"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)

0
votes
1answer
37 views

Toppling of a road cone that has an axis at an angle $\alpha$ to the horizontal.

A road cone consists of a $45cm$ x $45cm$ square base of height $10cm$, and a conical shell of radius $15cm$ and height $75cm$. The base has a circular hole through it , of radius $15cm$, to aid ...
2
votes
2answers
23 views

$\nabla \sqrt{\rho} \in L^2(\mathbb{R}^3) \implies \rho \in L^3(\mathbb{R}^3)$

I found this in the INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL XXIV, 250 (1983) inside the paper of Elliot H. Lieb with the title Density Functionals for ...
1
vote
1answer
28 views

n-dimensional wave equation proving the compactness of the support of the solution

The question is the following. Let $u\in C^2(\mathbb{R}^n\times[0,+\infty))$ be a solution of the problem \begin{cases} u_{tt}-\Delta u = 0\\ u(x,0) = \phi(x)\\ u_t(x,0)=\psi(x) \end{cases} where the ...
0
votes
1answer
39 views

Prove a particular property of Laplacian operator

I can't prove that Laplacian $\Delta(u(x))=\Delta(u(x_1,\ldots, x_n))=0$ also implies $$ \Delta\left(|x|^{2-n}u\left(\frac{x}{|x|^2}\right)\right)=0 $$ for $\frac{x}{|x|^2}$ in the domain of ...
0
votes
1answer
8 views

Finding the measured average angular velocities

Supposed I have a couple rows of data with recorded measured ratios $\omega_f/\omega_i$ and they ask me for the "Average Measured $\omega_f/\omega_i$ " This may seem like a really trivial solution but ...
2
votes
2answers
52 views

When is the Lagrangian a constant of motion?

It is known that when the hamiltonian is time independent, it also does not vary with time. That is, $\frac{\partial \mathcal{H}}{\partial {t}}=0$ implies $\frac{\mathrm{d} \mathcal{H}}{\mathrm{d} {...
0
votes
1answer
32 views

How to solve the possion equation with nolinear term by finite element method?

$$-\Delta u+u^3=f\ \text{in}\ \Omega$$ $$u=0\ \text{on}\ \partial{\Omega}$$ the difficult here is how to handle the nonliear term $u^3$?
0
votes
1answer
60 views

One parameter group of transformations and their infinitesimal generators

I have a differential map on manifolds $\psi: M_1 \to M_2$, where $\phi_1$ and $\phi_2$ are one parameter group of transformations on $M_1$ and $M_2$ respectively.Now if $\phi_{2t} \circ \psi = \psi \...
1
vote
1answer
35 views

Finding the limit of N approaching infinity: $N(x^\frac{1}{N}-1)\approx\ln(x)+\frac{1}{2N}\ln(x)^2+…$

I am having trouble understanding the linked exercise, final paragraph (not parts a or b) Entropy Calc Problem I understand this is a physics related exercise, however, my trouble comes in at the ...
6
votes
0answers
201 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
0
votes
0answers
58 views

What exactly is the diagonal subgroup of a group?

In specific consider the example of $SU(2)_a \times SU(2)_b$. What is the definition of the diagonal subgroup and how can one construct it from the generators of the group (or its algebra)? This ...
2
votes
0answers
39 views

Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
0
votes
0answers
13 views

How to infer $f$ from this operator: $g(k\nabla f - \nabla,.)$ where $g$ is the Euclidean metric, $k > 0.$

I have the following operator: $g(k\nabla f - \nabla,.)$ where $g$ is the Euclidean metric, $k > 0$, and $f$ is unknown. It acts on vectors in $\mathbb{R}^n$. What kind of informations can I obtain ...
1
vote
1answer
56 views

Is it correct to think of the Laplacian as the divergence of a gradient field?

Factoring out the notation, I see that $$\nabla^2(\phi) = \nabla \cdot \nabla(\phi) = \nabla \cdot (\nabla(\phi)) $$ which looks something like the divergence of the gradient of phi. Is it ...
0
votes
0answers
64 views

How is my proof that this vector field is identically zero?

EDIT: If my work is fine, I believe that the problem statement (an old exam question from 1992) has given one too many assumptions - namely, divF=0. I think towards the end of my proof, when I ...
0
votes
1answer
24 views

How to use the assumption that a vector field is curl-free in a “convex” region,

I don't seem to need this assumption in one of my proofs, but the problem statement gives it, so I think I had better try to use it. Does a convex region imply that it is simply connected (but that ...
1
vote
1answer
61 views

What does this gradient-like symbol mean?

If $\nabla \phi$ denotes the gradient of some scalar field $\phi$, then what does $\nabla^2 (\phi^2)$ mean? I don't think it means taking the gradient of a gradient (of a squared-scalar field), ...
0
votes
0answers
25 views

Covariance group of the functional equation of an L-function

These last few days, I've been wondering whether one could consider the parameters/variables $\chi$ and $s$ a Dirichlet L-function depends on as coordinates such that the pair of transformations $(\...
0
votes
2answers
50 views

Motion of a particle.

I first started by integrating both sides with respect to t (dt). It says that B is along the z-axis but how do I account for that.
1
vote
0answers
36 views

What it means to “put together all the maps” here?

I'm reading Spivak's Mechanics book and he says the following when talking about Hamiltonian Mechanics Given a Lagrangian $L : TM\to \mathbb{R}$, at each point $a\in M$ the restriction $L_a = L|...
3
votes
1answer
82 views

Advice on Mathematical Modeling with Differential Equations

I am on my fourth year studying in a bachelor program in applied mathematics and computer science and plan to write a term paper on mathematical modeling using differential equations. This will be the ...
-1
votes
1answer
29 views

when is a function changing by an order of 1,2,3…n

say for example we have the distance traveled by a vehicle as a function of time. if the speed(change in distance) is constant then this would be a linear function of order 1. if there was ...
4
votes
1answer
80 views

Lagrangian invariant under left and right multiplication by unitary matrices, slick way to see?

Is there a slick way to see that the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G\partial_\mu G^{-1}),$$where $G$ is an $N \times N$ unitary matrix, is invariant under left and right ...
0
votes
1answer
36 views

A bead is threaded on a friction-less vertical wire loop of radius $R$.

The question is the very last sentence at the end of this post. In this post, I'll demonstrate how I reach to a contradiction(the conditions mentioned in conjecture 1 should be satisfied by all ...
3
votes
1answer
38 views

Nonhomgeneous Linear Differential Equation: Harmonic Oscillator

Consider frictionless harmonic oscillator (w/ m = 1) driven by an external force $f(t) = A\sin{\omega t} $, so that $$\frac{d^2 x}{dt^2} + \omega_0^2x = A\sin{\omega t}. $$ Show that the particular ...
0
votes
0answers
28 views

2D reaction-diffusion Schnakenberg Model normalization

I need to normalice this Schnakegnber model in 2D: $u_t=D_u(u_{xx}+u_{yy})-u+av+u^2v\\ v_t=D_u(u_{xx}+u_{yy})b-av-u^2v$ We also know that $a,b>0$. Then I know that I'll applied separation of ...
1
vote
1answer
47 views

Bilaplacian - Explanation of the (Clamped) Plate Problem

I am studying the numerical aspects of fourth-order elliptic problems now, and I came across the plate problem: Let $\Omega\subset\mathbb{R}^n$ bounded domain with Lipschitz-Boundary. Find $u$ s.t. $\...
1
vote
0answers
35 views

How can I understand instantons as sheaves?

In specific, instantons are considered torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two moduli spaces (...
2
votes
0answers
21 views

Moving two links in a two-dimensional space

I have two robotic links. They're basically sticks. These two linked sticks can be controlled by changing their angle with respect to the former link. In this case, the first stick simply moves like ...
2
votes
1answer
89 views

Doppler effect: an understandable explanation for a mathematician

I have curiosity by Question. Can someone explain me, and to the audience too, the mathematical essence behind the so called Doppler effect? Thanks in advance. Then you have the ability to ...
3
votes
1answer
47 views

Show series representation of orthogonal polynomials

wikipedia has the following series expansion for hermite polynomials, namely: $$\exp \left\{xt-\frac{t^2}{2}\right\} = \sum_{n=0}^\infty {\mathit{He}}_n(x) \frac {t^n}{n!}.$$ Does anybody see how ...
2
votes
1answer
71 views

Energy functional and Euler Lagrange equation

We know that for potential energy functional, its derivative is called the Euler Lagrange equation and physically, it means that at the given point there is a force balance. Now if the energy ...
0
votes
0answers
19 views

Deducing the Equation of a Transformed Sinusoid

Given a wave, which you know to be a transformed sinusoid, how can you determine its equation? I have the following, which is a wave I obtained experimentally: It is a little off what we would ...
0
votes
1answer
42 views

Integration of motion using resistance and gravity.

I'm having trouble with a high school mathematics question. An object of mass $1kg$ falls from rest in a medium in which the resistance to motion is given by $r=kv^2$, where $k$ is a constant and $v$ ...
0
votes
1answer
21 views

Show a function behaves as a harmonic oscillator

We have a function $V(x)$ (potential energy) with $x$ being some variable. This function has a minimum at a certain $x_0$. We assume that $V(x)$ is an analytic real function of $x$ around $x_0$. ...
0
votes
1answer
50 views

Many worlds probability of getting cancer

I have first asked this question on physics.SE (where I personally believe it belongs), however it was suggested that this question better fits here, so here I am. My understanding of probabilities ...
3
votes
1answer
59 views

raising/ lowering indices

Here is my understanding of tensors: There is more than one way to think about tensors. One way is be thinking about tensors as objects with components which obey some transformation laws. For ...
2
votes
1answer
132 views

Why is $|\cos\theta d\omega|$ the projection of the differential solid angle $d\omega$ onto the $(x,y)$-plane?

Let $B\subseteq\mathbb R^3$ be the ball with radius $r>0$ around $0$ and $S_{\partial B}$ be the surface measure of the boundary $\partial B$. Given a piece of the surface $A\subseteq\partial B$, ...
0
votes
0answers
92 views

Trace of six gamma matrices

I need to calculate this expression: $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$ I know that I can express this as: $$ Tr(\gamma^{\mu}\gamma^{\...
0
votes
1answer
34 views

Norm of orthogonal matrices

Can someone help me with this problem. I have no idea how to solve it!! If A is a p×q matrix, U is a p×p orthogonal matrix, and Z is a q×q orthogonal matrix, prove that $||A||_2=||UAZ||_2$
0
votes
1answer
47 views

Finding the volume of a real egg if the volume of an egg shape(with different dimensions) on a graph is known

Equation of the egg shown above: If the volume of the egg show above is: $12.00405units^3$(found using calculus) if the volume of a real egg is $55cm^3$ Is there anyway of finding out the ...
7
votes
1answer
73 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
1
vote
1answer
40 views

How can I solve $\beta^2=\frac{m^2g}{h}\left(-\frac{\beta t}{m}+e^{\frac{\beta t}{m}}-1\right)$ for $\beta$?

This equation arose when I tried to find out how to derive $\beta$ in Stokes' Drag Force $F=\beta v$ as a function of the time $t$ it takes a mass $m$ to hit the ground after falling from a height $h$:...
0
votes
0answers
29 views

Special case of the inverse Ising problem with equal correlations

Let $s_1,\dots,s_N\in \{-1,1\}$ be $N$ binary spins. The problem of finding a symmetric interaction matrix $J=(J_{i,j})_{i,j=1}^N$ with zero diagonal and an external magnetic field $h=(h_i)_{i=1}^N$ ...
0
votes
1answer
39 views

How does the Pauli principle work?

Let $H$ be some Hilbert space. Now in general, in quantum mechanics, the vector space representing states of $n$ (non-interacting) particles is $H^{\otimes n}$, but if I consider these particles of be ...
0
votes
1answer
19 views

Christoffel connection

I am trying to determine the correct expression when expanding a contravariant derivative acting on another contravariant derivative acting on the Ricci scalar. $\nabla^a \nabla^b R = \partial^a \...
3
votes
1answer
80 views

Diffraction and Fresnel Integrals

Migrated from Physics SE due to mathematical content I am trying to derive the intensity variation function for a single slit diffraction. Sorry for the poor diagram... So I decided to take the ...
0
votes
0answers
58 views

The solution of Allen-Cahn equation?

$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ $$...
1
vote
1answer
62 views

Double-commutator $[f,[f, - \Delta]] = -2 |\nabla f|^2.$

This book (proof of Theorem 3.2) in chapter 3.1 claims click me that an easy computation shows that $$[f,[f, - \Delta]] = -2 |\nabla f|^2.$$ where $[.,.]$ denotes the commutator. Unfortunately, I ...
3
votes
0answers
47 views

Can I split this integral to a sum over three contours?

I have the following integral $$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$ and this integral has ...