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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3
votes
2answers
217 views

Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
95
votes
8answers
6k views

Importance of Representation Theory

Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I ...
61
votes
4answers
72k views

Teenager solves Newton dynamics problem - where is the paper?

From Ottawa Citizen (and all over, really): An Indian-born teenager has won a research award for solving a mathematical problem first posed by Sir Isaac Newton more than 300 years ago that has ...
44
votes
9answers
16k views

What is the meaning of the third derivative of a function at a point

(Originally asked on MO by AJAY.) What is the geometric, physical, or other meaning of the third derivative of a function at a point? If you have interesting things to say about the meaning of the ...
8
votes
3answers
2k views

Property of Dirac delta function in $\mathbb{R}^n$

How does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$ where $S$ is the surface inside $V$ where ...
17
votes
2answers
3k views

Physicists, not mathematicians, can multiply both sides with $dx$ - why?

The following question is asked without malicious intentions - it's not intended as a flamebait! In my physics textbooks (Young & Freedman in particular) I have often seen derivations of ...
8
votes
4answers
12k views

Calculating the probability of a coin falling on its side

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real ...
10
votes
2answers
4k views

What is the optimum angle of projection when throwing a stone off a cliff?

You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the ...
5
votes
2answers
316 views

Is there a mathematical theory of physical knots?

From the point of view of people tying real knots (canonically, sailors) mathematical knot theory ignores much of what makes the problem of knot-tying interesting. Some matters that come up in the ...
20
votes
6answers
1k views

What Mathematics questions can be better solved with concepts from Physics?

Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics ...
14
votes
1answer
2k views

Residue of $z^2 e^{1/\sin z}$ at $z=\pi$

A while back I was working through many problems in Mathews and Walker's Mathematical Methods of Physics. In the appendix is this problem: A-6. Find the residue of the function $z^2 e^{1/\sin z}$ ...
13
votes
4answers
3k views

Can this gravitational field differential equation be solved, or does it not show what I intended?

This is the equation I'm having trouble with: $G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$ That's the non-vector form of the universal law of gravitation on the left and Newton's second law of motion ...
14
votes
2answers
1k views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
10
votes
3answers
2k views

The vertices of an equilateral triangle are shrinking towards each other

For an equilateral triangle ABC of side $a$ vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex ...
7
votes
3answers
4k views

Dirac Delta Function of a Function

I'm trying to show that $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$ Where $a_{i}$ are the roots of the function $f(x)$. I've tried to proceed by ...
4
votes
2answers
172 views

Should isometries be linear?

Question Suppose $V$ is a (finite-dimensional) vector space over $F$ ($\operatorname{char }F\neq2$, due to user1551) equipped with a non-degenerate quadratic form $Q$, and $T$ is a ...
2
votes
2answers
682 views

Free-fall according to Newton's gravitation law

Most analysis of free-fall assume that bodies fall with constant acceleration. If however one analyses free-fall according to Newton's gravitation law, one is lead to a differential equation which I ...
3
votes
1answer
132 views

Solving Wave Equation with Initial Values

I am trying to solve the wave equation: $u_{tt}$ = $u_{xx}$ With initial values: $u(x,0) =\begin{cases} x^3 - x, &\text{for }|x|\le 1,\ \\0, &\text{for }|x|\ge1\end{cases}$ $u_t(x,0) ...
2
votes
1answer
239 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
1
vote
4answers
6k views

Converting a function for “velocity vs. position”, $v(x)$, to “position vs. time”, $p(t)$

Provided some initial point $x(0)$, how do I convert the function for velocity vs. position, $v(x)$, into a function for position vs. time, $x(t)$, with time derivative $v(x(t))$? Constant ...
5
votes
2answers
801 views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
4
votes
1answer
131 views

Finding $\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}$ in $SU(3)$

I know that in $SU(3)$ $$\mathbf{8}\otimes \mathbf{8} = \mathbf{27}+\mathbf{10}+\mathbf{\bar{10}}+\mathbf{8}+\mathbf{8}+\mathbf{1}. $$ How can one use this to compute $$\mathbf{10}\otimes ...
3
votes
1answer
656 views

Why does acceleration = $v\frac{dv}{dx}$

If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$ However I also understand $a=v \frac{dv}{dx}$. Can ...
2
votes
1answer
54 views

Find Distance Function from Acceleration Function

The (non-constant) acceleration as a function of time, $a(t)$, is defined and known over $[t_0, t_2]$. It is also known that $a(t)$ is integrable. Also, $a(t)=\frac{dv(t)}{dt}$ and ...
2
votes
0answers
85 views

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
2
votes
0answers
688 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
1
vote
2answers
145 views

What does adding $\sin\theta \cos\theta$ make my graph a linear relationship?

What is the point of adding sin n cos of theta when graphing range? e.g. I see on hyperphysics a graph of range vs sin n cos of theta and it makes the experimental data embody a linear relationship. ...
0
votes
2answers
116 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
0
votes
0answers
132 views

Derivation of the equations for non-uniform circular motion using complex analysis.

Here http://farside.ph.utexas.edu/teaching/301/lectures/node89.html They use complex analysis to derive the equations of non-uniform circular motion. My confusion is this: In the derivation they ...
0
votes
1answer
188 views

Center of mass in a straight rod

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $$\int_S x_1 \, dx_1 \, dx_2=0 $$ $$\int_S x_2 \, dx_1 \, dx_2=0 $$ ...
0
votes
1answer
140 views

Surface integral and the divergence theorem

I haven't done a surface integral in a while so I am asking to get this checked. $\mathbf{F} = \langle x, y, z\rangle$ and the surface is $z = xy + 1$ where $0\leq x,y\leq 1$. $\hat{\mathbf{n}} = ...
0
votes
2answers
874 views

Can somebody help and explain me with angular and linear Velocity textproblems?

So we got this as homework today, and I just don't understand it; how to start or how to do it. Explanations and/or setups would be great :) A wheel with ...
85
votes
19answers
4k views

Good Physical Demonstrations of Abstract Mathematics

I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to ...
29
votes
8answers
3k views

Very *mathematical* general physics book

I am searching for a book to study physics. So far, I've been suggested Resnick, Halliday, Krane, Physics, but it doesn't seem to be very suited for a math major. Can you suggest some more ...
36
votes
3answers
3k views

Intuitive reasoning behind $\pi$'s appearance in bouncing balls.

This video is about an interesting math/physics problem that when cranked out churns out digits of $\pi$. Is there an intuitive reason that $\pi$ is showing up instead of some other funky number ...
14
votes
5answers
904 views

In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
15
votes
2answers
3k views

What is the resistance between two points a knights move away on a infinite grid of 1-ohm resistors

On an infinite grid of ideal one-ohm resistors, what's the equivalant resistance between two nodes a knights move away? (please fix the tags, I didn't really know where to put it)
13
votes
2answers
2k views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
11
votes
2answers
1k views

Cross product and pseudovector confusion.

So called pseudovectors pop up in physics when discussing quantities defined by cross products, such as angular momentum $\mathbf L=\mathbf r\times\mathbf p$. Under the active transformation $\mathbf ...
8
votes
4answers
500 views

Consequences of solving the Halting problem

What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...
7
votes
2answers
419 views

Novel approaches to elementary number theory and abstract algebra

As a part of a university course, I'll have to study Herstein's Topics in algebra and Hardy&Wright's Introduction to the theory of numbers. Can you suggest some books (to be used as companions) ...
5
votes
1answer
2k views

Riemann, Ricci curvature tensor and Ricci scalar of the n dimensional sphere

I am calculating the Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar of the n sphere $x_0^2 + x_1^2 + ....+x_n^2=R^2$, whose metric is $$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 ...
7
votes
2answers
518 views

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding ...
7
votes
4answers
648 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
10
votes
2answers
229 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
10
votes
3answers
388 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following ...
6
votes
3answers
514 views

Learning about the universe or special/general relativity

I have done a standard course in differential geometry/Riemannian geometry. Am I now able to understand the concepts people talk about when they say things like "spacetime is curved" and when I see ...
3
votes
1answer
331 views

Complex parametrization of Airplane wing?

I read once about complex parametrization with fluid-dynamics objects such as airplane wings, something related Rieman Zeta function. What are the mathematical models this kind of things such as ...
7
votes
1answer
303 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
7
votes
4answers
300 views

Is length adimensional when space is not flat?

Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...