"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)

1
vote
2answers
59 views

How can $ \Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $ possibly be the Heaviside Step Function?

How can $$\Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $$ possibly be the Heaviside Step Function? What I'm looking for is a direct visualization or maybe an approximate ...
1
vote
1answer
85 views

Proving that $\int \delta \dot{x} dt = \delta x$

Everytime I've seen this I've assumed it was true. It seems plausible. But I would like to rigorously prove it. I think this is correct, but I would like another opinion because my mathematics is very ...
0
votes
1answer
31 views

Pendulum tension force

I realize this is physics related, although the problem is really about math so I thought it would be a good fit for this site. My illustration is supposed to depict a pendulum and the forces ...
2
votes
0answers
41 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
0
votes
2answers
40 views

Solving Bernoulli equation transformation

I'm trying to solve the Bernoulli's equation via perturbation method but I need some help understanding how its done: We start off with $y'=-y+\epsilon y^2$ with $y(0)=1$. Then how is it possible ...
3
votes
1answer
96 views

What is a good reference for rigorous Electromagnetism and Electrodynamics?

Is there any good book on Electromagnetism from a more mathematical point of view? By this I mean a book which makes use of differential forms and maybe De Rham cohomology. I was also searching for ...
0
votes
0answers
19 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
votes
0answers
18 views

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 ...
16
votes
3answers
1k 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
vote
1answer
36 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
votes
1answer
34 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
votes
0answers
37 views

Potential theory for LCA groups

I was wondering if there is a potential theory for locally compact abelian groups.
1
vote
0answers
28 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
votes
1answer
19 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
votes
1answer
50 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
votes
1answer
66 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
145 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
86 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
84 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
votes
0answers
21 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
votes
1answer
65 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
vote
0answers
44 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?
4
votes
1answer
102 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
37 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
21 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
67 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
vote
1answer
116 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
vote
1answer
40 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
72 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
71 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?
0
votes
2answers
159 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
vote
0answers
24 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
40 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
votes
1answer
30 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
votes
2answers
131 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 \ ...
-1
votes
1answer
29 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
votes
0answers
9 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
32 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
votes
1answer
70 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
54 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 ...
1
vote
1answer
44 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
vote
1answer
66 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
votes
0answers
57 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
vote
0answers
69 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
vote
0answers
31 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
votes
0answers
119 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
votes
0answers
87 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
votes
0answers
43 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
33 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
vote
1answer
48 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 $ ...