"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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61 views

Relation between a linear second order differential equation and Riccati special differential equation

Consider the following differential equation \begin{equation} \frac{d}{dx}\left[N(x)\frac{dw}{dx}\right]+\sigma^2\rho(x)w=f(x,\sigma),~~ 0<x<l, \end{equation} $0<N\in C^1(0,l)$, $0<\rho\in ...
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1answer
38 views

Finding the reflection of a plane wave from a sphere

The physical problem I'm trying to solve is this: I would like to find the "reflection" of a harmonic plane sound wave in a liquid, from a spherical air bubble. I'm modeling the problem as follows: ...
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31 views

Specifics on the Williamson normal form algorithm

I'm looking at the algorithm for Williamson normal form for symplectic diagonalization of positive-definite symmetric real matrices, given on pp.24 here: https://www.ime.usp.br/~piccione/Downloads/...
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3answers
35 views

Physics problem on derivatives and integrals

So there are a few basic formulas I'd like to start with, $W=\int_0^bFdx$, $F=ma$, and $a=\frac{d^2}{dt^2}x$. In words, Work $(W)$ is defined as the area under a Force versus Displacement $(F/x)$ ...
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36 views

Dynamics of fluid

While reading on Wikipedia about the partial differential equations (https://en.wikipedia.org/wiki/Partial_differential_equation), I wondered how dynamics for the fluid occur in an infinite-...
6
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92 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
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18 views

vertical projection with gravity as acceleration

$\mathbf Question.$ A stone projected vertically upwards with initial speed of $u\ m/s$ rises $70\ m$ in the first $t$ seconds and another $50\ m$ in the next $t$ seconds. $\bullet$Find the value of $...
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1answer
20 views

Find the equilibria

Consider the equation $\ddot s = s-s^3.$ Let $m=1.$ 1) Write this as a first order system. Let $\dot s=v.$ Then we get $\dot v=s-s^3.$ So first order system is $$\begin{pmatrix} \dot{s} \\ \dot{v} \...
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20 views

Find the potential energy and sketch it

I'm given the equation $\ddot s=s-s^3.$ I'm asked to compute the potential energy and sketch it. I'm also given $m=1.$ To do this I have done the following: $$F=ma=\ddot s=s-s^3=\frac{-dV}{ds}=\frac{...
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1answer
34 views

Book's for potential theory: single and double layer potential

Does anyone know recommend me some book about the theory of the potential, especially that concerning the layer potential. Besides the theoretical part in the higher dimension, if there are concrete ...
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0answers
12 views

The expectation number of collision to slow down below certain value

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward ...
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1answer
26 views

Units of $F(x) = x-x^3$

If we are given that $F(x) = x-x^3$ is a force function, which means it is in the units of $[M][L][T]^{-2}$, then how do we determine what kind of "unit units" participate in this function? Namely, ...
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1answer
20 views

another help with partial differential equation.

I am to solve next task. solve this PDE with boundary conditions. $\Delta u = \frac{64}{r^5}\sin\varphi, \quad 1<r<2,$ $u'_r|_{r=1} = 2\cos^2\frac{\varphi}{2}, \quad u'_r|_{r=2} = 4\sin^2\...
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1answer
39 views

Is there a general way to prove this Fourier transform property?

We know that one of the important Fourier transform properties is that, the Fourier transform of a narrow function has a broad spectrum, and vice versa, We can easily see this in this example, the ...
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20 views

How to model the following scenario with an ODE if possible

Consider a cylinder, full of charged particles travelling through. From the perspective of looking through the tube, you would see a circle of particles and obviously this circle continues down the ...
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0answers
13 views

Modelling a charged particle flowing travelling through a conductive pipe.

I'm on an internship and have a project to model how a charged particle might be affected by a conductive surface either side of it. Here's how I approached it: I assumed the particle had some charge ...
0
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1answer
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 ...
4
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1answer
52 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
49 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 ...
0
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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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1answer
114 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
45 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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0answers
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 ...
0
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1answer
26 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
25 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
37 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
81 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
26 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
71 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
24 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
35 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
52 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
25 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
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
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
872 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 equation)...
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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
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
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3answers
454 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 = \...
2
votes
2answers
33 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
votes
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'...