"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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Solving differential equation describing motion in a pendulum

I've been looking at Simple Harmonic Motion in particularly the period of a pendulum. This may seem like physics but my question is tailored towards mathematics. The differential equation is: ...
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Explanation Request for an alternative expression of a Gaussian integral over complex variables

We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...
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46 views

Runge-Kutta 4 in polar coordinates

How is the Runge-Kutta method implemented on this differential equation: $$ \frac{d^2 \theta}{dt} = -\frac{g}{l} \theta $$ (pendulum motion) which is in polar coordinates? Let: $c = \frac{g}{l}$ ...
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Velocity Verlet method: How to calculate acceleration

The velocity Verlet method algorithm is as follows: Calculate: $$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t)\, \Delta t+\tfrac12 \,\vec{a}(t)\,\Delta t^2$$ Derive: $\vec{a}(t + \Delta t)$ from ...
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11 views

Hamiltonian Elliptical Path

For a Hamiltonian of the form, $$ H = \frac{1}{2} p_i p^i - \frac{k}{\sqrt{q_iq^i}} $$ which is a Hamiltonian for a gravity system or something similar. These systems are know to have paths that ...
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38 views

Definition of Global Information and Local Information (CS)

I am a research student of computer science, I always feel like there are some thing missed when I am trying to define some concept mathematically. For example, I would like to define two concepts: ...
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34 views

Semi-infinite forms?

I am reading Vafa's paper 'Topological Mirros and Quantum Strings'(arXiv:hep-th/9111017). In this paper, the author says the Hilbert Space of a fermionic string theory corresponds to the space of ...
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22 views

Cyclic permutation

How did the author do the cyclic permutation? $\Gamma^k_{ij}g_{kl}+\Gamma^k_{lj}g_{ki}=\partial_jg_{il}$ We can cyclically permute these indices to generate two more equations: ...
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31 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$?

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where ...
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35 views

A computation from an article in computational neurosciences (from physical review) which doesn't fit

I am reading this article (with this erratum) in computational neuroscience, and there is a computation there that simply doesn't fit.. Maybe one of you can see something that I am missing? For the ...
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29 views

Simple Harmonic Motion; Tension in Elastic rope

I'm struggling to model this question out correctly. A glider and its pilot have total mass $230$ kg. The glider lands on a horizontal airstrip and when its speed is $16$ m/s it hooks on to the ...
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55 views

Calculating segment length on circle

I'm building a physical machine and I'm trying to figure out a geometrical problem. The machine is composed by a cylinder, and the wall of this cylinder is composed by many wooden boards, each of ...
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1answer
36 views

Page 72 of Courant and Hilbert's Methods of Mathematical Physics, Vol 1.

We have the following identities: $$ \beta_\nu = b_\nu -\frac{1}{2}(b_{\nu-1}+b_{\nu+1}),\ \ \ \ (\nu=2,3,4,\ldots)\\ \beta_1=b_1-1/2 b_2 $$ $$s_n(x)=\sum_{\nu=1}^n b_\nu \sin(\nu x) \\ ...
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77 views

Integration of the product of Hermite Polynomial and exponential function

how to proceed with these two integration.. $$\int^0_{−∞}e^{−ax2}H_{2k}(x)dx=?$$ $$\int^∞_{0}e^{−ax2}H_{2k}(x)dx=?$$ where $$H_n(x)$$ is the Hermite Polynomial (physicist's convention).
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63 views

Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
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28 views

A proof in Hilbert & Courant vol 1 of Weierstrass theorem.

My question is regarding a derivation of an inequality on page 67 of Methods of Mathematical Physics. Here's a scan of the book: ...
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1answer
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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29 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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54 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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42 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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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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54 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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60 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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20 views

Density matrix respect to Hilbert space

If there a two dimensional Hilbert sapce $H$ with the basis, $\{ e_1, e_2 \}$ and state $\psi = \frac{e_1 - e_2}{\sqrt{2}}$. How could we express it as a density matrix ?
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32 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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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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2answers
89 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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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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22 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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46 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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22 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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63 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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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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75 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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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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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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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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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
78 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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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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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 ...