"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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66 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
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21 views

Given the matrix representation what is the expectation value

For a particle with spin $\frac{3}{2}$, construct the matrix representation for $S_z, S_x$ and $S_y$. If the particle is in an eigenstate of $S_z$, what is $\langle S_x\rangle$ and $\langle ...
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1answer
34 views

Some derivation in mechanics

I have the following derivation in my physics book I don't know how did they derive them $\frac{d}{dt} \Sigma{_i}[(\vec{r}_{cm} + \vec{r_i})\times m_i(\vec{v}_{cm} + \vec{v_i})]$ = ...
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1answer
40 views

Modeling $k$-logistic diffusion process

I'm trying to model a diffusion process characterized by subsequent logistic diffusion processes. To give a better idea of what I'm trying to model, you can take a look at the attached figure. In such ...
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3answers
96 views

looking for help with a trace/norm inequality

I'm trying to understand a derivation that seems to claim that $\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where $\rho$ is Hermitian and has ...
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0answers
29 views

Movement of birds - Acceleration, Velocity, Time and Displacement. Needed for an assignment

Hi so there are a quandary of birds sitting on a tree.There are $3$ teams observing the movement of the birds. Team $1$ observes that on their first flight the birds move a short distance across a ...
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1answer
22 views

uniform angular distribution-change of origin

Given a variable which is uniformly distributed for $0<\theta<\pi$ on, let's say, a circle around the origin $O$ with radius $R$($\theta$ starting on the positive x-axis and turning ...
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0answers
14 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if $A$ and $B$ are two $n \times n$ Hermitian matrices, and $[A,B]=C$. I'd like a ...
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0answers
39 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
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1answer
50 views

Problem with second distributional derivative

I have the following function: $ f(x) = \begin{cases} \sqrt{x}, & \text{if $x>0$} \\ \sqrt[3]{|x|}, & \text{if $x<0$ } \end{cases} $. I have to find $f'(x)$, $f''(x)$ as ...
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1answer
38 views

Problem on string vibration

Given the standard wave equation for small amplitudes, we have been asked to find the position of string $y(x,t)$, given: $y(x,0)=\sin x$, and, $y'(x,0)=\cos x$, where $y'$ depicts partial ...
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35 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
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1answer
52 views

Changing the form of this equation

In quantum mechanics, a particle is described by its wavefunction, $y(x)$, which is related to the probability of finding the particle at position $x$ (roughly speaking). This wavefunction satisfies ...
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2answers
81 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the ...
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0answers
27 views

Algebraic determination of asymmetric unit (aka irreducible wedge) in Brillouin zone of lattice

In Solid State physics the reciprocal space is of utmost importance to predict the band structure and thus most of the electrical transport parameters like effective mass, etc. The First Brillouin ...
4
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1answer
123 views

Nonlinear Partial DE

In my work I have faced with following partial differential equation $$\left(\frac{\partial u}{\partial x}\right)^2-\left(\frac{\partial u}{\partial y}\right)^2+f(x,y)\frac{\partial u}{\partial ...
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2answers
120 views

In what sense does $\sum_{k=0}^{\infty} 2^{2k} = - {1 \over 3}$?

In The Road to Reality Penrose remarks on an identity written down by Euler which is "obviously wrong" and yet correct "on some deeper level". He makes reference to the series again when discussing ...
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2answers
56 views

Time-independent Schrodinger equation

the equation : (-h/2m)y'' + U(x)y = (E)y How do you put the time-independent Schrodinger equation in the form of: y'' + G(x)y' + P(x)y = 0
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1answer
65 views

Calculus of variations question with two variables

If $u(x)$ and $v(x)$ satisfy $u(0)=1$, $v(0)=-1$, $u(\pi/2) =0$, $v(π/2) =0$ on extremals of functional $$ \int_0^{\pi/2}\left[\big({\frac{du}{dx}\big)^2 +\big(\frac{dv}{dx}\big)^2 +2 \,u v ...
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1answer
43 views

Center of mass calculation

Calculate the center of mass for : The area bounded by parabola $y = x^2/b$ and the line $y = b$. I got the following integral I just need verification that my work is correct. First I got ...
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0answers
10 views

How can I find the domain of this diffeomorphism (coordinate transformation)?

I have been struggling with this coordinate transformation in $R^2$. $Q:\begin{bmatrix}\rho\\\phi\end{bmatrix}\to \begin{bmatrix}\cosh(\rho)cos(\phi)\\sinh(\rho)sin(\phi)\end{bmatrix}$ I am ...
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4answers
838 views

What is the motivation for analytic solutions in Mathematical Physics?

I am trying to understand why one cares about solving PDE's with an analytic/theoretical solution when one can use numerical methods? If you tell me, "only mathematicians try to find theoretical ...
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1answer
31 views

Quick formula rearranging

I'm having problems rearranging this formula to solve for c, could someone lend a hand please. It's a physics formula for projectile motion. ...
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1answer
46 views

Speed as a function

we were studing the rate of the function $\frac{f{x_1}-f{x_2}}{x_1-x_2}$ if it is positive so the fonction is growing if it is negative so the function is ascending . in this moment our teacher ...
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2answers
33 views

How can I show that the function is smooth?

I got an assignment which I just can't find the right way to solve. It goes like this: Let $\Omega \in R^n$ be a domain and $b_1,...,b_n:\Omega \to R$ smooth mappings (or functions, don't know the ...
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1answer
74 views

Trouble with a Statement in Arnold's “Mathematical Methods of Classical Mechanics”

On Pg 6 of Arnold's Mathematical Methods of Classical Mechanics (2nd Edition), there is a line which reads One can speak of two events occuring simultaneously in different places, but the ...
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0answers
30 views

Need help finding an equation of a boat.

The task is as follows: The flow rate of a river of width 2d is 0 at the river banks and linearly increases as you reach the center of the river to a maximum value of u. A boat crosses the river ...
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0answers
21 views

Forms and conservative forces

According to Tongs notes on Classical Mechanics; a force is called conservative when $F=-\nabla V$ And iff $\nabla \times F = 0$. This is in $R^3$. Also the potential $V=\int_{x_o}^{x^1} F(x)$ $dx$ ...
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1answer
45 views

Spinors and Möbius strips

Consider a Möbius strip; draw on one side of it an arrow aligned vertically; now take it for a trip by around the strip; then when it comes back to the same position it has flipped direction; another ...
2
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0answers
40 views

Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
2
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1answer
71 views

Do we deduce that the physical law isn't unit-free?

A small sphere with radius $1$ and density $p$ moves downwards with constant velocity $v$, under the influence of the gravity $g$, at a liquid of density $p_l$ and viscosity coefficient $\mu$. (The ...
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1answer
66 views

Finding potential of a given vector field

I am trying to solve the following problem: Let $ \textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2} $. Find an expression for a potential for $ \textbf{F}$. Find an expression also for ...
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1answer
23 views

According to Buckingham Theorem the rank of $A$ should be $2$

A physical system is described by a law of the form $f(E,P,A)=0$, where $E,P,A$ represent, respectively, energy, pressure and surface area. Find an equivalent physical law that relates suitable ...
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0answers
31 views

determining the fermi velocity via density of states

The problem is to determine the Fermi velocity for a fermion gas at absolute zero. the problem using integrating a function that looks like $$ v = \frac{4\pi V}{h^{3}} m^{3} \int_{0}^{\infty}{ ...
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0answers
28 views

How does the Schrodinger's potential transformer if the metric conformally transformers?

Given Schrodinger's equation $$ (-\nabla^2+V)\psi=E\psi $$ and the conformal transformation $\tilde{g}_{mn}=e^{2\phi}g_{mn}$, how does the Schrodinger's potential $V$ transformer if the metric ...
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0answers
53 views

How to prove the following determinant identity?

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model. For any even integer $N$, define two sets ...
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1answer
84 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
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1answer
68 views

What really is a path-ordered exponential?

In some texts about gauge theories in Physics I've found one object called a path-ordered exponential which I'm not sure what it means. As I understood, the idea is as follows: let $G$ be a Lie group ...
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0answers
92 views

Area enclosed by an equipotential curve for an electric dipole on the plane

I am currently teaching Physics in an Italian junior high school. Today, while talking about the electric dipole generated by two equal charges in the plane, I was wondering about the following ...
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0answers
323 views

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships [closed]

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics (e.g., electromagnetism, general relativity, gravitation, etc,...) are very huge, ...
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0answers
81 views

How to solve the biharmonic equation in this problem?

I'm struggling with the following Boundary Value Problem for some time. The problem is to solve the biharmonic equation $\nabla^4\psi = 0$ with $\psi$ dependent not just on the coordinates on the ...
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0answers
47 views

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$?

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$ ? To give context, this comes from: Dirac's Theory of General Relativity p19: http://imgur.com/mrkT5C7 I'm not comfortable with proofs regarding ...
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1answer
63 views

Why are nodes and nodal sets called this way?

Nodes of standing waves are points where they are zero. Generally, nodal sets of Laplacian eigenfunctions are the sets of points where they are zero. Why is this the name for them (that is, why is ...
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1answer
48 views

Newtons Law of Cooling Differential Equations

We have two differential equations, $$\begin{cases} {dT\over dt} = -\alpha(T-B)\\ {dB\over dt} = -\beta(B-T)\end{cases}$$ If $T(0) = 7$ and $B(0) =3$, determine the equilibrium temperature of the ...
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1answer
50 views

How to simplify the summation of log

I have a summation that involve log. I don't know how to solve this summation. I want to find an expression (even a good approximation is enough) for this summation. $\sum_{k=0}^{n}{log(a_k)}$ ...
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2answers
58 views

How can $ \Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $ possibly be the Heaviside Step Function?

How can $$\Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $$ possibly be the Heaviside Step Function? What I'm looking for is a direct visualization or maybe an approximate ...
1
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1answer
84 views

Proving that $\int \delta \dot{x} dt = \delta x$

Everytime I've seen this I've assumed it was true. It seems plausible. But I would like to rigorously prove it. I think this is correct, but I would like another opinion because my mathematics is very ...
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1answer
30 views

Pendulum tension force

I realize this is physics related, although the problem is really about math so I thought it would be a good fit for this site. My illustration is supposed to depict a pendulum and the forces ...
2
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0answers
39 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
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2answers
37 views

Solving Bernoulli equation transformation

I'm trying to solve the Bernoulli's equation via perturbation method but I need some help understanding how its done: We start off with $y'=-y+\epsilon y^2$ with $y(0)=1$. Then how is it possible ...