"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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0answers
13 views

Find out rotation and angle given a square

Consider I have a camera and a square, the projection of the square on the camera would give me a "distorted" square. Can I possibly find out the angle of the camera relative to the square and the ...
0
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0answers
9 views

Express 3 dimensional movement by using 2 vectors?

I have a system that can be described in 3 equations(Lagrange Euler equations) for each coordinate. $L_x, L_y, L_z$: $$\frac{d}{dt} \frac {\partial \mathcal{L}}{\partial \dot{L_x}} = \frac ...
0
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0answers
8 views

A collective spin motion, related to differential equations. - - how to prove y component of the field is zero throughout the motion?

This is a pure mathematical question, here is a little background for the interested reader, you can jump directly to the mathematical part if you are not interested. background Imagine we have ...
14
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3answers
915 views

A fun problem by Arnold using the Poincaré recurrence theorem

I came across this problem by V. I. Arnold while studying his classical mechanics book. Consider a sequence where the $n^{th}$ term is made up by considering the first digit of $2^n$, the first ...
1
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1answer
28 views

Convservation of Momentum

I am taking a course in fluid dynamics. I'm trying to establish the equality $$\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x, t)g(x, t)dx = \int_{a(t)}^{b(t)}\rho(x, t)\frac{D}{Dt}g(x, t)dx$$ I can use make ...
0
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1answer
18 views

Lagrangian of bead on a rotating hoop

I'm trying to find the Lagrangian for a bead on a rotating circular loop (constant angular velocity $\omega$, radius $a$) in two different ways and I'm unsure why these are giving different answers. ...
3
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0answers
32 views

Potential theory for LCA groups

I was wondering if there is a potential theory for locally compact abelian groups.
0
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1answer
34 views

Connection between Functional Analysis and Quantum Physics [on hold]

Hi does anyone have an idea of which specific areas of Functional Analysis and Topology have a strong connection with the study of Quantum Mechanics and Quantum Field Theory? Thanks.
1
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0answers
20 views

How are 2D collision forces calculated?

Between 2 circles of the same radii, how can I calculate the collision forces to apply to each of the 2 circles? I have position, mass, and velocity for each of the circles. Here's what I have ...
0
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0answers
12 views

Source sound position mutiple point

i want find sound source position like this picture : ![find source][1] But i just konw the delay I know the delay for position 2 and 3 (or more) from source after hits the first point. I don't ...
0
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1answer
17 views

What is the connection between $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ and $V(\vec x) = \frac {1}{2} x^TPx $?

For example, $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ has an equivalent representation $V(\vec x) = \frac {1}{2} x^TPx $ where $P$ is some matrix Can someone make this connection clearer for me ...
0
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1answer
48 views

Compactification of a straight line

Like in the case of mapping a infinite-plane to a sphere (Riemann Sphere), I can understand, that I can map the infinite line ($-\infty,\infty$) to a circle. Secondly, I can also map a finite line ...
0
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1answer
46 views

Separation of variables for the Laplace equation on a disk

I have the equation $$\bigtriangleup u=\frac{1}{r} \frac{\partial}{\partial r}(r\frac{\partial u}{\partial r}) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta}=0$$ where $0<r<1$ , $-\pi< ...
4
votes
2answers
56 views

Killing vector field along a geodesic

I was trying to show that a Killing vector field satisfies the Jacobi Equation for a geodesic, just by assuming that \begin{equation} \nabla_\mu X_\nu + \nabla_\nu X_\mu=0 \end{equation} Indeed, if I ...
5
votes
1answer
38 views

Debye Function Integral (BlackBody)

Show that $$ \int^{\infty}_{0} \frac{x^{3} \, dx}{e^{x}-1} = \frac{\pi^{4}}{15} $$ by expanding the integrand in powers of $e^{-x} $ and integrating term by term. Could anyone help with this one?
3
votes
2answers
69 views

Riccati Equation for falling particle.

I'am trying to solve the differential equation for a falling particle of mass 1 with air resistance proportional to $v^2$ (v is velocity): $$v'=g-v^2$$ This is a Riccati-Equation with stationary ...
0
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0answers
20 views

Is it always true that a linear map has a quadratic action?

If I have a linear map $A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, is it always true that the action of this map on a phase space of position and momentum $(q,p)^\text{T}$ is quadratic in $q_1$ and ...
0
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1answer
44 views

Finding speed of snowballs given initial velocity and angles

You and a friend stand on a snow-covered roof. You both throw snowballs from an elevation of $14$ m with the same initial speed of $12$ m/s, but in different directions. You throw your snowball ...
1
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0answers
21 views

formula of Gauss Legendre for 9 node element for 2D dimension?

This is maybe stupid question by im really noob in math. Can you show me formula of Gauss Legendre for 9 node element for 2D dimension?
3
votes
0answers
48 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
0
votes
1answer
21 views

Jets and vertical differential

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
0
votes
1answer
16 views

Little mistake with Levi-Civita symbol property

I have this equation $ \varepsilon_{ijk}B_k = \partial_iA_j - \partial_jA_i $ and I was asked to prove $\mathbf{B}=\nabla\times\mathbf{A}$, where $\mathbf{B}=B^i\mathbf{e}_i$ and ...
1
vote
1answer
48 views

Planetary Motion: A comet describe a parabola about the sun [closed]

A comet describe a parabola about the sun, show that the sum of the squares of the velocities at the extremities of a focal chord is constant. I have no idea how to solve. Please help. I only ...
1
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1answer
87 views

planetary motion: Particle describes an ellipse as a central orbit about a focus

A particle describes an ellipse as a central orbit about a focus. Show that the velocity at the end of the minor axis is the geometric mean between the greatest and least velocities. My attempt: ...
1
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1answer
33 views

Chronology condition and metric perturbations

Let $(M,g)$ be the quotient of the 2-dimensional Minkowski space-time by the discrete group of isometries generated by the map $f(t, x) = (t + 1, x + 1)$. Show that $(M, g)$ satisfies the ...
3
votes
1answer
58 views

The Poincaré dual of a space-time curve

We have a smooth space-time curve defined by $f:C{\mapsto}M$, where $M$ is a typical curved space-time manifold. ${\eta}^{(4)}$ is the volume 4-form defined on $M$ and ${\varepsilon}^{(1)}$ is the ...
1
vote
2answers
25 views

Normal matrix is diagonalizable

If $[A,A^*]=0$ ($A^*$ is a conjugate transpose of $A$), that is, $A$ is a normal matrix, How is $A$ diagonalizable? Or, this is just a definition of normal matrix?
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2answers
41 views

Formula to calculate password cracking time in years, taking into account Moore's law and known adversary guessing power [closed]

We know that the biggest human rights violators in human history are capable of one trillion password guesses per second as of approximately January 2013. Assume that the 1 trillion guesses per ...
1
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0answers
19 views

Quantization of Hermite differential equation

In the course of solving the time independent quantum harmonic oscillation Schrodinger equation $$ \Psi^{ \prime \prime} (y) +(2 \epsilon -2y^2) \Psi (y)$$ When we try ansartz $\Psi = u(y) ...
0
votes
0answers
32 views

How do you get the curvature tensor of the Schwarzschild Solution?

So, on the Wikipedia page on the derivation of the Schwärzschild solution , I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
0
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1answer
24 views

Diagonalising an infinite-dimensional Hermitian square matrix

I have a quantum state which takes the following form: $$\rho = \sum_{b, \,c \, = \,0}^\infty \frac{(-igt)^b(igt)^c}{\sqrt{b!c!}}\vert b\rangle\langle c\vert.$$ This is an infinite Hermitian matrix ...
6
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2answers
109 views

Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ ...
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1answer
25 views

Distance between a density operator and a pure quantum state.

Given density operators $\rho_1$ and $\rho_2$ and a pure quantum state $|\psi>$. It is promised that $|\psi>$ is in only one of the given density operators. How to find which density operator ...
0
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0answers
8 views

How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
1
vote
1answer
18 views

Trajectory Code Problem

Problem 2. “Pumpkin chucking” is a competition event to see which team can shoot a pumpkin as far as possible, usually with a pneumatic cannon. In this problem we’re going to write a computer program ...
2
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1answer
65 views

Could you offer another way to prove $e^{\hat{A}}\hat{B}e^{-\hat{A}}=e^{ad\hat{A}}\hat{B}$

My professor wants me to solve this identity in two ways. Sadly, I could only do one way and haven't figure out how to solve it another way. Here is my way, Denote ...
1
vote
1answer
50 views

Punctual Hilbert scheme of four points

I am looking at $\text{Hilb}^4(\mathbb{C}^2)$, which is the Hilbert scheme of four points on $\mathbb{C}^2$. In particular, I am just looking at four points collided (at the origin), and want to know ...
0
votes
2answers
71 views

Proof that Electromagnetic force is much stronger than gravity [closed]

The electromagnetic force is given by this formula: $$Fl=q(v\cdot\,B)$$ and gravity is given by $$Fz=m\cdot\,g$$ Now do I've to proof that the electrmagnetic force is much stronger than gravity but ...
1
vote
1answer
27 views

Notation in Reed/Simon Vol. IV (and possibly an earlier volume)

I'm wondering if there are any mathematical physicists/analysts out there that can help me with some notation I've seen in Reed and Simon's books on analysis. Unfortunately I don't have time to read ...
1
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1answer
43 views

FLATLAND's sphere intersection scenario, explored for four dimmensions

I recently finished this wonderful new vintage edition of FLATLAND. http://amzn.com/918775116X In 1884, Edwin Abbott wrote this strange and enchanting novella called FLATLAND, in which a square who ...
4
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0answers
46 views

Defining the quantum group $U_q(\mathfrak{sl}_2)$

I've seen two defining relation for $U_q(\mathfrak{sl}_2)$ by the Serre relations $$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$ or by taking $K=q^H$ $$KK^{-1}=K^{-1}K=1,\quad ...
1
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0answers
39 views

Personal Experiences with Probability Simulation

Simulations methods are increasingly used in theoretical and (especially) applied probability. Personally, I have used simulation for purposes that range from recreational Q&A to applications of ...
1
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0answers
28 views

Mean-pressure of an acoustic wave

I am looking at total internal reflection for an acoustic wave, defined in terms of its pressure such that $$p = p_1 \,exp\left[-i\{\omega t-\vec{k} \cdot\vec x\}\right]$$ Using the definition of ...
2
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0answers
110 views

Is this equal ? (I found it on this website)

I found this equation on this website! I would like to know it its true or not? And how can proof or disprove it?! Euler-Mascheroni constant expression, further simplification ...
0
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0answers
85 views

How can we proof that this is equal? About $ln(n)$

I found this on this website (Euler-Mascheroni constant expression, further simplification) without any explaining why this is equal can someone give me that? ...
0
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0answers
30 views

Green's function in the context of classical mechanics

I am following this paper entitled "The classical mechanics of non-conservative systems". I would like to discuss equation (2) since I cannot get what the autor says. This is the problem: let's ...
1
vote
1answer
25 views

Parameters in the Hamilton-Jacobi Equation

I'm reading through Gelfand and Fomin's 'Calculus of Variations', and they've just derived the Hamilton-Jacobi Equation: $$\frac{\partial S}{\partial x} + H \left(x, y_1, \ldots, y_n , \frac{\partial ...
1
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1answer
46 views

maximum position uncertainty of particle in a box

I want to verify mathematically for wave function $\psi(x)$ satisfying $\psi(x)=0$ for $\lvert x \rvert \ge \frac{L}{2} $ and $\int_{- \frac{L}{2}}^{\frac{L}{2}} \lvert \psi(x) \rvert ^2 dx = 1 $ ...
3
votes
1answer
78 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to ...
1
vote
3answers
72 views

Finding initial value of differential equation

Given, $$ mdv/dt = mg - kv $$ Question is: Find the velocity $v(t)$ that satisfies this initial value problem. Also, by letting $t$ approach positive infinity, determine the terminal velocity ...