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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43 views

Application of the Bernoulli Equation

I'm attempting a question on fluid dynamics and I'm using a rearranged form of the Bernoulli Equation. But I can't prove the equation for the velocity of water leaving the tap. It should equal $$v = ...
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1answer
79 views

Generators of a semi simple lie algebra must be traceless

Consider a semi simple lie algebra. Show that if $T_a$ are the generators of a semi simple Lie algebra then $\text{Tr}T_a=0$. Attempt: $[T_a, T_b] = ic^c_{ab}T_c \Rightarrow \text{Tr}[T_a, T_b] = ...
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0answers
44 views

What are Einstein's evolution equations for galaxies? [closed]

I'm researching galaxy distributions and have been tasked with solving Einstein's evolution equations for different levels of dark energy and matter. I've been told to do this numerically via Matlab, ...
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0answers
22 views

Reference request for complex scalar field, propagators worked out with path integral approach? [closed]

In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can anyone supply me a reference to where the ...
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0answers
71 views

Proving some simple formula with gamma matrices

a) Prove that $\text{Tr}(\gamma^5 \not a \not b) = 0,$ where $\not a = a_{\nu}\gamma^{\nu}$. b) Prove that $\text{Tr}(\gamma^5 \not a \not b \not c \not d) = 4i\epsilon_{\alpha \beta \gamma \delta} ...
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1answer
57 views

Differential equation free fall in gravitational field

For a physics problem I was told to set up a differential equation for the free fall in the gravitational field of the earth. The equation (via Newton) I've got is following: $$\ddot{r} = - G M \frac ...
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0answers
24 views

Potential energy of a Dipole interaction unexplained sign error.

Hello i've attactched a question as well as my attempt at an answer, I get the correct mathematical expression however I have a sign error for both the terms in the bracket. I can't work out what ...
2
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2answers
30 views

Solving Equation of the form $\sqrt{(a+\frac{b}{2})^2+L^2}-\sqrt{(a-\frac{b}{2})^2+L^2}=c$

I have been struggling to find the solution to one of my physics problems mathematically as this is the equation I arrive at where all of the values are known except $a$. I have tried solving for a ...
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1answer
26 views

Using epsilon to rewrite equations in terms of linear combinations

In physics, the damped oscillator is governed by the equation of motion: $\ddot{x} + 2\beta \dot{x}+\omega _{0}^2x=0$ where $\omega _{0}=\beta$ for an oscillator with critical damping. The solution is ...
6
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0answers
106 views

Stokes' Theorem and Vector Fields with Jump Discontinuities

What are the continuity requirements on a vector field $\boldsymbol{A}$ such that Stokes' theorem, $$ \iint_S\nabla\times\left[(\boldsymbol{\hat{x}}\cdot\nabla\phi)\boldsymbol{A}\right]\cdot ...
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0answers
31 views

Drawing Trajectories in a State Space (is Energy Conserved?)

There's something I'm slightly confused about regarding drawing the trajectory of a particle in state space $(x,v)$, where $v:=x'$. Here, I'm only working with $x\in\mathbb{R}$. Suppose a particle ...
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1answer
31 views

Calculation of inverse function

My question is physics related but the problem itself is just math. I have an expression for a refractive index depending on the wave length $\lambda$: $$ ...
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1answer
27 views

Solving differential equation and obtain expressions with unknowns?

I have the following differential equation $my'' + \beta y' + mg = 0$ , with condition $y(0)=0$. I need to solve the equation and obtain expressions for $y(t)$ and $y'(t)$. I have attempted to use ...
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0answers
21 views

Angular velocity and it's dependence on rotation vectors

This question is about rotational dynamics. Starting with an inertial and a body frame of reference, and using some sequence of 'euler' rotations, I can arrive at the expression of angular velocity. ...
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1answer
13 views

Invariance of radiative transfer equation in the absence of absorption and emission

The question is asking me to show that the line-of-sight intensity of radiation is invariant when there is no emission or absorption. Starting with the radiative transfer equation: $$ ...
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1answer
25 views

Show using centers of mass that the angle bisectors of ΔABC are concurrent at I, where I lies on bisector AP at a position such that:AI/AP = b+c/a+b+c

I found it quite easy to show that AP = a+b+c, because if b and c are the mass of points B and C respectively, P is the sum of their masses (b+c). Then segement AP has a mass of a+b+c because it ...
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0answers
53 views

Finding equations of motion for inverted pendulum using Lagrange

I studied engineering 10 years ago but am struggling to remember how to find the equations of motion for this project of mine. I have a two wheeled inverted pendulum that I want to balance. $M_w = $ ...
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2answers
18 views

Differential equation in Potential of circular symmetry "D

Given the potential $U_{pot} = \frac 1 2 k r^2$ where $\vec r = (x,y)$, $r=\sqrt{x^2+y^2}$ we can calculate the force $$\vec F = - \nabla U_{pot} = -2k \frac{1}{2} \frac{1}{\sqrt{x^2+y^2}}2(x,y) = ...
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0answers
18 views

Curious on how much force is exerted

This may not be the correct forum to ask, but may be a good starting spot. I saw on facebook this machine: (https://www.youtube.com/watch?v=j2SwTK6p72U) and was amazed that it could handle the torque ...
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0answers
86 views

On the Visual Manifestation of Curves in Nature

All sorts of curves are useful in modelling and describing phenomena we observe. Trig functions, logarithms, exponentials, polynomials, hyperbolas, circles, and so forth are all very useful in this ...
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1answer
30 views

Physics flow rate

Find the flow rate through a tube of radius $5~\mathrm{cm}$, assuming that the velocity of fluid particles at a distance $r~\mathrm{cm}$ from the center is $v(r) = 49-r^2~\mathrm{cm/s}$.
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2answers
106 views

Calculate the average acceleration and average speed of a particle

A particle has zero velocity initially (i.e., at time $t=0$) and its acceleration at $t$ seconds is $a(t)=72t−4t^3$ meters-per-second per second. During the time interval $[5,8]$ seconds find the ...
2
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1answer
108 views

When is $0$ ever used in real life?

I've just been going through an equation, which is as follows: $(x+4)^2 = 16$ Lets work through it: $$x^2 + 8x +16 = 16$$ $$x^2 +8x = 0$$ $$x^2 = -8x$$ $$x = -8$$ However, as i've just found ...
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1answer
34 views

Deriving an equation from known information.

Start with $$m_1 \, \frac{d^2r_1}{dt^2} = F_{12} \qquad m_2 \, \frac{d^2r_2}{dt^2} = F_{21}$$ and derive the equation $$\frac{m_1m_2}{m_1 + m_2} \, \frac{d^2r}{dt^2} = F_{21} $$ where $r_1$ and ...
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0answers
18 views

How is this radical transformed?

Wave equation derivation In the link shown above, there is a derivation of the wave equation. At line (1), the author says that he divides equation by $\Delta x$ and takes the limit as this quantity ...
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0answers
96 views

Differentiation under the integral sign for an electrostatic field

Let $\rho\in C(\bar{D})$ be a continuous function on the compact set $\bar{D}$ and let us define ...
3
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3answers
119 views

I don't know what this symbol means

I somehow made it to grad school without coming across this symbol: $\left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{array}\right) $ Here, $l_i$ and $m_i$ are all ...
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1answer
49 views

How to find the zeros of a fourth degree polynomial without integer coefficient

I am currently programming a simulation for a pinball game and want to calculate the time when the ball hits a circle (if they collide at some point). For the calculation part, i'm adding the radius ...
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2answers
61 views

Physics Homework - Final Velocity [closed]

So I'm doing my physics homework and I need some help on finding some answers. It's the last three questions on my sheet and I can't seem to get my head around it. A ball is projected at $20\ m/s$ ...
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2answers
44 views

High School physics/Kinematics/ statics question

A block rests on a plane which is inclined at 35 degrees. A force $F_b$ is applied to it at an angle of 40 degrees to the plane. $F_g = 980$ Newtons is the force due to gravity and $F_n$ if the normal ...
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41 views

Physics related integrals and sums. Black Body Radiation.

How were these calculated? $$\langle E\rangle = \frac{\displaystyle\int_0^\infty Ee^{-E/kT}{\rm d}E}{\displaystyle\int_0^{\infty}e^{-E/kT}{\rm d}E}=kT$$ $$ \langle E\rangle = ...
6
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2answers
116 views

Which theory is used to calculate the position and energy of a point source?

Consider an empty room with one point source that emits a stationary signal (constant sound, radioactive radiation, ...). The energy nor the position of the point source is known. We send someone in ...
2
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2answers
36 views

Conservative vector fields and Energy

If one knows that $\vec{F}=m\vec{a}$ is a conservative force, which means that $\vec{F}=\nabla\phi$, and that energy $E(t) = \frac{1}{2}mv(t)^2 + \phi(\vec{x}(t))$, where $v(t) = \|\vec{v}(t)\|$ then ...
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1answer
42 views

Is the derivative of the velocity with respect to time give us the sum of the forces? [closed]

is the derivative of the velocity with respect to time give us the sum of the forces, when the forces as function of the distance? I know the Acceleration is the derivative of the velocity with ...
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0answers
23 views

Change of the sequence of differentation in physics?

Assume to have a quantity A which is calculated from the formula $A=\frac{dB}{dC}$. dC can be written as dC=dEdF so $A=\frac{dB}{dEdF}$. I assume that the differential of A is also ...
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1answer
22 views

How to add Displacement using the Component Method?

How would I add the displacement using the Component method. So I am Given the Following: d = 20cm [N] d2 =50cm [S 35 E] d3 =100cm [ W 15 S] what I did was I first drew them and after that I added ...
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0answers
71 views

Transforming components of angular velocity vector in a rotating reference frame into time derivatives of Euler's angles

We've got a standard SO(3) setup: two Cartesian coordinate systems, one that is stationary (in the sense that we will refer everything to that one), and has the axes $X$, $Y$ and $Z$, and the second ...
2
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0answers
54 views

Function to describe teardrop shape

If I fill a plastic ziploc-shaped bag with water, the cross section profile should be sort of teardrop shaped (assuming we ignore the edge effects of the bag being sealed on the sides as well as the ...
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1answer
32 views

Find the maximum displacement given trajectory.

I'm trying to find the maximum displacement of a object given the trajectory. The trajectory is given by the equation: $$p(t)=-11.79t^2+25.9t+4.35$$ Looking around online, the closest article I've ...
2
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1answer
35 views

Finding electric flux given volume charge density

Question: Let $\rho_v = 8z(1 - z)$ C/m$^3$ for $0 < z < 1$ m, $8z(1 + z)$ C/m$^3$ for $-1<z<0$, and $0$ for $|z| > 1$. (a) Find $\vec{D}$ everywhere. (b) Sketch $\vec{D}_z$ vs. $z$, ...
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1answer
176 views

How to calculate moment of inertia of parabolic spandrel?

I want to find the moment of inertia of this spandrel along $DD'$ $$I=\int r^2 \,\mathrm dm$$ The area of the spandrel is $\frac{3m}{ab}$ so $I=\sigma\int r^2\mathrm dA $ where $\sigma$ is aerial ...
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0answers
29 views

Does $\{f,g\}$ mean anything when neither $f,g$ are the hamiltonian of a system?

Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...
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1answer
41 views

Recovering electric charge from potential

I have a region $\Omega \subset \mathbb{R}^2$ and a harmonic function $\phi$ on $\Omega$. I know that for some "charge" $Q:\partial \Omega \to \mathbb{R}$, $$\phi(q) = \int_{\partial \Omega} ...
2
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1answer
88 views

Flying Superhero

The path through the air of a flying superhero has the shape of the graph: $$y=\frac{1}{100}\cos\left(\frac{x}{2}+3\right)+\frac{1}{4}$$ A sidekick on the ground measures the displacement of the ...
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1answer
48 views

Legendre functions $Q_n(x)$ of the second kind

Legendre functions $Q_n(x)$ of the second kind \begin{equation*} Q_n(x)=P_n(x) \int \frac{1}{(1-x^2)\cdot P_n^2(x)}\, \mathrm{d}x \end{equation*} what to do after this step? how can I complete ? I ...
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0answers
19 views

Help with derriving and understanding an equation

In this paper about bottle rockets, the second page starts with an equation derrived using the transport theorem about conservation of momentum \begin{equation} \boldsymbol{F_s} - \boldsymbol{W} - ...
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1answer
78 views

Physics related initial value problem (horizontal spring mass system)

Consider the horizontal spring-mass system where the spring-force is the only force acting on the mass. Suppose that a mass is initially at $x=x_0$ with an initial velocity $v_0$. Show that the ...
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1answer
26 views

Question about angular velocity?

If $z=r\cos\theta +ir\sin \theta$, show that $\dfrac{dz}{dt}=r\dfrac{d\theta}{dt}(-\sin\theta+i\cos \theta)$ and that if $\dfrac{d\theta}{dt}$ is constant, then $\arg\dfrac{dz}{dt}=\theta-\pi/2$. I ...
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1answer
25 views

How do divergences of vector fields generate distributions?

Just to clarify the title before I start, there are some "fuzzy" words that I want to get out of the way: Divergence here is in the sense of the divergence theorem, the operator sometimes written ...
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1answer
45 views

Can an object be in freefall if it is traveling upward? [closed]

Can an object be in freefall if it is traveling upward? I'm thinking the answer is no?