-\sin(\alpha)}$$ Let ...
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0answers
31 views

How define the entropy of heat equation?

Today, I report a paper about Ricci flow, I saw entropy. As I know, entropy is a physical term.And I know it is used to describe how far the system from heat death.But I don't know the equation of ...
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3answers
39 views

Integrate $dx$ over interval $a\le x \le b$ instead of just $b-a$

In the Wikipedia article on the wave function it's stated that the probability of a spin-less particle in 1D space being found in the interval $a\le x \le b$ at time $t$, where $x$ is the position, ...
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1answer
45 views

Integral for Biot-Savart

What strategy is the quickest for solving the following integral? Note: this integral is generated by the need to determine the magnetic field at a point along the z-axis generated by a wire of length ...
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0answers
50 views

How can we describe the diffusion of “things” injected into a fluid?

Let $d\in\left\{2,3\right\}$ and $\Omega_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $c\in\Omega_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto ...
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2answers
53 views

Elastic Strings and Simple Harmonic Motion

The Ceiling of a hall is 15m above the floor. A vertical elastic string of natural length 5m and modulus of elasticity 6N has one end attached to the ceiling and the other end attached to the ...
2
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0answers
80 views

Find the terminal velocity of skydiver using differential equations

I am studying differential equations in university and I came across this problem: A parachutist whose mass is $75$ kg drops from a helicopter hovering $4000\hbox{m}$ above the ground and falls ...
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26 views

Hamiltonian system - find gradient of vector

A particle velocity in $(x_1,x_2)$-plane is called $p=(x_1',x_2')$. The particles total energy can be written as $$H(x,p)= \frac{|p|^2}{2} + v(x).$$ A particle that moves a long the orbit $X(t)$ and ...
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2answers
60 views

Solving time needed to travel a given distance, given simulated (not real physics) properties of acceleration of object

For a small personal project I'm looking at travel time of objects in a video game called EVE-Online. I need to calculate time it will take object to travel from stand-still, constantly accelerating, ...
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1answer
58 views

How can we describe the evolution of a density “injected” into an incompressible Newtonian fluid?

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. The evolution up to time $T>0$ of an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity ...
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1answer
32 views

How to verify correctness of my numerical method for Allen-Cahn equation?

\begin{equation}\label{Parabolic} \frac{\partial \phi(\mathbf{x},t)}{\partial t} - \Delta\phi(\mathbf{x},t)+\frac{f(\phi(\mathbf{x},t))}{\epsilon^2}=0 \end{equation} \begin{equation}\label{boundary} ...
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58 views

How to get asymptotic form of the integrals with special functions?

I got difficulty when I try to plot I(x) for $m=1$ and $t=0.2$. The questions is how to get the asymptotic form of the following integral? $I(x,t)=\int_{0}^{\infty} \frac{f(y)}{2 \sqrt{\pi t}} ...
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2answers
37 views

Integral of an operator

In quantum mechanics we know that if $q$ corresponds to a complete set of parameters characterizing a quantum system, then the state vectors $|q\rangle$ satisfy the following identity: $$\int ...
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2answers
41 views

Prove that if a particle travels a unit of distance in one unit of time starting and finishing in repose it has in a moment an acceleration $\ge 4$

How would you solve this problem? Prove that if a particle travels a unit of distance in one unit of time starting and ending whith velocity $0$ it has in a moment an acceleration $\ge 4$ (positive ...
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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 ...
2
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0answers
30 views

Lorentz transformation proof

For an occurence, we can choose coordinates ct and x (calculating both time and space in length) ct and x thereby take on the form of a two dimensional vector. Show that the same coordinates in S' ...
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0answers
50 views

Affine geometry book for physicist

I'm looking for a textbook to help me with understanding the geometry of Galilean relativity and the Galilean group. The reason is that I tried going through V.I. Arnold's Mathematical Methods, but ...
2
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0answers
37 views

Numerical solution of the stationary Navier-Stokes equations

Let $d\le 3$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I'm considering an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity $\nu$. In this case, the stationary ...
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0answers
106 views

Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists ...
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1answer
42 views

Manifolds as Homogeneous Spaces

With very little effort one can, for example, show that $S^n$ can be written as a homogeneous space as $S^n\cong G/H$, where $G$ is the group of all rotations in $\mathbb{R}^{n+1}$ about the origin ...
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5answers
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find distance traveled of object that is slowing down

I apologize in advance because I feel like this problem is fairly simple but I can't seem to figure out what the formula would be. Essentially, if I had an object that were traveling 60cm/second, but ...
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1answer
28 views

Find a coordinate to make a vector perpendicular to other

I have two vectors $(2,5,7)$ and $(-1,x,3)$. I need to select an $x$ in a way that the second vector will be perpendicular to first. The angle of two vectors is described as $cos(\phi)=\frac{a \cdot ...
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1answer
29 views

Finding a composite solution to an ODE (boundary layer problem)

Given $\epsilon \frac{d^2u}{dt^2}-a(t)\frac{du}{dt}+b(t)u=0$, where $a(t)>0$, $u(0)=1$, $u(1)=1$, and assuming that the boundary layer is at $t=1$, and the boundary layer variable is ...
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2answers
21 views

Constant of motion that is not the hamiltonian

Given the lagrangian $L(x,\dot x)=\frac12 (\dot x_1^2+\dot x_2^2)-\frac k 2(x_1-x_2)^2$, we know that its hamiltonian is a constant of motion. (See here) Is there another function $f$, not of the ...
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0answers
38 views

Nonlinear Schrödinger Equation

I have to find equation and starting condition to solve Nonlinear Schrödinger Equation with periodic edge condition. This method should control the propagation of fiber optical signal. In details I ...
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1answer
35 views

Shape Of A Blimp.

Was playing around with solids of revolution, the shape given by rotating $y=\sqrt{\sin x}$ about the $x-$axis seems to resemble a blimp. The only thing I can find out about the natural shape of ...
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35 views

Rope wrapped around pole Friction

A rope is wrapped $M$ whole turns round a cylindrical post, the two ends of the rope going in opposite directions. The coefficient of friction between rope and post is $0.25$. It is desired that by ...