"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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Find the energy for which the motion under the central force is circular

I am told that a particle moves under the action of an attractive central force $F=\frac{-k}{r^2}\hat r,$ with angular momentum $L.$ I am asked to find the energy for which the motion is circular and ...
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23 views

derive solution of possion equation ; electrodynamics problem

Hi I had posted the same post 2 days ago but I am posting it again because of my bad handwriting. I apologize to the man who wanted to read my post. I am not familiar with the tool which is used in ...
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1answer
24 views

Cauchy Problem (Waves with a Source)

Solve: $$u_{tt}=c^2u_{xx}+x t,\quad u(x,0)=0, \quad u_t(x,0)=0$$ The final answer should be $u=xt^3/6$. I keep getting $xt^3/2$. How I did the problem: 1/2[phi(x)+phi(-x)]+1/2c int(x+ct,x-ct, 0) dy ...
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1answer
79 views

Arnold's proof of Liouville's Theorem on integrable systems

My question happens to be almost identical to the one left unanswered/closed here, which gives a bit of background information - it may not be necessary. I hope the reason it was closed on ...
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42 views

Rotating Tube Mechanics

I've asked this question before, but it was closed down as I didn't show any working. I have now completed all of the question apart from (bii). I think that the polar coordinates are: $$x=l \sin(\...
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23 views

What is spectral flow symmetry?

I can't find much about this, and am looking into this to satisfy personal curiosity. I will like to know what spectral flow is, and what spectral flow symmetry is. I tried looking for this on ...
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31 views

Mechanics, trying to find the absolute velocity of a bead on a tube

A rectilinear tube of length 2l rotates with a constant angular speed ω around the vertical axis through the middle of the tube at a constant angle αα!=0 with the tube. The tube does not move up or ...
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39 views

Kinematics Motion along a circle, trying to find the absolute velocity of a bead on a tube.

A rectilinear tube of length 2$l$ rotates with a constant angular speed ω around the vertical axis through the middle of the tube at a constant angle $α$!=0 with the tube. The tube does not move up or ...
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1answer
33 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...
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1answer
30 views

Einstein Summation Convention Minkowski Metric

Picked up a book on General Relativity for Mathematicians, but I'm a bit unclear on some of the tensor notation. For example, the Minkowski Metric $$\eta_{\mu \nu} (\Delta x^\mu)(\Delta x^\nu)$$ ...
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21 views

Lifting the Einstein-Hilbert action into the frame bundle

If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some ...
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1answer
54 views

Milne-Thompson Theorem with a Vortex

I'm doing a problem related with Milne-Thompson theorem which tells that: "A cylinder of radius $a$ is immersed in a counter-clockwise whirlpool, which we model here as a potential vortex of intensity ...
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1answer
19 views

Showing that the integral of one equation yields another.

Background: The equations are derived from a Physics 2 Lab circuit that has a resistor and a capacitor Problem: Show that the integral of equation 5 yields equation 2. I'm given: $I(t) ...
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3answers
32 views

Showing that one physics equation 'satisfies' another

Background: This is from a Physics 2 Lab. The equations come from a circuit that has a resistor and a capacitor I'm given these two equations $V - \frac{dq}{dt} R - \frac{q}{C} = 0$ <== Eqn(2) ...
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1answer
42 views

Solving a differential equation using $F=ma$

A body with mass $m = \frac{1}{2}$ = kilogram $\left(kg\right)$ is attached to the end of a spring that is stretched two meters by a force of $100$ Newtons. It is set in motion with an initial ...
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1answer
29 views

Using linear algebra to find resonance frequency and normal oscillations and motion

I am stuck part way through the following and not sure how or if finding eigenvalues will help with finding modes of oscillations: Consider the system of three masses and two ideal elastic bands: $(...
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2answers
110 views

Where should the Lorentz transformations fit into this?

I am trying to figure out how to "see" things in relativity via a toy model. With a pinhole camera I'd like to capture a relativistic scene consisting of a vertical marked stick which is moving ...
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3answers
40 views

Trying to understand the Nabla Operator

I'm trying to wrap my head around the following line done in my physics textbook: $\vec\nabla f(r) = \begin{pmatrix} f'(r) \frac{\partial r}{\partial x}\\ f'(r) \frac{\partial r}{\partial y}\\ f'(r)...
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2answers
140 views

Uniqueness of a periodic solution for nonlinear pendulum

I am working with the system of ODE's or second order differential equation representing the nonlinear pendulum with constant torque and damping. \begin{equation*} \theta'=v \end{equation*} \begin{...
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22 views

Probability distribution obtained by repeatedly sampling $S_x,S_y$ on a spin-$S$ system

While trying to rework an upcoming quiz problem for a quantum physics course, I came up with the following question which turned out to be harder than I expected. (Note: I take $\hbar =1$ in all ...
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3answers
29 views

Is the following derivation of how to find $v$ given $a=v'$ wrong?

My physics professor did the following: Let $a(t)=v'(t)$ be a given function. Suppose $v(0)$ is known, then $$ \int_{v(0)}^{v(t)} dv=\int_0^ta(t)dt \iff v(t)=v(0)+\int_0^ta(t)dt $$ I believe ...
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35 views

elastic strings and springs mechanics problem.

This is an example given in Edexcel M3. In question below length =1m and λ=10N but the given answer(Circled in red) it looks like the value of λ multiplied by 2. I couldn't figure it out why? Need ...
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26 views

On Hyper-geometric Functions and its recurrence relation

I research in generating functions of Hyper-geometric functions $_2F_1(a+n,b;c+n;x)$ using Lie group theoretic method and so the recurrence relation is important in this method. I want recurrence ...
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27 views

Schrödinger's equation denumerable eigenvalues

The Schrödinger's equation can be written in this form: $-u''(x)+V(x) u(x) = E u(x) $ $V(x)$ is a function that is defined on the real line. We know ${u}^{2}$ is integrable on the whole real line. ...
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44 views

Static Friction

The coefficient of static friction between car’s tires and a level road is 0.80. If the car is to be stopped in a maximum time of 3.0 s, its maximum speed is (a) 2.4 m/s (b) 23.5 m/s ...
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94 views

Intuition behind definition of spinor

Some time ago I searched for the definition of spinors and found the wikipedia page on the subject. Although highly detailed the page tries to talk about many different constructions and IMHO doesn't ...
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1answer
57 views

Finding the Velocity of a Particle after an Impact

If a particle of mass $m$ has velocity $v$, its momentum is $p=mv$. In a game with balls, one ball of mass $2g$ springs with velocity $2m/s$, it hits two balls, both of which have mass $1g$, and ...
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17 views

Identity in continuum mechanics

For a problem in the textbook I am reading, I need to prove that $\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS$, where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a ...
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3answers
74 views

Falling objects - finding the speed [closed]

I am trying to work out how fast water will be falling by the time the water hits the ground. If it starts 100m high how fast would it be travelling and why? With the acceleration because of gravity ...
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185 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
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1answer
106 views

Introductory book on probability for physicists

I'm a physics student looking to start learning more about probability. Is there some introductory book on measure theoretical probability theory that includes sections on quantum probability? To ...
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3answers
47 views

Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
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1answer
93 views

Motion of a pendulum with air resistance

I am trying to model the motion of a pendulum with air resistance. I have resolved perpendicular to the direction of motion to get this equation where $m$, $g$, $p$, $C_D$ and $A$ are constants: $$mg\...
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42 views

How can I solve this partial differential equation?

I'm modeling the dynamic localization, after solving the Helmholtz equation I obtained this partial differential equation, if anybody can give me a guideline I would be truly grateful. $$ i\alpha(...
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1answer
216 views

In what sense does analyticity guarantee the following equality?

I was reading a paper$^1$ on particle physics, and at some point it is stated that, provided $f(x)$ is analitic, we have $$ f(x)-f(0)=\frac{x}{\pi}\int_0^\infty \frac{\text{Im}\;f(y)}{y(y-x-i\...
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37 views

Wick renormalization of stochastic integral

I am trying to understand a paper that summarizes some results concerning Wick renormalization of some stochastic integral. In the last few lines of the paper the authors say: In Euclidean ...
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2answers
39 views

What is the speed of the car given the time taken to receive an echo?

I am trying to solve this question- The driver of an engine produced a whistle sound from a distance $800m$ away a hill to which the engine was approaching.The driver heard the echo after $4.5s$....
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1answer
31 views

The inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$.

I am reading the book. On page 80, there is a concept the inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$. Here $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra $...
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18 views

Notation in Belavin-Drinfeld's classification of solutions to classical Yang-Baxter equations.

I am reading the paper, on page 6, equation (3.5), there is a notation $(1 \otimes \alpha)r_0$, where $r_0 \in g \otimes g$, $g$ is a semisimple Lie algebra, $\alpha$ is a root. For example, suppose ...
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1answer
25 views

Magnitude of velocity and acceleration around a track?

A car travels around a circular track with a radius of $r=250m$. When it is at point $A$ then $V_a=5m/s$ which increases at a rate of $\dot{v}=(0.06t)m/s$. Determine the magnitude of its velocity and ...
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20 views

What is the relation between solutions of classical Yang-Baxter equations and solutions of modified Yang-Baxter equations.

Let $g$ be a Lie algebra. The classical Yang-Baxter equation (CYBE) is: $$ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. $$ The modified classical Yang-Baxter equation (MCYBE) is: $$ [r_{...
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22 views

Associated Laguerre Polynomials negative indices?

For the associated Laguerre polynomials/functions, it is taken (specifically when solving for the eigenstates of Hydrogen in QM) that the associated Laguerre functions with negative indices (and also ...
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1answer
27 views

Double integral multivariable calculus

Consider the following integral $$\int_0^1 dx_1 \int_0^{1-x_1} dx_2 \, (1-x_1-x_2)^{-\epsilon-1} (-sx_2 - x_1p_1^2)^{-\epsilon-1}$$ where $s$ and $p_1^2$ are to be treated as constants throughout the ...
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1answer
40 views

Interpretation of a reaction diffusion equation

I have a reaction-diffusion equation in 1-dimensions of the typical form: $$\frac{\partial }{\partial t} u(x,t)= \frac{\partial^2 }{\partial x^2} u(x,t)+ \alpha(x) u(x,t), \,\qquad (x,t)\in (0,1)\...
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1answer
52 views

Limit of probability density function as random variable approaches +/- infinity

Consider a complex-valued function $\Psi(x,t)$ such that $|\Psi|^2$ is a probability density function for $x$ (for any time $t$). In his introductory Quantum Mechanics book, David J Griffiths writes ...
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22 views

The sum of two r-matrices.

Let $g$ be a Lie algebra. Suppose that $r \in g \wedge g$ satisfy the condition: $[[r, r]] = [r_{12}, r_{13}]+[r_{12}, r_{23}] + [r_{13}, r_{23}]$ is a non-zero unique, up to scalar multiple, ...
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1answer
112 views

Origin of delta

Why does delta mean change? What is the origin of delta? I understand that upper-cased delta is used in this way and that delta is the fourth letter of the Greek alphabet. I also read that delta is ...
4
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3answers
174 views

Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
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43 views

Why is $J_n$ not symmetric, for $n\notin\mathbb Z$, while Bessel's equation is still symmetric?

Bessel's equation, $$x^2y''+xy'+(x^2-n^2)y=0,$$ has even parity, regardless of the value of $n$. So a solution of this equation must be even or odd. However, the Bessel functions $J_n$, which are ...
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1answer
16 views

Double random number from a gaussian, how to evaluate the skewness

I have a question for an application in physics. So my description will be really concrete, sorry. It's about the estimation of a systematic error from a calibration system. I have a LED with an ...