"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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63 views

Deriving Euler-Lagrange Equation

I have just started studying Calculus of Variations, and need some help about deriving the Euler-Lagrange equation. In the book I'm reading, the writer starts by imposing the following inner product ...
3
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1answer
58 views

A mirror focusing beams at one point

How can I find a shape of a mirror which focuses all parallel beams in one point? I tried to do it in this way: The mirror must be symmetric hence I assumed it has a center in the point $(0,0)$. The ...
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1answer
21 views

Proof involving Poisson bracket

Not being able to understand how each term has been simplified to get from the third step to the fourth step. So how did 1/2m become 1/m and {qj,plpl}pk become {qj,pl}plpk and how did k/4 become ...
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2answers
34 views

Differentiation with polar coordinates

I'm sorry if this is supposed to be something basic but I'm not being able to understand if r is as given above, how have they worked out r dot? What have they differentiated the x,y and z ...
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1answer
42 views

Constant Force Pendulum (undamped)

How does one sketch the derivation of the equation of motion for a planar pendulum of length l and mass m in constant gravity g, subject to a constant torque force F (directed along the tangent to the ...
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0answers
24 views

Understanding derivation of expression for magnetic flux in cylinder from Ampere's law

I'm looking at a paper using Biot-Savart and Ampere's law to determine the induced magnetic field within a conducting cylinder. By inserting $$J_z = \frac{I}{2\pi} \int_0^\infty \lambda J_0(\lambda ...
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1answer
40 views

A question regarding Eigenvalues

Note: $\psi,\psi^{\dagger} :\Bbb{R} \to \Bbb{C}$ and $x, \lambda_i , \hbar, m \in \Bbb{R}$ Say we know that $\lambda_1$ is a solution to the eigenvalue equation: $$\hat{\Pi}\psi(x)= \lambda_1 \psi(x) ...
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18 views

Modeling the Motion of a Particle where $ ||\vec{f_i}|| = \frac{k_i}{r_i^2} $

At the origin of an $n$-dimensional space, there exists a single free-moving particle ($\gamma$) with a known mass ($m$) and velocity ($\vec{v}$). There also exists $p$ number of fixed points with ...
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1answer
27 views

Justification of manipulations used to solve a physics problem.

Problem. A particle moves in a deaccelerated manner, describing a circular trajectory of radius $r$, having an initial speed $v_0$. Suppose $a_n=-a_t$ (normal acceleration and tangential ...
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35 views

I'm not certain this makes any sense: Matrix Multiplication of Metric Tensor for calculating arclength

I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Arclength Where in it gives the euclidean measure of distance as $$ ds^2 = E du^2 + 2 F du dv + G dv^2 $$ Equivalently as $$ ds^2 ...
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5answers
860 views

Why does adding a term $5f'(t)$ to $5f''(t)+10f(t)=0$ cause damping?

So we have a differential equation to model an oscillator: $$5f''(t)+10f(t)=0$$ Where the initial conditions are $f(0)=0$ and $f'(0)=4$. It is given that $f(t) = \frac{2\sqrt 2}{5}\sin\sqrt2 t$. ...
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53 views

Why do we need in general mathematical physics only orthogonal transformations.

Why do we need in mathematical physics (as I know in English it is called Partial Differential Equations) orthogonal transformations coordinates? (for example, the heat equation and the wave ...
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0answers
37 views

String theory without physics

Is there some aspects of string theory that are possible and worth learning without the physics background? Which are the most approachable ones? What are some resources for learning some of the ...
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0answers
28 views

What are examples of multi-valued mappings in the real world? [closed]

I would like to know about some examples of multi-valued mappings in the real world. Like for example, a function that relates the set of signals emitted by bats and the echo received from nearby ...
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0answers
8 views

On the algebra of functions of an embedded manifold

We know that we can embed a manifold $\mathcal{M}$ of dimension $n$ in $\mathbb{R}^m$ with $m$ sufficiently high and specify the embedding using $n-m$ relations for the ambient coordinates. The ...
3
votes
3answers
453 views

Find equation for mass in gravity

A satellite is moving in circular motion round a planet. From the physics we know that $$\Sigma F_r = ma_r = \frac{GMm}{r^2}$$ So I wanted to find the equation for $M$ knowing also that $$v = ...
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2answers
30 views

Find an function that oscillates between a given upper and lower envelope

Suppose I'm given two real, continuous functions $f(x)$ and $g(x)$ such that $f(x)\ge g(x)$ for all real $x$. I'd like to determine an oscillating function $h(x)$ that has $f(x)$ as its upper-envelope ...
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0answers
19 views

Reference for Green's Functions

Some time ago I've studied Green's Functions in one dimension. In that case we had one differential operator $L$ and the differential equation $$Lf = g,$$ and we had some boundary conditions. To ...
2
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2answers
54 views

Second-order equation

$u''_{xy}+2xyu'_y-2xu=0.$ solve it for $u(x,y)$. I received the following equations: $u=\frac{1}{2x}v'_x+yv,$ $v''_{xy}+2xyv'_y=0.$ where $v=u'_y$. All my following tryings are worthless. I ...
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1answer
80 views

Find the energy required for the motion to be circular

A particle of mass $m$ moves under an attractive central force $Kr^4$ with angular momentum $L.$ For what energy will the motion be circular and what is the radius of the circle? In order to find the ...
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0answers
66 views

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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1answer
22 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 ...
2
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1answer
22 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) ...
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1answer
60 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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0answers
40 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 ...
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0answers
22 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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0answers
29 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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0answers
35 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
26 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
21 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
45 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
18 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: ...
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3answers
30 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 $V - \frac{dq}{dt} R - \frac{q}{C} = 0$ <== ...
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1answer
35 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
25 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
102 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
38 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}\\ ...
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2answers
135 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*} ...
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0answers
19 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
27 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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0answers
26 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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0answers
24 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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0answers
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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3answers
42 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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1answer
81 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
56 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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0answers
15 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 ...
3
votes
3answers
70 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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173 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 ...