"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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Finding a composite solution to an ODE (boundary layer problem)

Given $\epsilon \frac{d^2u}{dt^2}-a(t)\frac{du}{dt}+b(t)u=0$, where $a(t)>0$, $u(0)=1$, $u(1)=1$, and assuming that the boundary layer is at $t=1$, and the boundary layer variable is ...
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21 views

Constant of motion that is not the hamiltonian

Given the lagrangian $L(x,\dot x)=\frac12 (\dot x_1^2+\dot x_2^2)-\frac k 2(x_1-x_2)^2$, we know that its hamiltonian is a constant of motion. (See here) Is there another function $f$, not of the ...
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39 views

Nonlinear Schrödinger Equation

I have to find equation and starting condition to solve Nonlinear Schrödinger Equation with periodic edge condition. This method should control the propagation of fiber optical signal. In details I ...
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35 views

Shape Of A Blimp.

Was playing around with solids of revolution, the shape given by rotating $y=\sqrt{\sin x}$ about the $x-$axis seems to resemble a blimp. The only thing I can find out about the natural shape of ...
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36 views

Rope wrapped around pole Friction

A rope is wrapped $M$ whole turns round a cylindrical post, the two ends of the rope going in opposite directions. The coefficient of friction between rope and post is $0.25$. It is desired that by ...
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27 views

Confusion on Theorem in Kato's book

On page 432 (pdf-page: 455) of Kato's book perturbation theory of linear operators, I do not understand why in Theorem 1.15 $$H_n = \int dE_n(\lambda)$$ instead of the ususal thing $$H_n=\int ...
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I am calculating the Shannon entropy and stuck on this integral

I am calculating the Shannon entropy of $\left|\Psi_{+}\left(x_{+}\right)\right|^{2}=\frac{1}{\sigma^{3}_{+}\sqrt{2\pi}}x^{2}_{+}\exp\left\{-\frac{x^{2}_{+}}{2\sigma^{2}_{+}}\right\}$, which is given ...
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38 views

Green`s function of bending equation with damping term

Suppose I have Green`s function of initial-boundary value problem $$ \frac{\partial^4w}{\partial x^4}+\alpha^2\frac{\partial^2w}{\partial t^2}=f(x,t),~ \alpha\neq 0,~ 0<x<l,~ t>0, $$ $$ w = ...
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I am having some difficulties deriving the Wigner function

Forgive me for asking this question. I am deriving the Wigner function, $$ W\left(x_{1},p_{1},x_{2},p_{2}\right)=\frac{1}{4\pi^{2}}\int ...
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1answer
17 views

Boundary conditions on periodic Sturm Liouville Problem

I arrived at the following problem when using separation of variables to solve a PDE on $\mathbb{R}^2$ using polar coordinates. In that case I needed to impose the condition that $u(r,0)=u(r,2\pi)$ ...
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40 views

Centre of Mass of a Quadrant of a Square from which a Quadrant of a Circle is Cut

The wing of a hang-glider is a uniform lamina, formed by removing from a square of side $l$ a quadrant of a circle of radius $l$, with its centre at one corner of the square. Find the distance of the ...
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19 views

How to deal with this boundary condition when using separation of variables?

Consider Laplace's equation $\nabla^2u = 0$ on the region $A = [0,a]\times [0,b]\times [0,c]\subset \mathbb{R}^3$ and suppose we impose the boundary conditions: $$u(0,y,z) = \sin \frac{\pi y}{b} \sin ...
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2answers
79 views

Understanding Bell's inequality vs. quantum mechanics

I have difficulty to understand how Bell's inequality rules out local hidden variable theory. It seems to me that there is some hidden variable in the Kolmogorov's axiomatization of probability ...
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25 views

General Solution to linear Schrodingers Equation

I am trying to find a solution to $$\displaystyle \left[-\frac{1}{2}\nabla^2 - \frac{2}{r} + C(r)\right]\phi(r) = E\phi(r)$$ where C(r) is a known function of r. I am just looking for some help on ...
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20 views

How to find friction?

The question is as follows: A straight uniform beam of length 2h rests in limiting equilibrium, in contact with a rough vertical wall of height h with one end on a rough horizontal plane and with the ...
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1answer
29 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 ...
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2answers
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$\nabla \sqrt{\rho} \in L^2(\mathbb{R}^3) \implies \rho \in L^3(\mathbb{R}^3)$

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 ...
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24 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 ...
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1answer
34 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 ...
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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 ...
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2answers
39 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} ...
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1answer
24 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 $u^3$?
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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 ...
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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 ...
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192 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 ...
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31 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 ...
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32 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 ...
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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 ...
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1answer
51 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 ...
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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 ...
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1answer
15 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 ...
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1answer
58 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), ...
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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 ...
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2answers
46 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.
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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 = ...
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1answer
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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 ...
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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 ...
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1answer
66 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 ...
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1answer
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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 ...
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1answer
35 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 ...
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16 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 ...
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1answer
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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. ...
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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 ...
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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 ...
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1answer
82 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 ...
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37 views

Converting meter per second to km/h [closed]

Is there's a formula for converting Meter/s to Km/h? Example I need to convert 1200M/s to Km/h.
3
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1answer
45 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
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1answer
44 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 ...
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18 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 ...
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1answer
41 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 $v$ ...