"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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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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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
84 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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31 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 ...
3
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29 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 ...
2
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1answer
46 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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0answers
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 ...
2
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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
86 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
45 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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92 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
13 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
38 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 ...
2
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1answer
24 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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0answers
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, $A$ = ...
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1answer
87 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
33 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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0answers
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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0answers
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&...
1
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1answer
27 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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38 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
30 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 $...
2
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0answers
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
40 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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0answers
36 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/...
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3answers
39 views

Physics problem on derivatives and integrals

So there are a few basic formulas I'd like to start with, $W=\int_0^bFdx$, $F=ma$, and $a=\frac{d^2}{dt^2}x$. In words, Work $(W)$ is defined as the area under a Force versus Displacement $(F/x)$ ...
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0answers
37 views

Dynamics of fluid

While reading on Wikipedia about the partial differential equations (https://en.wikipedia.org/wiki/Partial_differential_equation), I wondered how dynamics for the fluid occur in an infinite-...
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0answers
99 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
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0answers
20 views

vertical projection with gravity as acceleration

$\mathbf Question.$ A stone projected vertically upwards with initial speed of $u\ m/s$ rises $70\ m$ in the first $t$ seconds and another $50\ m$ in the next $t$ seconds. $\bullet$Find the value of $...
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1answer
20 views

Find the equilibria

Consider the equation $\ddot s = s-s^3.$ Let $m=1.$ 1) Write this as a first order system. Let $\dot s=v.$ Then we get $\dot v=s-s^3.$ So first order system is $$\begin{pmatrix} \dot{s} \\ \dot{v} \...
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0answers
21 views

Find the potential energy and sketch it

I'm given the equation $\ddot s=s-s^3.$ I'm asked to compute the potential energy and sketch it. I'm also given $m=1.$ To do this I have done the following: $$F=ma=\ddot s=s-s^3=\frac{-dV}{ds}=\frac{...
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1answer
35 views

Book's for potential theory: single and double layer potential

Does anyone know recommend me some book about the theory of the potential, especially that concerning the layer potential. Besides the theoretical part in the higher dimension, if there are concrete ...
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12 views

The expectation number of collision to slow down below certain value

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward ...
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1answer
27 views

Units of $F(x) = x-x^3$

If we are given that $F(x) = x-x^3$ is a force function, which means it is in the units of $[M][L][T]^{-2}$, then how do we determine what kind of "unit units" participate in this function? Namely, ...
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1answer
20 views

another help with partial differential equation.

I am to solve next task. solve this PDE with boundary conditions. $\Delta u = \frac{64}{r^5}\sin\varphi, \quad 1<r<2,$ $u'_r|_{r=1} = 2\cos^2\frac{\varphi}{2}, \quad u'_r|_{r=2} = 4\sin^2\...
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1answer
43 views

Is there a general way to prove this Fourier transform property?

We know that one of the important Fourier transform properties is that, the Fourier transform of a narrow function has a broad spectrum, and vice versa, We can easily see this in this example, the ...
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0answers
21 views

How to model the following scenario with an ODE if possible

Consider a cylinder, full of charged particles travelling through. From the perspective of looking through the tube, you would see a circle of particles and obviously this circle continues down the ...
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0answers
13 views

Modelling a charged particle flowing travelling through a conductive pipe.

I'm on an internship and have a project to model how a charged particle might be affected by a conductive surface either side of it. Here's how I approached it: I assumed the particle had some charge ...
0
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1answer
17 views

MOI about a diagonal

If by taking a thin rod, and finding its Moment of Inertia about an axis, say through the mid point of its side, one can observe that stretching the rod uniformly along the axis of rotation will give ...
4
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1answer
56 views

“Flow lines” of “dust” are geodesics?

The stress-energy tensor representing "dust" takes the form$$T_{ab} = \rho u_au_b$$where $u^a$ is a unit timelike vector field, i.e., $u^au_a = -1$. Does it necessarily follow that in any solution to ...