"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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35 views

Srednicki's QFT - chapter 2 - understanding from a mathematician's point of view

I am reading the first chapters of Srednicki's Quantum Field Theory book, trying to understand them from a mathematician's point of view. In particular, I'm interested to what happens when you try to ...
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14 views

wouldn' a discretized space-time violate pontryagin duality?

While this question regards physics, it is more of a mathematical question, so here it is. One often hears about attempts to model space time with tilings or some type of discretized structure. ...
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1answer
35 views

Summation of $A\cos (\omega n+\phi)$ [closed]

I'm trying to evaluate the following summation: My original problem is $$\lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^N \left|A \cos(\omega n+\phi)\right|^2$$ Now I'm stuck at calculating the ...
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58 views

Intuitive, short explanation of differential forms and exterior calculus

Are there any introductory lecture notes on differential forms and exterior calculus, preferably aimed at physics students studying General Relativity and Black holes? I have some familiarity with GR ...
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11 views

Bolzmann machine cost function

Quote from Haykin book: Neural Networks and Learning Machines Third Edition From an analogy with thermodynamics, the energy of the Boltzmann machine is defined as follows: $$E(x) = -\frac {1}{2} \...
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1answer
42 views

Finding $x$ and $y$ components of Navier Stokes

An incompressible viscous fluid of constant densite and kinematic viscosity occupies the space between porous walls at $y=0$ and $y=d$. The steady two dimensional flow is subject to a constant ...
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1answer
63 views

Why are these distributions positive?

I am trying to understand some calculations in a paper by Sidney Coleman. He is showing that certain distributions are positive. The paper can be found here. What I am talking about is happening at ...
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1answer
18 views

How to calculate rotation rates of a rotating body relative to another rotating body?

I have two 3D bodies A and B, each of them is rotating around its own Z-axes with an angular velocity (e.i. yaw rate) of $\dot{\alpha}_A$ and $\dot{\alpha}_B$, respectively, relative to an absolute ...
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1answer
47 views

Diagonal operators on infinite dimensional Hilbert spaces

the following is a short question regarding a theorem from a quantum mechanics book I am working through but the question is a mathematical one. There is a theorem which states: Theorem: The ...
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2answers
60 views

Circuit Analysis problem (find the problem)

In this question, I know that $\text{C},\text{R},\text{T},\text{A}\in\mathbb{R}^+$ I've this circuit (the bottom of the resitor is connected to earth ($0$)): When I use Laplace transform I can find ...
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0answers
33 views

Time-$t$ map of a Hamiltonian flow: how to check twist property?

I would like to obtain a general formula to verify if a certain time-$t$ map of a Hamiltonian flow is twist. I have a Hamiltonian $1$ degree of freedom system $H=H(q(t),p(t))$, such that all orbits ...
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11 views

How to model a point force with uncertain concentration point?

I consider a beam which is bent under influence of a point force concentrated at some point $\xi$ of the beam. The exact co-ordinate of $\xi$ is not known, but it is known a neighbourhood $(l_0,l_1)\...
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1answer
26 views

Laplace transform of a square wave function

What is the right way to find the Laplace transform of this function: The thing I noticed was: $$f(t)=\text{A}\space\space\space\space\space\space\space\space\space\space 0\le t<\frac{\...
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0answers
4 views

Calculate Universal Time for when an object in orbit reaches a given radius / altitude?

Assuming that an object in orbit WILL reach a given radius / altitude at some point in the future, how can I work out the exact time it will reach that point? Assume that the object is a Satellite in ...
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1answer
43 views

Invere Laplace transform of a function (related to circuit analysis)

I'm studying circuit analysis. I've to solve this inverse Laplace transform to see the response: $$\mathcal{L}_{s}^{-1}\left[\frac{r}{r+\frac{1}{cs}}\cdot\frac{k\tanh\left(\frac{as}{2}\right)}{s}\...
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14 views

How to make good approximation for a sum of squared expression?

In both expression, n is integer and nmax is the maximum n and can be very large. How to use nmax to approximately and analytically to express these two expressiones? Are there any analytical ...
2
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1answer
30 views

Unique ground state of Schrödinger Operators

I'm reading a book and there is an argument that the ground state of a Schrödinger operator is unique. The problem is I think the argument is complete non-sense! These are lecture notes by Witten, I ...
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0answers
24 views

prove de Rham cohomology of S,the “spherical universe,” is 0-dimensional?

How to prove de Rham cohomology of S,the "spherical universe," is 0-dimensional?(Here, S is a rectangle where if you exit the right, the enter from the top and if you exit the left, the enter from the ...
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1answer
79 views

Employing Newton's Laws with differential equations [closed]

Going through some problem sheets from previous semesters and can't find a full solution for this question so was wondering what the answers might be. A particle of mass $m$ moves on the $x$ axis ...
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0answers
11 views

Regularity of Fourier transform?

Let $ |k|^n\cdot\,\hat{u}(k) \in L^2(\mathbb {R} ^d)$. Can we make a statement about the regularity of the $u$ itself? The idea would be to use the differentiation rules for the Fourier transform, ...
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7answers
5k views

What is “Bra” and “Ket” notation and how does it relate to Hilbert spaces?

This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my ...
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30 views

Seperation of variable Heat equation

Consider a copper bar of length $L = 100cm$ which is kept at the temperature $u = 0\space °C$ at one end, and is perfectly insulated at the other end. The bar is initially heated according to the ...
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28 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 ...
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1answer
45 views

Ellipsoid moment of inertia matrix

Some background info: torque $\tau$ is defined as $$\tau = I*d\omega$$ Where $I$ is the moment of inertia matrix and $d\omega$ is an object's rotational acceleration. As I understand it, the inertia ...
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1answer
28 views

Calculate position with increasing acceleration.

So if calculating the change in an object's position (with a constant acceleration) is done with this equation: $o = vt + (\frac12)a t^2$ $o$ is offset from original position $v$ is starting ...
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21 views

Writing PDE in the form of convervation law

What does one need to know in order to write $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial x^3}=0$ in the form of a conservation law, which contains the ...
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1answer
22 views

Computate the commutator $[p^n,x]=-ihnp^{n-1}$

Computate the commutator of $[p^n,x]=-ihnp^{n-1}$. With $p=-ih \frac{\delta}{\delta x}$ the impulse operator. $h$ stands for $\frac{h}{2\pi}$. Answer: I do it with induction over $n$. For $n=1$ it ...
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1answer
84 views

Figuring out velocity,acceleration, work of a particle given that we know its position vector.

Recently this question came up in a problem class of mine. A particle moves in such a way that its position vector at any time $t$ is $\vec{r}(t)=\pmatrix{A\sin{\omega t}\\A\cos{\omega t}\\Bt^2}$, ...
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1answer
40 views

Mcgehee transformation, conversion to polar coordinates and blowing up the singularity

I am looking for any reference on the above topics as I am struggling to convert the below to polar coordinates in phase space: The system is: \begin{equation*} x''=\frac{-\mu x}{(\mu x^2 + y^2)^{...
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1answer
18 views

What's the value of $\int f(x)\delta(x-a) dx$ if $a$ is not in the domain of integration?

A problem occurs when I was solving an exersice of perturbative kind. The delta function has the fundamental property that \begin{align} \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) \end{align} ...
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85 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(\...
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2answers
33 views

Integral path between 2 points

so I need your help calculating the next inegral: Calculate the integral $$\int(10x^4-2xy^3)dx -3x^2y^2dy$$ at the path $$x^4-6xy^3=4y^2$$ between the points $O(0,0)$ to $A(2,1)$ please explain me ...
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0answers
12 views

how to integrate a equation in Ito formulation

Does any one know how to integrate the differential equation in Ito formulation. I have following stochastic equation in Ito form: $$dx=x(1-x)dW(t)$$ where $dW(t)$ is the Weiner increament such that $\...
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1answer
36 views

need help for integrating the differential equation

I am not able to integrate the following stochastic equation. The equation is $$\frac{dx}{dt}=g(1-x^2)x+\sqrt{g}(1-x^2)\xi(t)$$ $g$ is a constant and $x$ is defined between $-1$ and $+1$. $\xi(t)$ is ...
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1answer
18 views

Symmetry of Green's function on the general case

Let's consider the differential equation $$\nabla\cdot[p(\mathbf{r})\nabla u(\mathbf{r})]-s(\mathbf{r})u(\mathbf{r})=-f(\mathbf{r}).$$ I want to show that the Green's function is symmetric, so that $...
3
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0answers
51 views

How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
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25 views

how do I resolve equations that are both dependant on each other

I'm working on a project concerning the ideal power equation of aerodynamic bodies seen here: $$P = \frac{1}{2}C A D v^3 + \frac{W^2}{Db^2v}$$ where $P$ = power, $C$ = coefficient of drag, $A$ = ...
3
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1answer
63 views

Helmholtz decomposition of a vector field on surface

Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and ...
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1answer
26 views

Uniqueness for Dirichlet problem in exterior domain

I have the following problem: $\Delta u =0$ in $\Omega_e = \mathbb{R}^3 - \overline{\Omega}$, and with condiction $u=0$ on $\partial \Omega$ and $u=o(1)$, that is $\lim_{r \rightarrow 0} u(x) =0$. ...
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22 views

Converse of this theorem about existence of Green's function

I've been solving some problem which asks us to find the Green's functions for some problems when it exists. Now, there's a theorem which allows us to guarantee that it exists. The theorem is as ...
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1answer
52 views

Interpretation of Equations of Motions

I started a lecture on differential equation with following example. If a body is moving in a straight line in plane with constant speed, how can we describe this motion mathematically? To answer ...
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0answers
63 views

Cubes in cubes in cubes in… ad infinitum.

Suppose I have a cube with one open side (with a volume of let's say $1\ m^3$) for the sake of simplicity; the problem is scale invariant) made from a material that makes the cube just float in water ...
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31 views

Heat problem with an internal source of heat for which the maximum principle doesn't hold.

Heat problem with an internal source of heat for which the maximum principle doesn't hold. The problem is the following and honestly I don't know how to solve it... $$u_{t}=u_{tt}+2(t+1)+x(1-x) , 0&...
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1answer
26 views

finding curvature radius

given a projectory equation of the form $ y=y(x) $find the curvature radius as a function of $x.$ a projectory equation , hence $ x=x(t)$, input that in y and we get $y=y(x(t))$, which is what one ...
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36 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 ...
2
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1answer
29 views

Projectile motion: Proving:$ x^2 + 4 \left(y-\frac{v^2}{4g} \right)^2 = \frac{v^2}{4g^2} $

Question: Projectiles are fired with initial speed $v$ and variable launch angle $0< \alpha < \pi$. Choose a coordinate system with the firing position at the origin. For each value of $...
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80 views

how to solve this integral ? It seems bounded and well defined integral but I don't know how to solve this

how to solve the following integral ? It seems well defined i.e. bound but I could not solve it. I tried by expanding series expansion of tanh[x] but after that I got a series as an answer, which I ...
2
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0answers
61 views

Relation between a linear second order differential equation and Riccati special differential equation

Consider the following differential equation \begin{equation} \frac{d}{dx}\left[N(x)\frac{dw}{dx}\right]+\sigma^2\rho(x)w=f(x,\sigma),~~ 0<x<l, \end{equation} $0<N\in C^1(0,l)$, $0<\rho\in ...
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1answer
38 views

Finding the reflection of a plane wave from a sphere

The physical problem I'm trying to solve is this: I would like to find the "reflection" of a harmonic plane sound wave in a liquid, from a spherical air bubble. I'm modeling the problem as follows: ...
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31 views

Specifics on the Williamson normal form algorithm

I'm looking at the algorithm for Williamson normal form for symplectic diagonalization of positive-definite symmetric real matrices, given on pp.24 here: https://www.ime.usp.br/~piccione/Downloads/...