"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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16 views

Dividing before and after integration give different results

I'm having a physics exercise, but the question is more of math. Assuming I have the following constants: $m_1, m_2, \alpha, V_0$ and two variables: $v, t$. (v as velocity). I reach the following ...
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30 views

Difference between 'principal of indifference' vs 'the assumption of equal a priori probabilities'?

Is there a difference between the "principal of indifference" and "the assumption of priori probabilities" and if so what? If there is no difference why the use of two different terms? EDIT I have ...
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1answer
58 views

Question: Fourier transform

I need to calculate the (distributional) Fourier transform of $$ f(x) = \frac{x^2}{x^2+1}. $$ I unsuccessfully tried to find a differential equation for $f$ in order to solve the Fourier-transformed ...
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13 views

Reference request: 2D conformal field theory and the honeycomb lattice

Would anyone know what is meant by "conformally invariant" functions defined on the plaquettes of the honeycomb lattice (ie the function is defined on the vertices of the dual tringular lattice)? ...
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44 views

How to compute Casimir elements of $g \otimes g$?

Let $g$ be a Lie algebra. How to compute Casimir elements of $g \otimes g$? I am asking this question because in the book a guide to quantum groups, page 80, there is an equation $r_{12} + r_{21}=t$, ...
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17 views

What is the difference between a first order compartment and diffusion

Biologists use Compartment models to represent the flow and storage of fluids in an animals body. In tissue (like muscle) the diffusion of blood is more accurately represented by the diffusion ...
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49 views

What is the mathematical understanding behind what physicists call a gauge fixing?

I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an ...
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1answer
55 views

Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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62 views

How to use Runge-Kutta 4th order method without direct dependence between variables

Following equation shall be solved using Runge-Kutta method of 4th order: $$ \frac{\partial E(z,t)}{\partial z} = \frac{\partial P(t)}{\partial t} $$ $P(t)$ is given as an array, so that the ...
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33 views

Rewrite a Lagrange function to Euler-Lagrange equation in polar coordinate

If we have a Lagrange function in the form $L(p, q) = \frac{p^2}{2} + q^2$, how could it be re-written as a form of Euler-Lagrange equation in polar coordinates ?
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17 views

Transformation of Graph

Hello all, I tried to solve this transformation and my answer was $-(x+3)^3+2$ my reason for thinking: reflect cubic power, shift to the left $3$ units, move up $2$ units. $-(x+3)^3+2$ However, ...
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31 views

Conversion of polar equations when you change the position of the origin

I'm working on a physics problem that is described as follows: "I am standing on the ground beside a perfectly flat horizontal turntable, rotating with constant angular velocity w. I lean over and ...
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92 views

Simple, stable $n$-body orbits in the plane with some fixed bodies allowed

I'm working on a visual simulator for the $n$-body problem in the plane (here). The goal is to show how complex behavior can arise from the simple inverse-square law of gravity. To that end, I want ...
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1answer
32 views

Examples of self-adjoint operators on $L^2(\mu)$

I'd like to come up with a number of simple examples of (formally) self-adjoint operators on $L^2(\mu)$, where $L^2(\mu)$ denotes $L^2(\mathbb R)$ with respect to the Gaussian measure $d\mu$ ...
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57 views

Is it possible to perform Integration in this equation?

I have been working on a problem for a long time and have finally arrived at this differential equation. The problem is simple, which surfaces obey the Reflection Property. Now there are several ...
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1answer
23 views

Probablity distribution for two particles to decay?

Let us say I have the probability distribution of the decay of one particle as: $$f(t)=\frac{1}{\tau}e^{-\frac{t}{\tau}}$$ Then how would I find the probablity distribution for the time it takes two ...
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48 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 ...
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31 views

Suppose that you measure three independent variables as…

Suppose that you measure three independent variables as $x = 6.5 \pm 0.8; y = 3.1 \pm 0.3; \theta = 40^\circ \pm 3^\circ $ and use these vales to compute $$q = \frac{x^2 + y\sin\theta + 2}{x + ...
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63 views

How to solve an ODE with $y^{-1}$ term

My major is not Mathematics, but I came across the following ODE for $y(x)$: $$\left(y^3y^{\prime\prime\prime}\right)^\prime+\frac{5}{8}xy^\prime-\frac{1}{2}y+\frac{a}{y}=0,$$ where the prime denote ...
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52 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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23 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
64 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
43 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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61 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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76 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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74 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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43 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
53 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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25 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
81 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
31 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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47 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
55 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
52 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
32 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
57 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
38 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 ...
2
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1answer
53 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
37 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 ...
2
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2answers
77 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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35 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
37 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, ...
2
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1answer
62 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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58 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 ...
2
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2answers
85 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 ...
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0answers
153 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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31 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 ...