"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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22 views

Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...
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34 views

Size of square formed by soap in a cube frame

So through the work of Plateau (as I understand it), we know that soap tries to find the shortest connection between points. At least, that's what I was taught. With this in mind, I had to solve the ...
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1answer
36 views

Why are the uncertainties so different?

Here is my scenario: I am trying to calculate the uncertainty of the function $y=x^2$, that is, I want to find $\Delta y$, and I found that we can get a great difference in the $\Delta y$, depending ...
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165 views

Solution of the equation.

I have the following equation and I am interested in to find out the value of $r$, $(1-r)^3+3(1-r)h^2-3h(1-r)^2-\dfrac{wh^3}{KM}=0$ I simplified this equation to the following equation, ...
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1answer
56 views

Scalar Product Conditions

Let $x$ and $y$ be two vectors, $x\cdot y$ their scalar product, $\beta$ the angle between the vectors, and $|x|$ and $|y|$ their absolute values. Then we have $$|x| |y| \cos \beta =x \cdot y \quad ...
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55 views

Finding acceleration at a certain velocity

A race car starts from rest and travels east along a straight and level track. For the first $5.0s$ of the car's motion, the eastward component of the car's velocity is given by ...
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1answer
189 views

Find acceleration at the first instant when a car has zero velocity.

The position of the front bumper of a test car under microprocessor control is given by: $x(t)=2.17m+\left(4.8\frac{m}{s^2}\right)t^2-\left(.100\frac{m}{s^6}\right)t^6$ Find its acceleration at the ...
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35 views

Special Relativity Dilation problem

I've been given the following scenario: Observer $B$ is in the center of a train carriage which is moving at velocity $v$ with respect to an observer $A$. Two light signals are emitted from ...
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1answer
96 views

Transpose of the gradient of a vector field.

Whereas I understand what the gradient of a vector field means physically, I am having difficulty understanding what its transpose actually is. I came across it in the context of defining strain in ...
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1answer
99 views

How are Lagrangian mechanics equivalent to Newtonian mechanics?

I didn't study Lagrangian mechanics yet but I did study Newtonian mechanics, and someone said to me that later we would study analytic mechanics (which contain Lagrangian mechanics) and that it ...
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92 views

Integrate the product of two exponential functions

I am trying to solve the following integral: $ \int^{\infty}_{-\infty} \exp{-3/2L [(r^{(0)}- \epsilon)^{2} + \sum_{a=1}^{n}(r^{(a)}-\Lambda \epsilon)^{2}]} d^{3}\epsilon$ I have try to use the the ...
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33 views

Can you add potentials if charge redistributes?

Let say we have charged conductor $M$ and we know its potential energy function $V_m(r)$ when $M$ is isolated from any charges. We also have charged conductor $N$ with potential energy function ...
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132 views

SO(2) group generator Lie Algebra

For the $2 \times 2$ orthogonal group of matrices which for the $SO(2)$ group, there is only one free parameter in the group element and hence only one generator for the group. Which is, $$ X_g = ...
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83 views

What is the reason for normalizing eigenvectors?

In Linear Algebra, when we have found eigenvectos related to specific eigenvalues, we normalize the eigenvectors. If I want to normalize eigenvectors, why do I need to normalize the eigenvectors?
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1answer
65 views

proof of $\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial \frac{\partial f(x,y)}{\partial y}}{\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial ...
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1answer
42 views

The “computability” of fundamental physical constants

I would like to ask if any of the fundamental physical quantities like the speed of light or plancks constant (all measured according to a common standard of of units) can be classified as computable ...
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2answers
31 views

Finding the magnitude of a vector product between two vectors?

Vector $\overrightarrow{A}$ has magnitude $11.0m$ and vector $\overrightarrow{B}$ has magnitude $16.0m$ . The scalar product $\overrightarrow{A}\bullet \overrightarrow{B}$ is $79.0m^2$. What is the ...
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3answers
63 views

Boxes on a slope [closed]

A box with friction slides down a slope and takes 2 times longer than a similar box with no friction takes to slide the same slope. What is $ \mu $ (the coefficient of friction)? I'm pretty lost. I ...
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1answer
27 views

Trying to understand free body diagram [closed]

Please consider the following image: Now I'm just trying to understand how exactly this thing is rotated...I'm looking at it exactly like on the image of the car...So the normal force is slightly ...
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3answers
72 views

Online resources for special relativity

I wasn't sure where to post this, but I'm on a mathematics course that has basically brushed over special relativity. I'm also doing an out of department module called philosophy of physics and as you ...
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50 views

Proving commutation relation in Algebraic Bethe Ansatz

I have a problem with proving a certain commutation relation. For my Bachelor's thesis I give a more mathematically rigurous 'treatment' of a select set of chapters of a paper by L.D. Faddeev. Noting ...
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1answer
167 views

Applications of infinity in real life [duplicate]

I am writing a mathematical essay and would like to focus on the concept of infinity. I am not sure of any real life applications of infinity to write about or some way to narrow down the topics. Does ...
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1answer
82 views

A simple physics related algebra question.

This is making me feel like an idiot. I'm given this answer for a question but I don't understand it. $$y =\rm (-12.9\, m/s)(3.27\, s) + 1/2(9.81\, m/s^2)(3.27\, s)^2 = 105\, m = 0.11\, km$$ I ...
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41 views

pierre simon laplace and his knowledge of the (Laplacian) matrices

so as we all know, there is a graph matrix called the Laplacian that is used in some eigenvalue/eigenvector/graph theory/spectral theory problems. i'm wondering if the name of this matrix is ...
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1answer
111 views

Generators of Translation - Lie Algebra [duplicate]

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
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16 views

Using Invariance of Lorentz interval and constant speed of light to prove the Lorentz transformations

By the invariance of the Lorentz interval and the fact that the speed of light is the same in both frames we have \begin{align*} -c^2 dt^2 + dx^2 = -c^2 dt'^2 + dx'^2 \end{align*} By considering the ...
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40 views

Help interpret Sommerfeld radiation condition.

I am studying the Sommerfeld radiation condition. In potential theory, a solution $u(r,\theta)$ to a partial differential (such as the Helmholtz equation $\Delta u(r,\theta)+\lambda^2 u(r,\theta)=0$) ...
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2answers
50 views

Why can't we say that all PDEs of a specified order require a fixed number of boundary conditions?

For an $n$th order ODE we always need $n$ boundary conditions (right?). But, as I've seen somewhere, for 2nd order PDEs there are many possible situations and a general answer to the question of ...
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1answer
39 views

Determine the motion for all time

In the frame $F=[0,\hat{k}]$, a particle of mass $m$, whose trajectory $[0,\infty)\xrightarrow{\rm r}\mathbb{R}$ is $r=z\hat{k}$ moves in response to a force ...
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28 views

An algebra associated with an important function

In the paper here the authors make a claim that the Natanzon potential (an implicit potential very important in mathematical physics) follows an $SO(2,2)$ algebra. This potential defined as : $$ ...
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48 views

Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} ...
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41 views

Angle that car is at after angular acceleration

A car starts from rest on a curve with a radius of $150m$ and tangential acceleration of $\displaystyle 1.5\frac{m}{s^2}$. Through what angle will the car have traveled when the magnitude of its ...
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65 views

Definition of logarithmic capacity

In the definition of logarithmic capacity of a compact set $E$ in the plane, the Robin constant is defined to be $V(E)=inf\int_E\int_E log\frac{1}{|z-w|} d\mu(z)d\mu(w)$ where $inf$ is taken over all ...
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2answers
99 views

Particle Motion

So this is a simple problem but I'm just getting stumped. The question is: A particle not connected to a spring, moving in a straight line, is subject to a retardation force of magnitude ...
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14 views

Fourier Operator and roots of Identity operator

I have seen that if Fourier operator is defined by $$ h(k) = \hat F(g(x)) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty} dx\:g(x)\:e^{ikx} $$ then $$ \hat F^2\{g(x)\}=g(-x) \implies \hat F^2 ...
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1answer
43 views

E. Artin theorem? (Ergodic theory)

In the framework of mathematical cosmology, Bianchi IX model has great importance due to its stochastic properties. I'm reading a publication in which is claimed The use of the invariant measure ...
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29 views

Holonomy of Maurer-Cartan 1-form

I am studying the book Sternberg (2012): Curvature in Mathematics and Physics; I am also doing research on LQG. I was wondering: if on a 4-dimensional spacetime one defined the Maurer-Cartan 1-form, ...
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19 views

Differential Realizations of certain algebras

I'm a first year graduate student in Mathematical Physics, and I am trying to generalise a certain method involving the so-called "Differential realizations" of certain algebras. The problem I'm ...
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1answer
97 views

Will a knot tied in a hanging, frictionless rope slip out under the force of gravity?

I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless ...
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2answers
76 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
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30 views

Algebraic formulation of quantum mechanics and unbounded operators

Posted in the physic site: In AQFT one specifies the structure of the observables as a $C^*$-algebra. This seems to excludes algebras that don't have a norm, such as the Heisenberg algebra. ...
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73 views

Geodesics Through a Singularity

A singularity on a manifold with metric is defined to be a point at which some geodesic cannot be continued through. For example in Schwarzchild spacetime, $r=0$ defines such a point. Is it the case ...
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1answer
652 views

Solving a distance/time/speed problem using the quadratic formula. [closed]

"The distance between Toronto and Ottawa is 352.72 km. The speed on a road trip from Ottawa to Toronto was double of the return, and therefore the drive took 2 hours less. What was the speed on the ...
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38 views

What does “The Hilbert space carries a representation of […] group” means?

Often, in quantum mechanics I found the sentence "The Hilbert space carries a representation of $SU(2)$ group" (in particular when dealing with anglar momenta). Effectively, I know that this means ...
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1answer
47 views

Theorems on orthonormal bases and spectrum

Are there theorems similar to the following: If $T$ is symmetric and $D(T)$ contains an ONB of eigenvectors of $T$, then $T$ is essentially self adjoint and the spectrum of $\bar{T}$ is the closure of ...
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85 views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk ...
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1answer
61 views

Fourier transform of Legendre

I am trying to figure out the Fourier transform of Legendre polynomial $P_\ell [\cos(\theta-a t )]$: $Q(\omega)=\int_{-\infty}^\infty P_\ell [\sin\phi\cos(\theta-a t )] e^{i \omega t} dt,$ where ...
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2answers
72 views

Obtaining explicit solutions of the differential equation $\left(\frac{dy}{dx}\right)^{2}=\frac{1}{ay^2+by+c}$

I'm trying to see if it is possible to obtain an explicit form of the following differential equation $$\left(\frac{dy}{dx}\right)^{2}=\frac{1}{ay^2+by+c}$$ where $a,b$ and $c\in\mathbb{R}$\{$0$}
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1answer
89 views

Center of mass in a straight rod

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $$\int_S x_1 \, dx_1 \, dx_2=0 $$ $$\int_S x_2 \, dx_1 \, dx_2=0 $$ ...
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543 views

Stacking Cylinders Mechanics Question (from brilliant.org)

Three cylinders, all of the same mass, are stacked on a table as shown in the figure. There is enough friction between the cylinders and the table such that the cylinders remain at rest. Let Fh be the ...