"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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7
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87 views

Existence of classical solution for equation $\Delta u + v(x) u = 0$

It is known that if $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$ then the Dirichlet problem $$ \begin{array}{rl} \Delta u & = 0 \quad \text{in $D$}, ...
0
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100 views

Stability of Lax-Wendroff Approach for Advection Equation

The Problem: I am attempting to solve the following problem in 1D over a periodic region: "In one dimension, the mass density $\rho$ is advected with velocity $v$, so that it follows the equation: ...
0
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38 views

Solve the following IVP for the wave equation in 2D with numerical methods

I must resolve the following IVP for wave equation in 2D: $$\begin{cases} u_{tt}=u_{xx}+u_{yy} & (x,y,t)\in\mathbb{R}^2\times(0,+\infty)\\ u(x,y,0)=f(x,y)= \frac{1}{1+x^2+y^2} & ...
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1answer
36 views

well-posedness of a mathematical model

what is the meaning of Well-posedness of a mathematical model of a physical phenomena for example stokes equation in fluid dynamics ? what is the necessity to prove that a model is well-posed? how ...
1
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1answer
56 views

The Hodge dual and the Moyal product related or just notation?

The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an inner product space $V$ of dimension $n$. So we can we write; \begin{equation} \lambda\in ...
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29 views

Derivatives : trouble to understand formulas

My teacher gave us some useful formulas, but honestly I don't know how to understand it. gradient of a scalar field : $d_{x}i{V^{i}}f(M)\varepsilon ^{i}$ gradient of a vector field : ...
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29 views

Stochastic resonance

Having a look at the book Dynamical Cognitive Science, by Lawrence M. Ward, MIT PRESS, I encounter something which might be useful for my research, namely what he calls stochastic resonance, of great ...
3
votes
1answer
87 views

Total thrust on the face of a vertical dam

"A vertical dam is a parabolic segment of width $12m$ and maximum depth $4m$ at the center. If the water reaches the top of the dam, find the total thrust on the face." Is it possible to answer this ...
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62 views

Algebraic treatment of Feynman potential wells?

This question refers to the discussion here: Chomsky, Feynman, Thom I will try the divide-and-conquer strategy to try to make some progress in the problem dealt with in the above-mentioned link. My ...
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1answer
240 views

Can someone identify the following key equations in physics? [closed]

I need help identifying the following equations in physics. Most equations are related to quantum mechanics, a few is from relativity and electromagnetism. Thanks
1
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1answer
26 views

Charge given to the electroscope

The question is: What is the charge given to the electroscope? Also each ball has a weight of 25g. Here's how I started. First I draw the scheme on the forces acting one one ball (I took the one ...
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44 views

Finite difference scheme for the continuity equation

I am currently trying to solve a system of PDE's numerically, one being the equation; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 $$ I have been reading up on ...
1
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1answer
42 views

Help for understanding a vectorial equation found in a paper.

Trying to implement a code for the algorithm described in this paper I found something not very clear to me that leads me to misunderstand the whole concept. To calculate the vector $\vec{b_{3d}}$ ...
1
vote
1answer
63 views

Basis for Clifford algebra $Cl^2 (W)$ and quotient space $Cl^3(W)/Cl^2(W)$

Consider a basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W=V \oplus V^*$. I need help to write the basis for Clifford ...
2
votes
1answer
118 views

An elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system

Arnold in his essay On teaching mathematics made the following statement: The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only ...
6
votes
1answer
179 views

How to find the integral curves that are orbits of one-parameter groups?

Consider $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta ...
3
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91 views

Functional derivatives on manifolds

This might be more of a physics question, but it is mathematics-related, I hope I am not out of place with this. Let $(M,\mathcal{S},g)$ be a smooth, $n$-dimensional manifold equipped with a Riemann ...
1
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1answer
43 views

time delay in a system of difference equations

I was looking for some advice in regard to incorporating a system of difference equations with a delay (sort of?). I'm not a mathematician and just started to play a little bit around with systems of ...
3
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0answers
162 views

Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 ...
5
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0answers
66 views

The metric and Kronecker's delta

I am reading some lecture notes for GR and it is currently showing how we are going to derive the field equations using a metric for a massive free particle with a metric ...
0
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0answers
24 views

Does an arbitrary product of $f$ and $f^\dagger$ belong to a universal enveloping algebra of the Heisenberg algebra?

The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons $[f,f^\dagger]=1$. $f$ is called an annihilation operator in physics ($f^\dagger$ creation operator). ...
2
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1answer
84 views

Reference: Bethe Ansatz Equations

Could someone show me good references to find solutions of the Bethe Ansatz Equations, for simple cases (using algebraic geometry or others interfaces with mathematics)? For example in the case of ...
0
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2answers
66 views

How do you keep track of what vectors nabla ($\nabla$) should be working in on?

Take the following example: $$\vec\nabla\times(\vec A \times \vec B)$$ I assumed that this worked out to: $$\vec A(\vec\nabla.\vec B) - \vec B(\vec\nabla.\vec A)$$ Where, in both terms, Nabla ...
0
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1answer
65 views

Interchange of derivatives

Given Euler-Lagrangian equation $$\frac{d}{dt}\frac{\partial L}{\partial \dot q}-\frac{\partial L}{\partial q}=0$$ Can I equivalently write as $$\frac{\partial \dot L}{\partial \dot q}-\frac{\partial ...
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24 views

How to describe the motion of a mass point?

Consider a mass point moving around a fixed point on a circle with radius $r$ with constant angular velocity $ω$. At a certain moment of time, the connection is removed, and the point mass is flying ...
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0answers
41 views

Interior boundary value problem for the Helmholtz equation

Let $D \subset \mathbb R^d$ be a $C^2$ bounded domain. I consider the following boundary value problem for the Helmholtz equation $$ (\Delta+k^2)u = 0 \quad \text{in $D$}, \\ u|_{\partial D} = ...
6
votes
1answer
115 views

Dirac Gamma matrix identity

In my library's (old -- 1996) copy of Peskin and Schroeder, there's an identity I'm struggling to prove. In my copy it occurs on page 42, between equations 3.28 and 3.29, but I don't know how well ...
3
votes
3answers
109 views

A problem on infinite domain diffusion equation

Consider the following problem $$u_t-u_{xx}=p(x,t), -\infty<x<\infty,t>0$$ $$u(x,0)=0$$ $$u\rightarrow0 \text{ as } x\rightarrow \pm \infty$$ This can be solved using many sub problems as ...
3
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1answer
70 views

Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
0
votes
1answer
77 views

hanging pictures: a practical question about horizontal centering on a wall

Here's a little math/physics problem I just ran into with some house maintenance. Suppose you want to hang a heavy picture/mirror in the center of a wall. However, the studs are not arranged in a way ...
1
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1answer
30 views

Notation in Srednicki's QFT

In the book Quantum Field Theory by Srednicki, equation 21 for the commutators of the generators of the Lorentz group is ...
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0answers
20 views

Period ground state 1-dim Ising model

Good morning! I'm at the beginning of my study about the Ising model and it has been proposed to me this problem: Find all periodic ground-state configuration for the following one-dimensional Ising ...
3
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2answers
161 views

Is “mixed math” a useful way to learn math?

I was reading a book about how mathematics was taught in Cambridge in the 19th century, and it struck me how much physics was included in the syllabus, and it wasn't optional but everyone had to learn ...
3
votes
1answer
52 views

Over what distance is transporting a box of DVDs faster than a 100Mbps connection?

This questions blends a bit of math and computer science, but I thought this would be the most appropriate SE board for it (if not please guide me to what you believe is the most appropriate board for ...
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0answers
113 views

how to solve Schrödinger equation

I would like to solve a complete solutions of the Schrödinger equation for a particle with time & position dependent mass ($m(x,t)$) moving in a potential $V(x,t)$. Any suggestions to solve ...
0
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2answers
66 views

How to visualize(inside ones brain) the Four-dimensional_space

Can the fourth dimension https://en.wikipedia.org/wiki/Four-dimensional_space be visualized intuitively by the humans. Does the professional mathematicians can do this ? If so what are the things to ...
2
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0answers
36 views

BVP eigenvalue problem

I am working on the following problem and I am completely stuck: Show that the eigenvalue problem $$ -u''+4\pi^{2}\int_{0}^{1} u(x)\,dx=\lambda\,u $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ has ...
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0answers
68 views

Green function for a second-order elliptic PDE

Let $L = L^*$ be a second-order elliptic PDE with smooth and bounded coefficients in some bounded domain $\Omega \subset \mathbb R^d$, $d \geq 3$, with smooth boundary. Let $G(x,y)$ be a Green ...
0
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0answers
87 views

Overlap of Planets in Elliptical Orbit

I'm investigating further into my orbital overlap problem. I've already looked into the overlap ($0°$ angle between the two orbits) of two planets in a circular orbit around the sun. I'm now trying ...
0
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1answer
154 views

Angular Velocity around an ellipse [duplicate]

I'm investigating into the angular velocity of a planet in its elliptical orbit. I have these variables defined: speed of planet. speed of planet at perigee and apogee. length of orbit. ...
3
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0answers
57 views

Simply-connected Lorentzian manifold and event horizon

Can a simply connected Lorentzian manifold admit an event horizon? Or does the event horizon makes it non-simply connected?
3
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0answers
40 views

12 in the definition of Virasoro algebra and Regge symmetry

In the definition of Virasoro algebra, there is a following condition on the generators: $[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}$ Now, Regge symmetry is the following ...
2
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0answers
61 views

Doubt in the derivation of the field Euler-Lagrange equations

I'm looking at a derivation of the Euler-Lagrange equations in a field setting, and one step in the proof is continually eluding me. Let $\phi(\vec x,t)$ be a field and $\mathscr ...
3
votes
1answer
92 views

How could I find the vector potential in cylindrical coordinates?

In a physics problem I'm asked to find the vector potential $\vec{A}$ given that magnetic field is $\vec{B}=\dfrac{k \mu_0 s^3}{4}\hat{\phi}$ where $k$ and $\mu_0$ are constants. I know that $\nabla ...
2
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2answers
63 views

Is it possible to build a fiber bundle whose fibers are different? (Or we should not call it a fiber bundle?)

Suppose there is a fiber bundle $E$. The base space is $M$ so that $\pi:E\rightarrow M$ is the projection. By the definition, the bundle has a typical fiber $F$ such that the local trivialization over ...
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45 views

Thomson problem vs. maximizing sum of distance

Given $N$ equally charged points lying on the unit sphere ("electrons"), the Thomson problem consists of finding the configuration of these points such that the electrostatic potential energy $$ ...
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0answers
76 views

Calculating force per unit width

Question: A line source of strength $2πm$ is located a distance $a$ above a horizontal plate. Find the force per unit width on the plate, ignoring gravity and taking the pressure below the plate to be ...
1
vote
1answer
88 views

A difficult question on mathematical physics

Let $TQ^*$ be equipped with its standard symplectic structure and let $X_H$ be a Hamiltonian vector field which is tangent to the fibers of $\pi: TQ^* \to Q.$ I need to show that $$H=h \circ \pi = \pi ...
0
votes
2answers
42 views

Does Runge Kutta need future state of system?

In order to use the RK methods, you need to know the state of the system at future time-steps which can be expensive to compute (e.g., in physics simulations). As a simple example I'll use RK-2: In ...
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0answers
35 views

Why does this graph produce a straight line? [duplicate]

When we graph the sin and cos of theta against the range of a projectile, we get a straight line. When we graph range against angle, we get a hyperbola. Why does the sin and cos of theta against ...