"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 ...

learn more… | top users | synonyms (1)

1
vote
1answer
44 views

Question about moment of inertia calculation and logic

Question: Determine the moment of inertia for a quadrant of a uniform circular lamina of radius b. Here I saw the answer that,however I don't understand it first of all here is the answer and I ...
2
votes
1answer
171 views

Explaining the differential operator found in Physics equations.

I'm new to this exchange so please bear with me regarding notation. I would like to know what the differential operator $d^n$ means as seen in some physics equations. Normally, one would have an ...
2
votes
1answer
36 views

Indices at the left of a tensor in mathematical physics/differential geometry?

I am a mathematician and I am reading a paper in mathematical physics and I found the following notation: Let $Y$ be a two–form on $M$ such that $$\nabla({}_iY_j)_k = 0.$$ Here, $\nabla$ is ...
1
vote
1answer
61 views

How to find $r$ in an equation like this: $r^3= xr+y$

Can anyone give me an an idea how to solve this and find $r$, where $r^3= xr+y$ and $x$ and $y$ are known numbers?
3
votes
0answers
86 views

Upper bound on the Lipschitz constant of entanglement entropy

I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full ...
1
vote
0answers
33 views

How can I calculate the most efficient movement vector when already moving?

So, for context, this isn't a math problem in a text book, but an issue I've run into while coding a video game. However, due to the nature of the problem, I felt posting this in the mathematics ...
1
vote
0answers
43 views

Is there any some straightforward calculation of finite rotation operator?

Rotation with axis $\hat{k}$ and angle $\theta$ in $\mathbb{R}^3$ is represented by $$ R = I + (\sin \theta) K + (1-\cos \theta) K^2 $$ where $K$ is the matrix for left cross product by $\hat{k}$. ...
0
votes
0answers
6 views

Dual lattice and extreme value

Suppose I have a periodic function which has minimums only at lattice sites. Say the lattice is a honeycomb lattice, do I have my maximums at sites of a triangular lattice (which is the dual lattice ...
3
votes
2answers
60 views

Can we apply Buckingham $\pi$ theorem?

A physical system is described by a law of the form $f(E,P,A)=0$ where $E,P,A$ represent, respectively, enery, pressure and area of surface. Find an equivalent law that relates suitable dimensionless ...
2
votes
2answers
55 views

How to find the poles of a green function?

I am trying to construct a green function for $y''+\alpha^2u=f(x), u(0)=u(1), u'(0)=u'(1)$. For that I am trying to follow the procedure described here:(Construct the Green s function for the ...
3
votes
0answers
40 views

Solvability of an integral equation

Is the following integral equation solvable ? $$ F(x)-\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\frac{\sin \gamma(x-y)}{\pi(x-y)}$$ and $$f(x)=e^{i\gamma x}$$ and $\gamma$ is a parameter.
0
votes
1answer
26 views

Lorentz transformation of electromagnetic field tensor

I need to calculate: $f^{\mu'\nu'}=L^{\mu}_{\kappa}L^\nu_\lambda f^{\kappa\lambda}$ Where $L^\nu_\lambda$ is the usual Lorentz transformation matrix I thought that I just needed to do some normal ...
3
votes
1answer
38 views

I need some help understanding proofs for an upside-down cycloid being the tautochrone curve. Could someone show me or point me to a simple proof?

The tautochrone curve has fascinated me since I first heard about it and I want to share it with my Calculus class as an end of the year project. I think something similar to this (Demonstrating that ...
1
vote
1answer
32 views

Light attenuation through water at an angle

I know that light intensity decreases exponentially governed by \begin{equation*} \frac{dy}{dx} = -ky \end{equation*} where $y$ is the intensity and $x$ is the distance. Now what happens when light ...
1
vote
1answer
68 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 ...
0
votes
0answers
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 ...
0
votes
1answer
35 views

Some derivation in mechanics

I have the following derivation in my physics book I don't know how did they derive them $\frac{d}{dt} \Sigma{_i}[(\vec{r}_{cm} + \vec{r_i})\times m_i(\vec{v}_{cm} + \vec{v_i})]$ = ...
2
votes
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 ...
1
vote
3answers
99 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 ...
0
votes
0answers
31 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 ...
0
votes
1answer
25 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 ...
2
votes
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 ...
4
votes
0answers
41 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 ...
0
votes
1answer
51 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 ...
0
votes
1answer
43 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 ...
3
votes
0answers
37 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 ...
3
votes
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 ...
2
votes
2answers
85 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 ...
1
vote
0answers
29 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
votes
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 ...
7
votes
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 ...
-1
votes
2answers
61 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
4
votes
1answer
70 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 ...
3
votes
1answer
47 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 ...
0
votes
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 ...
15
votes
4answers
849 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 ...
0
votes
1answer
34 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. ...
0
votes
1answer
47 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 ...
-1
votes
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 ...
3
votes
1answer
77 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 ...
0
votes
0answers
32 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 ...
1
vote
0answers
22 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 $V=\int_{x_o}^{x^1} F(x)$ $dx$ ...
2
votes
1answer
47 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
votes
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
votes
1answer
72 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 ...
1
vote
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 ...
2
votes
1answer
24 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 ...
0
votes
0answers
33 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}{ ...
0
votes
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 ...
2
votes
0answers
65 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 ...