Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

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Integrating the equation for change in entropy

In physics, the change in entropy is defined as $$dS = \frac{dQ}{T}$$ I tried integrating this. $$\int ds=\frac{1}{T}\int dQ$$ $$S=\frac{Q}{T}+C$$ This essentially converts the change in entropy to ...
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17 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
21 views

Which functions can be the resistance of a network?

Say we have $n$ resistors, with unknown resistances $r_1,\ldots,r_n$. We build a network using these, along with any finite number of other resistors of known resistance. We then measure the ...
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22 views

Time taken for a Mass on a Spring to Travel 7mm with constant friction and a force acting against it.

So I have been driving my self mad with this problem and I cant seem to deduce a logical answer any help would be greatly appreciated. Problem: I have a mass on the end of a compression spring. It is ...
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1answer
42 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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1answer
108 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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2answers
47 views

How to solve $a = \cos x - b\sin x$ where $a$ and $b$ are real numbers?

I found this equation when solving a physics problem related to finding an angle when entering a river, that has a known current, and trying to get to a specific point on the other side. I'm not sure ...
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1answer
25 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 ...
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0answers
74 views

Diffeomorphisms vs symplectomorphisms / volume conserving diffeomorphisms in an application

This question needs a bit of background: one way to study the mechanics of deformation of a continuous solid body is by defining a reference body $B_0$, a connected, well-behaved subset of $R^2$ or ...
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1answer
29 views

Tips to find magnitude of 2 forces when given the magnitude of their resultant

Forces A and B has a resultant force C with magnitude of 200N. The magnitudes of A and B have the relation of 2||A||=3||B||. $\theta$ is the angle between A and C, and the angle between B and C is ...
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1answer
23 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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19 views

Notation for variables representing complex numbers

Is there a standard way to indicate that a variable represents a complex number? In physics, it is convenient to analyze oscillating systems using complex numbers. The authors of one popular ...
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4answers
134 views

A Proportion Question

If $a$ is directly proportional to $b$ and also directly proportional to $c$, is it true that $a$ is directly proportional to $bc$.(It seems like it is true) Here is what I did and I have a feeling ...
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0answers
20 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 ...
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0answers
33 views

Interesting derivation of the moment of inertia of a Ball

I was attempting to calculate the moment of inertia for a ball of radius R about the z axis when a mistake still led me to the correct answer and the seemingly correct method leads to the wrong ...
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1answer
25 views

Stuck in using Stirling's approximation to show and justify an approximation of the number of permutations with and without ordering

This is a problem from my applied mathematics class where we are currently working on using Stirling's approximation which is: $ n! \sim (\frac{n}{e})^n \sqrt{2 \pi n} $ and the context of this ...
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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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1answer
60 views

A problem with calculus..

A body is dropped in a well and it travels $p$ depth in $t$ time interval. The relation is $$p(t)=\frac4{4+t^2}+0.8t-1.$$ Find the velocity and acceleration. Now if I differentiate the ...
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2answers
48 views

What's a method for computing the indefinite integral $\int \dfrac{dz}{(a^2 + z^2)^{3/2}}$?

This integral occurs in EMFT when computing $\overline{E}$ due to an infinite line, uniform charge distribution. I'm trying to figure out the formula for $\int\dfrac{dz}{(a^2 + z^2)^{3/2}}$, using ...
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0answers
22 views

How to do this Poisson bracket proof

For the proof of the above equation, I understand the first step which has been obtained from the definition but in the second step I don't understand why they are summing over $j$ first ...
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1answer
19 views

Poisson bracket proof

For this question I understand the first line of the solution which they have obtained from the definition but how have they simplified each term to get to the second line from the first line? The ...
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1answer
99 views

Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains $$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$ The standard ...
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0answers
21 views

decreasing travel time over an arc

I am trying to write a program that travels along the same arc but each iteration the travel time needs to decrease. I have been trying to do this by changing the starting velocity and acceleration. ...
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0answers
18 views

Elliptical orbits and differential unit angle? What's $d\hat{\theta}/d\theta$

I was reading a short article on the derivation of the elliptical orbits of bodies under the influence of inverse-square gravity, and it goes something like this: $m\frac{d\vec{V}}{dt} = ...
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1answer
32 views

Poisson bracket proofs

I understand the first sentence you wrote for the need of a different summation index. However, i'm still not able to understand the individual steps. Like how in the first line we have four ...
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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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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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3answers
22 views

Basic problem on deceleration

A car accelerates from rest for $15$ s with a uniform acceleration of $1.5$ m/s^2 and immediately decelerates with a uniform deceleration of $5$ m/s^2. How long does deceleration take? I used ...
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1answer
35 views

Can we interchange derivatives when multiplied?

Although this looks like a physics question, this is more of a Math question, I was reading the Energy-Mass relationship derivation, it goes as follows, Force $F$ is given by ...
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1answer
49 views

Escape velocity

Calculate the escape velocity from a white dwarf and a neutron star. Assume that both the white dwarf and the neutron star is 1 solar mass. Let the white dwarf’s radius be $10^{4}$ kilometres and the ...
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0answers
20 views

What is the gravitational potential of an ellipsoid at $0$?

What is the gravitational potential of an ellipsoid at $0$? Given an ellipsoid: $\frac{x^2+y^2}{a^2} +\frac{z^2}{c^2} \leq 1,$ and a uniform density of mass: Using spherical coordinates, and the ...
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1answer
55 views

Simplifying a complex trigonometric expression

Context: In a previous question , I've stated I'm making a program that will be used for calculating stuff with Statics of a particle. I've come across another scenario in which there's three forces ...
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1answer
26 views

How to make vector from azimuth and speed?

I don't know how to make $3$D vector of speed from azimuth and speed. Azimuth is in degree and speed is in m/s. Azimuth is angle on $X$ and $Y$. Thank you for your help!
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2answers
34 views

Least squares regression with two predictor variables (exponential functions of time)

Question cropped from textbook (Apologies for the link- I don't have enough rep to post the actual image.) [Now pasted below. Ed.] I've come across a question in a textbook (linked above) requiring ...
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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
20 views

Normalizing the Simple (Quantum) Harmonic Oscillator for $n=2$

I am trying to find the constant for the second excited wave state that will normalize it. The equation for the second excited state is $$\psi_2(x)=A_2(1-\frac{2m\omega ...
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2answers
31 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
13 views

Simulation missiles - initialization vector

I'm trying to implement a simple simulation of the trajectory of bullets. Only for school project. I am using formula: ...
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1answer
16 views

How to calculate work done on standing a rod upright with arbitrary mass distribution m(x)

I was way overthinking how to calculate calories that you would do performing a pullup, which is a straightforward mgh. But with a sittup, you have some non-uniform torso that's being lifted to ...
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0answers
35 views

Hydrostatic force on a dam

A dam is inclined at an angle of 30° from the vertical and has the shape of an isosceles trapezoid 200 ft wide at the top and 100 ft wide at the bottom and with a slant height of 160 ft. Find the ...
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1answer
34 views

Rocket simulation. Are my equations correct?

I am simulating a rocket launch (two dimensions) and I am a little unsure about my implementation of Euler Forward. I will only give the $x$ component since $y$ is done the same way. I first ...
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3answers
65 views

Having trouble simultaneous trigonometric equations

I'm racking my brain trying to solve some formulae that I will need to implement into a program I'm making. The program is based around statics of a particle, as in that all forces acting on the ...
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0answers
18 views

Solving the finite square well graphically (Transcendental Equations)

I am attempting to comprehend how they are solving these transcendental equations here. The equations they derive are $\tan(ka)=\frac{\alpha}{k}$ which gives the even solutions and ...
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2answers
27 views

Find distance as function of time.

A car starts from rest and accelerates in $a = \frac{2\cdot m}{3\cdot s^3}t$, After $3$ seconds, The car will be $27$ metres from beginning. Find distance as function of time. I know i have to ...
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3answers
36 views

Free fall of 2 water drops

Cloud is showering 2 water drops from 1000 meters above the ground in 1 second difference. Let be Gravitational acceleration = $10\frac{m}{s^2}$, and Air resistance is negligible. What will be the ...
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13 views

Energy Conditions

Hi guys im trying to solve the energy conditions for a specific stress energy tensor but have come up at a stumbling block. How would i calculate the energy conditions for the following stress ...
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8 views

Entropy fix for 1-d conservation law with convex flux functions

Consider the following Riemann problem for a convex function $f$ \begin{eqnarray*} \partial _t q + \partial _x f(q) &=& 0 , \\ q(x,0) &=& q_0(x) =\left\{\begin{array}{ll} q_l ...
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32 views

Modelling diffusion in composite media (PDEs)

I have to construct and solve a system of PDEs for drug delivery through a transdermal patch. I have been given that the epidermis has thickness ${\delta}_1$ and coefficient of diffusion $D_1$, the ...
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8 views

Greens Function for poisson equation in cartesian along boundary

I am tryng to solve $$\nabla ^2 \phi = -\rho(\vec x)$$ Where $\phi(0) =0$ and $\phi(L)=\phi_0$ and $\rho$ is an arbitrary functionAccording to wolfram we can write G in terms of spherical harmonics, ...
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50 views

Angular moment (Hibbeler's book question)

This question's from the book HIBBELER, R. Engineering Mechanics: Statics; chapter 4. The curved rod lies in the x-y plane and has radius $3m$. If a force $F=80N$ acts at its end as shown, determine ...