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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Integrate $\int_{-\infty}^{+\infty} \frac{1}{\sqrt{P(x)}}e^{-ax^2 - bx - c}dx$ where $P$ is a polynomial of degree $6$

From a physics problem I'm interested by a closed form of this integral : $$\int_{-\infty}^{+\infty} \frac{1}{\sqrt{P(x)}}e^{-ax^2 - bx - c} dx$$ where $P(x) = \lambda_6 x^6 + ... + \lambda_0$ I ...
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19 views

How do I find the angular momentum and the energy of a central force?

I've been studying classical mechanics with Symon's book and I'm having trouble when I have to find the energy and angular momentum for a given potencial if the particle moving in a circular orbit, ...
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1answer
67 views

Heat equation — Modelling a real-life situation

I have read through a lot of books and lecture notes that cover the heat equation and I am still not sure how I would model the easiest real world situations. For example, take a rod at constant ...
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1answer
36 views

Defining a partial derivative with respect to an antisymmetric tensor/matrix

I'm looking at some nonlinear electrodynamics, and have been following a textbook which contains a primer on some of the stuff I'm interested in following up. However, I seem to have fallen at the ...
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1answer
36 views

Impact of two bodies problem

A body of mass $M$ moving with a velocity $u$ collides with another of mass $m$ which rests on a table. Both the balls are perfectly elastic and smooth and the the body of $m$ is driven in a ...
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1answer
13 views

Curvature at a point in a vector valued function

I am trying to determine the curvature when $t=2$ of the function $r(t)=<t^3,3t^2,8t>$ So I found $v(t)=<3t^2,6t,8>$ and $a(t)=<6t,6,0>$. So now that I have these two functions, I ...
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2answers
51 views

If you flip a quarter, what are the odds that you will land on that little piece on the edge? [duplicate]

If you were to flip a quarter what is the probability you will get the quarter to land on it's little edge? How would you calculate this? Assuming the probability is not $0$ and is instead a really ...
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1answer
29 views

How do I calculate the gravitational force exerted by a thin ring of uniform mass M?

I'm working on a problem and can't seem to get it. Find the gravitational force exerted by a thin uniform ring of mass M and radius a on a particle of mass m lying on a line perpendicular to the ...
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1answer
27 views

unit vectors: solving with component method and graphical method

P and Q are vectors in the X , Y plane, have the same magnitude, and are perpendicular to each other. If Q=3.0i+4.0j. What is P?
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1answer
67 views

A problem on collision of two elastic spheres

Two elastic spheres, each of mass $m$ collide directly. Show that the energy lost during the impact is $m(u^2-v^2)/4$, where $u$ and $v$ are their relative velocities before and after impact. ...
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1answer
62 views

Stuck on computing distance travelled from velocity and yaw rate.

I am somewhat stymied on what appears to be a simple formula. Here is the problem statement: Assume that a rigid body is traveling with constant velocity $v$, and is rotating with a constant yaw rate ...
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1answer
42 views

Methods of Calculating Position

Suppose we have the following code (Euler Method?) to determine Position versus Velocity, Acceleration and Time: ...
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1answer
31 views

Sphere intersecting a triangle

I'm studying "game physics" programming, and in this article, at page 14, I don't understand how is the author right. ... if the sphere does indeed collide with the inside of the triangle then a ...
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2answers
59 views

Functional Derivative ${\delta q_a(t)}/{\delta q_b(t')}$

$\newcommand{\fdv}[2]{\frac{\delta #1}{\delta #2}}$ $\newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d}#2}}$ $\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}$ I'm from a physics background and ...
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1answer
19 views

How to generate the icosahedral groups $I$ and $I_h$?

The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their ...
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1answer
32 views

Finding constant acceleration required to hit a target where mag(a) = n

I am currently using the equation: $$ \vec p=\frac12\vec at^2+\vec vt+\vec x$$ to find the acceleration I need to move from one point to another. where: $\vec p$ = target position (point 2d) ...
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1answer
24 views

Mechanics: Projectiles involving a ball shot out of a cannon, moving in the opposite direction of the shot

A child is playing with a toy cannon on the floor of a long railway carriage. The carriage is moving horizontally in a northerly direction with acceleration $a$. The child points the cannon ...
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1answer
45 views

Significance of 'faces' in Stress tensor components?

I am trying to understand what the significance is of the face for which a force is acting on when talking about a stress tensor. Say we consider the components $T_{xx}$ and $T_{zx}$ of the stress ...
4
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1answer
67 views

The real equation of a pendulum

In physics I never solve the equation $\ddot\theta = \sin(\theta)$. Instead, we used the approximation $\theta = \sin(\theta)$ for small angles and then it was easy to solve. I didn't do any physics ...
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3answers
68 views

Figure out the component of a value in X and Y coordinates using trigonometry.

Alright. It's been long that I studied trigonometry and did Laws of Motion and Free Body Diagrams, and I was decent good at them, but somehow I am having trouble in understanding the following. Note ...
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55 views

A problem on Constrained Motion

Q. A particle is moving in a smooth curve under gravity and its velocity varies as the actual distance from the highest point. Prove that the curve is a cycloid. Attempt: The eq. of motion is ...
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71 views

Why do people prefer cosine to sine when speaking of harmonic oscillation?

In almost all of the physics textbooks I have ever read, the author will write the oscillating function as $$x(t)=\cos\left(\omega t+\phi\right)$$ My question is that, is there any practical or ...
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1answer
29 views

solving equation in terms of $w_1$ and $w_2$

I have a a physics problem involves the following equation $$\tan(\alpha) = \frac{(w_1 + w_2)^{1/2}}{w_3}$$ from a certain set of equations that I use I derive the following equation: ...
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2answers
68 views

Practical use for negative $dt.$

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would ...
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1answer
66 views

What's wrong with my math in this function to update the position of a planet near a star?

Initially the code seems to work as the planet curves toward the star, but then as it should either get pulled into the star or make an orbit, it just gets pushed away in the opposite direction. What ...
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1answer
34 views

Connecting a mathematical solution to a differential equation with it's physical solution

I have seen this question in a neuroscience course: It is given after the lecture with these and these slides. I have no background in physics. However, I do know how to solve a differential ...
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1answer
12 views

Setting constant as a function of all the variables expect the one we integrate on

I am having trouble fully grasping the concept of setting a constant as a function of other variables: I would like to use a particular example where I could explain my thought process. Hopefully you ...
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1answer
32 views

Following a simplification of expression

I am struggling with this expression: In particular I get stuck with the simplification from the first to the second line. As far as I can see they replace $\text{m$\ell $}=\mu$. Does the new ...
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2answers
142 views

what would a planetary orbit look like if gravity had constant magnitude?

Consider a unit-mass particle that is always experiencing a single unit-magnitude force towards the origin. This is a central force, but it is not one of the familiar ones, e.g. gravity whose ...
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1answer
70 views

What is the ratio of the intensities of the two sounds?

1. Suppose that a jet engine at 50 meters has a decibel level of 130, and a normal conversation at 1 meter has a decibel level of 60. What is the ratio of the intensities of the two sounds? we ...
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3answers
72 views

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$? Is integration the only way? Homogeneous charge distribution ...
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1answer
42 views

How to derive the weak form of the PDE?

I have some difficulties solving the weak form of the PDE: The proof of the preceding statement is elementary. The weak form of the PDE $\nabla \cdot (A(x) \nabla u) + \omega^2 q(x) u = 0$ for all ...
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1answer
81 views

Counter exchanging limit and integral

Background I came across this answer on Math SE which claimed it made a lot of sense to switch limit and integral. In response I came up with the following counter-examples: $\lim_{w \to 0} ...
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2answers
55 views

Why is the potential function defined differently in physics and calculus?

I am very familiar with the concept of a potential function, and potential energy, from calculus-based physics. For instance, if we have the familiar force field $\mathbf{F} = -mg \,\mathbf{j}$, then ...
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2answers
118 views

Trigonometric identities — a parallel RLC circuit connected to an AC-supply [closed]

An RLC-circuit is connected to an AC-supply as in the figure below. $I_{tot}(t)=I_0sin(\omega t+\phi)$ (denoted as $I_{ges} ( t)$ in the picture), $\phi$ is the phase angle between ...
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1answer
48 views

Can I calculate the scalar potential from the electric field using: $\phi = \int \nabla \phi = - \int \vec{E}$

If I have a relation between the electric field and the radius, can I calculate the relation between the scalar potential and the radius using: $\phi = \int \nabla \phi = - \int \vec{E}$? $$\nabla ...
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1answer
39 views

Failing to calculate earth's standard gravitational parameter

I am using this equations from Wikipedia: $$\frac{4\pi^2 a^3}{T^2}=\mu$$ Where: $\mu$ = standard gravitational parameter ($\mbox{km}^3 \mbox{s}^{−2}$) $a$ = the orbiting body's semimajor axis (AU) ...
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1answer
18 views

Hamiltonian Constant on integral curves

Let $H \in C^{2}(\mathbb{R}^2)$ and let $(x(t),y(t))$ be a solution to the equations $$\frac{dx}{dt} = \frac{\partial}{\partial y} H(x(t),y(t))$$ $$\frac{dy}{dt} = -\frac{\partial}{\partial x} ...
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2answers
44 views

Determining movement time with uniform acceleration/deceleration

Consider some movement along a path segment $s$ with constant acceleration/deceleration (see figure below). The initial speed is $v_0$ and the final speed is $v_1$. The constant acceleration is $a$ ...
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20 views

When is the density matrix not diagonal?

The density matrix (density operator) in quantum mechanics is defined as $$\hat{\rho} = \sum_{i} p_i |\psi_i\rangle \langle \psi_i|\, ,$$ where the $|\psi_i\rangle$ are a full orthonormal system and ...
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1answer
36 views

Derivation of the momentum operator

\begin{align*} \langle p \rangle &= \int_{-\infty}^\infty \frac{d p}{2\pi \hbar}\, \phi(p, t)^\ast \, p \, \phi(p, t) = \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} \int_{-\infty}^\infty dx' ...
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1answer
45 views

Mass of Ocean to Atmosphere [closed]

This is a bizarre question, but here goes... If all of the water in the oceans were boiled into steam by the newly forming molten earth, could the atmosphere retain the steam? In other words, ...
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1answer
36 views

Fourier transform of a Green's function

I was studying for an exam and I found this question which has caused me a bit of trouble: Given the Green's function that satisfies the equation $$\Box ...
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1answer
43 views

Bezier curve and deceleration

I have a question regarding calculation of a cubic Bezier curve. I'm programming an app where in there's continuous straight line motion of a vehicle at a constant speed. (Let's call it $u$). When the ...
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38 views

Decomposition of acceleration into normal and tangential components

If the velocity $v=\|\mathbf{v}\|$ of a point having position $\mathbf{x}(t)$ at time $t$ is never null, then acceleration $\mathbf{a}:=\frac{d^2\mathbf{x}}{dt^2}$ can be written ...
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1answer
37 views

Rewrite the system in the form $\dot x = Ax + bu.$

$Ml\ddot\theta = (M + m)\dot x + mL\ddot\theta = u$ $M\ddot x = u - mg\theta$ Using the variables $x_1 = \theta, x_2 = \dot\theta, x_3 = x, x_4 = \dot x$ Rewrite the system in the form $\dot x = Ax ...
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3answers
586 views

Trying to understand the true meaning of integral and Derivative in calculus [duplicate]

I'm solving a physics question, and i just encountered some question i had no idea how to start, i just got the right answer and inside it it has something in math i never thought possible, I know ...
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0answers
56 views

Why we have to take the vector $\overrightarrow{e}$ ?

The differential equation of the balance of the momentum is $$\rho \frac{\partial{\overrightarrow{u}}}{\partial{t}}=-\rho (\overrightarrow{u} \cdot \nabla )\overrightarrow{u}-\nabla p+\rho ...
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27 views

Law of hydrostatic pressure

For a calm fluid of uniform density $\rho_0$, that occupies the space $W \subset \mathbb{R}^3$, and is subject to massive forces (per unit of mass) $\overrightarrow{b}(\overrightarrow{x})$, write the ...
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1answer
18 views

Null velocity and piecewise smooth path

On texts of multivariable calculus and real analysis I have always seen the work made by $\mathbf{F}$ along the path $\gamma$ defined as the ...