"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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89 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. ...
0
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0answers
30 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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28 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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0answers
17 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 ...
1
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2answers
30 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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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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0answers
7 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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34 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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28 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 ...
2
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1answer
21 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 ...
1
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2answers
21 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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0answers
15 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 ...
2
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0answers
49 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 ...
1
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0answers
47 views

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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0answers
19 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 ...
2
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1answer
50 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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0answers
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 ...
0
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0answers
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 ...
2
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2answers
37 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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0answers
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: ...
0
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0answers
22 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) ...
0
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2answers
39 views

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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72 views

Does number 1 really exist? [closed]

1) As per decimal system when we start numbering we can start from 0.000000.....1 or the number before 1 ie .99999999999......9 so since .00000... can be infinity we dont even start with the first ...
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41 views

I need your help 😞 [closed]

Prove p = 1 c2 (v ×m) = 1 c2 vm0 ˆy.
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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
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 ...
0
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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= ...
0
votes
1answer
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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0answers
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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0answers
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 ...
2
votes
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 ...
0
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0answers
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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0answers
6 views

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 ...
0
votes
1answer
39 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 ...
0
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1answer
28 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: ...
2
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0answers
55 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
2
votes
1answer
39 views

Are Friedmann equations linear or nonlinear?

I'm trying to improve my understanding of cosmology, and these 2 equations are basic . You can find them here: https://en.wikipedia.org/wiki/Friedmann_equations Also, if you could tell why they are ...
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votes
1answer
46 views

Unable to understand a solved problem: Impulse & Impulsive force. [closed]

Why the acceleration $f$ of $m$ to the attached solution of the problem has been taken as $\frac{m-M}{m+M}g$.
0
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1answer
41 views

Another problem on Impulse & Impulsive Forces

A bullet of mass $125$ gms strikes a fixed block of wood horizontally with a velocity of $100$ metres/seconds. The resistance of the wood is $500$ kg-weight. Show that the distance through which ...
5
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0answers
46 views

Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and ...
1
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2answers
89 views

A problem on Impulse & Impulsive Forces [closed]

A jet of water leaves a nozzle of $1$ inch diameter at a speed of $50$ ft/sec and impinges on a plate fixed at right angles to its direction. What pressure is exerted on the plate? If the ...
3
votes
2answers
190 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 ...
1
vote
1answer
33 views

Scale change properties in $R^n$ for a kinetic energy functional

I'm working on a project and for it I needed to read and understand a paper by Berestycki-Lions about the existence of Nonlinear Scalar Field equations. (If someone has it or is further interested in ...
1
vote
3answers
75 views

Stone's theorem for bounded operators

Let $H$ be a Hilbert space (assume separable if you like), and let $(U_t)_{t\in\mathbb{R}}$ be a unitary representation of $\mathbb{R}$ on $H$. Let us assume that $t\mapsto U_t$ is continuous, where ...
0
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0answers
17 views

Projector method for tensor and double groups

I'm currently trying to understand a computation in my script. The setup is the following: We are looking at the double group of $C_{3v}$, i.e. $C^D_{3v}$. The character table is given by the ...
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2answers
45 views

Challenging circular question (Empty bobbin)

An almost empty bobbin is pulled along a flat surface by a thread which is wrapped around it. The diameter of the inner reel is 5cm and that of the outer wheels is 10 cm. Assuming no slipping or ...
3
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0answers
15 views

Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
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0answers
19 views

An integral of modified bessel functions

How to integrate the expression ${(\frac{\omega \rho}{3c})}^2(\frac{1}{\gamma^2}+\theta^2)^2[K^2_{2/3}(\xi) +\frac{\theta^2}{\frac{1}{\gamma^2}+\theta^2}K^2_{1/3}(\xi)]$ w.r.t. $d\theta$ for the ...
0
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1answer
36 views

Calculating maximum velocity in simple harmonic motion

I'm a bit confused about simple harmonic motion... If a particle is in simple harmonic motion, to calculate the maximum velocity can I use either displacement = 0 or acceleration = 0, since i know in ...
2
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0answers
22 views

Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated ...