"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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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 ...
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1answer
46 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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21 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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25 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} = ...
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1answer
92 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 ...
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3answers
102 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 ...
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1answer
46 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 ...
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1answer
33 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 ...
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1answer
20 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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14 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
129 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 ...
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1answer
47 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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101 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 ...
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2answers
59 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 ...
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0answers
32 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
30 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 ...
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46 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 ...
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1answer
36 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. ...
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47 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?
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25 views

Kinematics(Newtons laws of motion - acceleration) - Practical

Note: initial velocity = 0ms^-1 , (S= displacement) I am trying yo plot a graph of a practical i have earlier done( varying inclination angle and measuring time taken to displace the whole incline), ...
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32 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 ...
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0answers
35 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
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1answer
75 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
47 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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22 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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42 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 ...
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1answer
75 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 ...
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2answers
37 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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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 ...
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2answers
76 views

What does adding $\sin\theta \cos\theta$ make my graph a linear relationship?

What is the point of adding sin n cos of theta when graphing range? e.g. I see on hyperphysics a graph of range vs sin n cos of theta and it makes the experimental data embody a linear relationship. ...
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4answers
192 views

Is $\nabla$ a vector?

The following passage has been extracted from the book "Mathematical methods for Physicists": A key idea of the present chapter is that a quantity that is properly called a vector must have the ...
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1answer
32 views

Pure Point Spectrum implies Spanning Eigenfunctions

If $H$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$, and the spectrum of $H$ is a pure point spectrum, i.e., the spectrum consists of discrete eigenvalues (perhaps with multiplicity ...
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39 views

How to check that this is an orthogonal linear map with $\det (A) = 1$, so it is a rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
3
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2answers
77 views

what is scattering theory?

I often read the the words "scattering theory", "scattering data", "scattering matrix", scattering XXX ... in my math lecture, but I realised that I am not able to define it correctly. A short search ...
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0answers
25 views

How to find the axis of the rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
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58 views

Neutron density PDE

On Mathews and Walker's book exercise (8-2) We are given that the neutron density n inside $U_{235}$ obeys the differential equation $$\nabla ^2u+\lambda u=\frac{1}{k}\frac{\partial{n}}{\partial{t}} ...
2
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1answer
136 views

Tough second order differential equation

I can't figure out this diff equation (in cylindrical coordinate). How can I solve it ? Any comments appreciated $$ \frac{1}{r}\frac{d}{dr}(r\frac{dE}{dr})+\frac{d^2E}{dz^2}+(\epsilon_0 ...
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0answers
78 views

Characteristic class integral: on what manifold does $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$ hold?

Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds? Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd ...
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0answers
41 views

Using Feynman's Subscript Notation

I have a homework problem that wants me to calculate the force $\vec{F} = \vec{\nabla}_{\vec{X}}U + \frac{\mathrm{d}}{\mathrm{d} t} \left(\vec{\nabla}_{\dot{X}} U\right)$ where $U(\vec{X}, \dot{X}, ...
3
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3answers
104 views

Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm ...
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1answer
17 views

How to find out the asymptotic behavior of a Bessel function?

How can one find out the asymptotic behavior of a Bessel function? If we start from $z= 0$, we can get a Taylor series. But in physics, we have to know the asymptotic behavior of the solution at both ...
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1answer
53 views

Metric and Convariant Tensor

$g_{ij}$ is the metric tensor. Show that $g^{ij}$ which satsifies $g_{ij}g^{jk}=\delta_i^k$ is a covariant tensor of rank $2$. I am not sure how to show this? Does it instead mean to show that ...
2
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2answers
119 views

Find $t$ in $i = 50\sin\left(120\pi t -\frac{3\pi}{25}\right)$ where $i = 25$

An alternating current generator produces a current given by the equation $$i = 50\sin\left(120\pi t - \frac{3\pi}{25}\right)$$ where $t$ is the $\text{time}$ in $\text{seconds}$. (Q) Find ...
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3answers
42 views

Addition in linear vector spaces

In the definition of linear vector spaces, one of the axioms is that the addition must be commutative and associative. The addition of scalars and matrices are both commutative and associate. Can ...
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2answers
38 views

Center of mass Double Integral?

Can you help me with this problem? Find the center of mass of a lamina whose region $R$ is given by the inequality: $$|x|+|y|\le 1,$$ and the density in the point $(x,y)$ is : ...
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3answers
60 views

Calculate center of mass multiple integrals

Can you help me with this problem? Find the center of mass of a lamina whose region R is given by the inequality: and the density in the point (x,y) is : The region r is this one: Is this the ...
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1answer
87 views

Determining when these two waves separate

There's probably something really obvious I should be getting, but I haven't yet developed the intuition for working with the wave equation. Suppose we're given the wave equation $u_{tt} = c^{2} ...
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1answer
32 views

No non zero solution to E.V.P in $L^p$

Can you show that: If for some $1\leq p\leq \infty$ function $f\in L^p(\mathbb{R}^n)$ solves $\Delta f-\lambda^2 f=0$ then $f\equiv 0$. (This is essentially uniqueness of solution to homogenous ...
3
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0answers
42 views

Taking a stationary phase approximation of a multidimensional integral

I'm looking for a way to take a stationary phase approximation of an integral of the following form: $$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - ...
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1answer
24 views

A ball that is thrown upward.

A ball is thrown upwards from a point on the ground, with an initial velocity $v_0$. the ball is affected by earth's gravity, and force of fraction with air that depends on the velocity of the ball. ...