"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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17 views

MOI about a diagonal

If by taking a thin rod, and finding its Moment of Inertia about an axis, say through the mid point of its side, one can observe that stretching the rod uniformly along the axis of rotation will give ...
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1answer
56 views

“Flow lines” of “dust” are geodesics?

The stress-energy tensor representing "dust" takes the form$$T_{ab} = \rho u_au_b$$where $u^a$ is a unit timelike vector field, i.e., $u^au_a = -1$. Does it necessarily follow that in any solution to ...
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0answers
31 views

Equivariant Cohomology and Mayer Vietoris sequence [closed]

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
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3answers
50 views

Moment of Inertia (Square Laminas)

If I have a uniform square lamina of side length 2a and intend to find its Moment Of Inertia about a perpendicular axis to its plane, is there a general formula for this? If there isn't, I have tried ...
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1answer
21 views

Hamiltonian mechanics: constant energy hypersurfaces with $dH \neq 0$

I read substantially the following sentence in Frankel's "Geometry of physics": Look now at the level set $$V_{E}=\left\{(p,q)\in T^{*}M:H(p,q)=E\right\}$$ where $T^{*}M$ is the cotangent space, $p$ ...
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116 views

Identity in general relativity, not sure if true or not

Let $(M, g_{ab})$ be a spacetime and define a new metric, $\tilde{g}_{ab}$, on $M$ by $\tilde{g}_{ab} = \Omega^2 g_{ab}$, where $\Omega$ is a smooth, positive function. Let $\nabla_a$ denote the ...
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1answer
47 views

Circular orbit problem

A particle moves under the action of the central force $Kr^4$ with angular momentum $l$. Find the energy for which the motion is circular and find the radius of that circular orbit. From a previous ...
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1answer
26 views

Does it necessarily follow that the integral curves of $k^a$ are null geodesics?

Let $f$ be a function on a spacetime $(M, g_{ab})$ whose gradient, $k_a = \nabla_a f$, ie everywhere null, i.e., $k_ak^a = 0$ throughout $M$. Does it necessarily follow that the integral curves of $k^...
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1answer
49 views

Proving solutions to the anisotropic kepler system that meet certain constraints lie on the position axes of configuration space

The system is: \begin{equation*} x''=\frac{-\mu x}{(\mu x^2 + y^2)^{3/2}} \end{equation*} \begin{equation*} y''=\frac{-y}{(\mu x^2 + y^2)^{3/2}} \end{equation*} With $\mu>1$ a constant ...
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26 views

Is there a way to directly recover differentials from an integral; anti-separation of variables?

I have an expression that relates the tension in a string, moving in two dimensions, to its acceleration. I'm sure that it's a solved problem, but in my exploration I saw that twice the tension is the ...
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1answer
29 views

How to diagonalize a Hermitian matrix using a quasi-unitary matrix?

I met a problem requiring the diagonalization of a $2n\times 2n$ Hermitian matrix $H$ in the following way: $U^{*} HU=D$, where $D$ is diagonal, $U^*$ is the transpose conjugate of $U$. The matrix $...
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0answers
24 views

Fourier series method of solving inhomogenous wave equation on infinite interval?

My brain is muddled today and this is bothering me. I seem to remember a method of solving $$u_{tt}-u_{xx}=f(x,t)$$ on the interval $[0,1]$ with one's pick of Dirichlet, Neumann, or mixed boundary ...
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0answers
27 views

Infinitesimal canonical transformation

I'm not able to understand how they have simplified both the computations from the second line to the third. So in the first computation how did {ri,pl} become 1 in the third line and how did {pi,rk} ...
2
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1answer
38 views

How to compute the fourier transform of $\operatorname{sgn}$ directly?

I've been trying to compute the fourier transform of $\operatorname{sgn}(x)$, but I'm having trouble with the complex exponential at infinity. The issue is the following: by definition we have $$(\...
2
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1answer
83 views

Quantum Mechanics Project Ideas!!! [closed]

I am in my first year in uni and I have to write a project in Quantum Mechanics. But I have been struggling with an idea for the project since I have recently started studying quantum mechanics and my ...
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1answer
28 views

Acceleration problem involving a toy rocket launching upward

So, I have attempted this problem several times, yet I fail to get the correct answer. My question implies that a toy rocket is basically being launched ground up, so y (height) would be 0.00 m and ...
3
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1answer
74 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
59 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 ...
0
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1answer
25 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 k/2 ...
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2answers
36 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 coordinates ...
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1answer
56 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
26 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 r)...
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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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0answers
19 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
28 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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0answers
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
875 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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0answers
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 equation)...
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0answers
30 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
9 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 ...
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3answers
455 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
35 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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21 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
55 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 can'...
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1answer
82 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
68 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 ...
0
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1answer
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 ...
2
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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
81 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
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(\...
0
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0answers
25 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
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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0answers
40 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
37 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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0answers
23 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 ...
0
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1answer
20 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
33 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
45 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 ...