"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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Introductory Reference for Mathematical Physics

I'm a senior undergraduate student studying differential geometry. I have experience with smooth manifolds, some elementary theory of Lie groups, and a little multi-linear algebra. I understand this ...
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38 views

Issue in first order differential equation

I've tried many times to reach the solution of a first order differential equation (of the last equation) but unfortunately I couldn't. Could you please help me to know how did he get this solution. ...
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10 views

Boundary conditions for a radiative heat transfer problem

Consider the heat equation $$ \frac{\partial T}{\partial t} - a\Delta T + \mathbf v \cdot \nabla T = S $$ where $S$ is a source term dependent of the radiation intensity $I$ and the temperature $T$. ...
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25 views

How to scale a problem involving heat and the flow of a viscous fluid?

I recently got set this problem and I was wondering if anyone would be able to give me some help on the later parts. An incompressible thermal conducting fluid is contained between two infinite ...
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1answer
24 views

Evolution operator always in $SU(n)$?

Think about the evolution operator $U$ in Quantum Mechanics for finite dimensional systems. Then this operator satisfies an equation $$U'(t) = -iHU(t)..$$ Here, I assume that $H$ is ...
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36 views

AP Physics: What topic is this question? [on hold]

So I have a summer assignment for AP Physics, and we haven't really learned any of this material. The first question is this $$T_s = 2\cdot 3\sqrt{\frac{4.5\cdot 10^{-2}\ \mathrm{kg}}{2.0\cdot ...
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23 views

Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...
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33 views

Simple Harmonic Motion under Periodic disturbing force

A particle of mass $m$ is executing a SHM in a straight line under an acceleration $n^2 \times (distance)$. If a periodic force $mk \cos{pt}$ be introduced and the time period of forced vibration ...
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91 views

Fourier-Bessel series of $f(x)=x^2$

I'm trying to calculate the expansion of $f : [0,1]\to\mathbb{R}$ given by $f(x)=x^2$ in a Fourier-Bessel series of zeroth order. In that case let $J_0$ be the $0$-th order Bessel function and ...
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41 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
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33 views

How to deal with this type of integral?

When we have a complete orthogonal set of functions $\{\phi_n\}$ on a certain interval $I$ we might want to expand a certain function $f : I\to \mathbb{R}$ in a generalized Fourier series. In that ...
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459 views

Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...
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228 views

What are super-translations?

There's been a lot of news lately about a possible solution to the black hole information paradox from a presentation given by Stephen Hawking to the KTH Royal Institute of Technology in Stockholm. ...
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34 views

Solving an integral that includes an exponential function and the error function

This question contains all the values needed to compute an equation. My question is, do you get the same result I get? Or do you get the result in the paper I've linked to? I'm trying to decipher ...
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2answers
32 views

Triangle of forces

Forces equal to $5P$, $12P$ and $13P$ acting on a particle are in equilibrium ;find ,by geometric construction and by calculation ,the angles between their directions? I have an problem that, With ...
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42 views

Schwarzschild solution question

Since we set the Ricci tensor to be zero everywhere, why is it still a solution if it doesn't apply to the point where the point mass exists? Shouldn't it apply also to that point as well, or am I ...
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27 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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21 views

What is a conjugate weight?

The authors here write that the longest element of the Weyl group is $$w_{\max} = - id$$ except for $E_6$, $A_r$ and $D_r$ with $r$ even. There they write that $w_{\max}$ acts on a weight $\lambda$ ...
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11 views

Does complexification make a self-conjugate representation non-self-conjugate?

I recently learned that a non-self-conjugate representation is not the same as a complex representation. Given a real representation $\pi$, with highest weight $\mu$ $$\pi : \mathfrak{g} \rightarrow ...
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37 views

Scaling Two Equations

I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated. An incompressible thermal conducting fluid is contained between two infinite ...
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32 views

Its physics that lead to development of maths or its maths that lead to the development of physics?

Like calculus(which is math) was discovered from physics. Are there theories in which math leads to the discovery of new physics theories? So is it physics that lead to development of maths or its ...
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22 views

Einstein summation convention: Del operator and dot product

Now, I am aware of the summation convention for the dot product $$\mathbf{a} \cdot \mathbf{b} = a_i b_i$$ But I am unsure about how to represent $(\nabla \cdot \mathbf{a}) \mathbf{b}$ and ...
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1answer
22 views

Sketching a Graph of a Particle Trajectory

How can I sketch the trajectory of a particle of mass $m$ with a position vector $\mathbf{r} = \cos(\omega t)\,\hat{\mathbf{i}} + \sin(3\omega t)\,\hat{\mathbf{j}}$ ? Will this be a three ...
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20 views

Flow between two infinite horizontal plates

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
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52 views

QFT and topology

I have had a course in topology, I have heard of homotopy quantum field theory and topological field theory, but I dont know anything about QFT, what would be a good starting point to learn about the ...
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Trouble understanding Poisson Brackets

I'm looking at page 94 here - I understand the definition of Poisson brackets at the top of the page (which uses summation convention) but I don't get why the calculations in (4.61) are true. I'm ...
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20 views

Uniqueness of the Green's function

Given a linear operator $L$, a Green's function $G(x,s)$ is any solution of $$\tag{1} LG(x,s) = \delta(x-s)$$ where $\delta(x-s)$ is the Dirac Delta function. The Green's function can also be used in ...
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1answer
52 views

Are the physics and math definitions of a complex representation equivalent?

I was astonished to read at Wikipedia that The term complex representation has slightly different meanings in mathematics and physics. In mathematics, a complex representation is a group ...
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28 views

What does it really mean to complexify the $10$-dimensional representation of $ \mathfrak{so}(10)$?

A commonly used "trick" in $SO(10)$ Grand Unified Theories is to use a "complex" instead of a "real" $10$-dimensional representation for the Higgs fields. My problem is understanding what this ...
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23 views

Expanding E, B in post-Newtonian Gravitational Potential

Thanks to someone who can help me with this particular equation. I've been trying to take a stab at these equations by myself, though I realized I need to seek some help. I'm currently trying to ...
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37 views

Why is a weight automatically a complex weight?

EDIT: I think the partial answer to my question is that in order to talk about weights we always need a complex Lie algebra. If the Lie algebra is real, we use the complexification, This is necessary, ...
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38 views

Weighting In a Function

What is the intuitive explanation of weighting factor $\alpha$ and $1-\alpha$ in the equations such as score, optimization, smoothing etc, that takes the form below: $$ f(\alpha) = \alpha \cdot A ...
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47 views

Angular Velocity calculation

I am trying to calculate the time derivative of the quaternion from the following paper: Robotics and Biomimetics (ROBIO) See equation 1 below: ...
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24 views

Euler-Lagrange equation of motion for tensegrity

I have read this paper “Dynamic equations of motion for a 3-bar tensegrity based mobile robot” (1) and this one “Dynamic Simulation of Six-strut Tensegrity Robot Rolling”. 1) ...
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1answer
295 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
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2answers
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Decomposition of an unitary operator by simple operators

For quantum computation, it's well known that any unitary operator can be approximated with an arbitrary accuracy by simple operators, for example to approximate an unitary operator on n qubits by no ...
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22 views

Fourier methods and a conductor bar

I was doing this question bellow: I tried: Could you help me in the 3 (second Picture) and how to solve the problem?
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1answer
29 views

Gaussian optics

We are given $\frac{n_1}{l_0}+\frac{n_2}{l_i}=\frac{1}{R}(\frac{n_2s_i}{l_i}-\frac{n_1s_0}{l_0})$ $l_0=\sqrt{R^2+(s_0+R)^2-2R(s_0+R)cos(\phi)}$ $l_i=\sqrt{R^2+(s_i-R)^2+2R(s_i-R)cos(\phi)}$ $h= ...
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1answer
62 views

Finding Equations of Motion

A package is dropped from an aeroplane travelling horizontally at speed $U$ at time $t_0$ and height $z_0$. The package experiences acceleration due to gravity $ \boldsymbol{F_g} = $ $m$ $\boldsymbol ...
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41 views

Finding angular acceleration

Given: $\mu_B=0.52$ $\theta=30^{\circ}$ Weight- $25$ lb $\omega=0$ $l=6$ ft $1/\kappa=3\sqrt 2$ radius of curvature. Find $\alpha$ My Equations of motion are the following: $\xleftarrow{+}\sum ...
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1answer
50 views

Conformal group in two dimensions

In Conformal field theory, physicist says, the conformal group in two dimensions is infinite dimensional, so the associated with the infinity of generators and infinity conserved charges provided. Is ...
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191 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty ...
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30 views

Collision of Inelastic ball above the ground [migrated]

An inelastic ball of mass $m$ is dropped from a height $h$ above the ground and at the same time a second ball of mass $m_1$ projected vertically upwards to meet the former. Show that in order that ...
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28 views

A question in Special Relativity. [migrated]

In books the equation for length contraction is derived by supposing that the velocity of the spacecraft is the same for both observers. So the question is that, is the velocity really the same for ...
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1answer
37 views

Find a ratio of velocities.

The following image shows a circular disk rolling on a surface. If the velocity of a point on the edge of the circular disk is $V{p}$ and the velocity of the center of the disk is $V_{cm}$ then find ...
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43 views

Analytical solution to $mx''(t)+b(x'(t))x'(t)+k(p)x(t)=0, p(t)=k(p)x(t)/A$

I have the following differential equation $$mx''(t)+b(x'(t))x'(t)+k(p)x(t)=0$$ As can be seen, "attenuation term" is dependent of velocity $x'(t)$. Also stiffness term $k(p)$ is dependent on the ...
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Optimal set of generators of conformal group in 2D

Can we write Lorentz transformations and dilations in terms of translations and special conformal transformations? In V. Kac's book "Vertex algebra for beginners" 2nd edition, on p.7, Kac writes that ...
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1answer
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Three cylinders on two inclined planes, each inclined at an angle alpha.

Two smooth uniform right circular cylinders, each of mass $m$ and radius $a$, are placed symmetrically in contact with each other and with 2 planes, each inclined at an angle $\alpha$ to the ...
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1answer
40 views

Writing an expression for a change in angular velocity of an angle

Let $AB$ is rotating at $\omega_{AB}=4$ rad/s. Find $\omega_{CD}$ when $\theta=\pi/6$. So the first thing I did was wrote an express for $CD$ call it $r$. $\phi$ is Angle $CAB$ for reference. By ...
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1answer
29 views

Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: ...