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
48
votes
3answers
63k 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 ...
61
votes
8answers
3k 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 ...
31
votes
8answers
5k views
What is the meaning of the third derivative of a function at a point
What is the geometric, physical or other meaning of the third derivative of a function at a point?
(Originally asked on MO by AJAY)
If you have interesting things to say about the meaning of the ...
10
votes
1answer
1k 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
2answers
510 views
Why are harmonic functions called harmonic functions?
Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
7
votes
2answers
1k 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 horizontal ...
5
votes
2answers
208 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 ...
0
votes
2answers
67 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 ...
0
votes
4answers
292 views
Show the negative derivative of a function.
A type of interaction between atoms in a molecule is called a Van der Waals interaction. This can be described by the potential energy function;
$$U= ...
69
votes
18answers
2k 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 ...
30
votes
2answers
2k 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 ...
11
votes
2answers
944 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 ...
9
votes
2answers
271 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 ...
9
votes
3answers
632 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
4answers
263 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$. ...
4
votes
2answers
363 views
General solution for $y^{iv}+ 2y''+y=\cos x$
Here is another problem from Mathews and Walker that has given me some trouble.
1-18. Find the general solution of
$y^{iv}+ 2y''+y=\cos x$.
Note: Thanks, everyone, for clearing up the ...
3
votes
3answers
456 views
Cat Dog problem using integration
Consider this equation :
$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$
where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$.
Now how to proceed ?
This equation ...
3
votes
2answers
345 views
What strategy do you use when solving vector equations involving $\nabla$?
$\Phi, \Lambda$ are both scalars dependent upon, and $\mathbf u$ is a vector independent of coordinates. I'm trying to express $\Lambda$ in terms from $\mathbf U \cdot \nabla\Lambda = \Phi$ and to ...
0
votes
1answer
86 views
Complex parametrization of Airplane wing?
I read once about complex parametrization with fluid-dynamics objects such as airplane wings, something related Rieman Zeta function. What are the mathematical models this kind of things such as ...
0
votes
3answers
89 views
Resource request: history of and interconnections between math and physics
Reading this article I became curious to learn more of (- study more thoroughly and *seriously*$^{\star}$-) the topic.
Is / are there some good references - either papers, books and/or other ...
0
votes
1answer
633 views
Maximum range of a projectile (launched from an elevation)
If a projectile is launched at a speed $u$ from a height $H$ above
the horizontal axis, and air resistance is ignored, the maximum
range of the projectile is $R_{max}=\frac ug\sqrt{u^2+2gH}$, ...
5
votes
4answers
378 views
What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?
I find that many authors write the Yang-Mills action as follows:
$$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$
I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
5
votes
3answers
415 views
What is a physical “dimension” - in the sense of “dimensional” analysis?
Mathematically speaking, what does it mean to say that a physical quantity is some numerical value with a “dimension” associated with it? When we say that the velocity of light is some constant, c ...
4
votes
3answers
212 views
Learning about the universe or special/general relativity
I have done a standard course in differential geometry/Riemannian geometry. Am I now able to understand the concepts people talk about when they say things like "spacetime is curved" and when I see ...
3
votes
4answers
580 views
Hydrostatic pressure on a square
Vertically inserted into the water I have a rectangle 6 feet wide and 4 feet high that is submerged under the water with 2 feet of water above it.
Using a riemann sum how do I find the pressure? I ...
3
votes
3answers
237 views
Numerical computation of the Rayleigh-Lamb curves
The Rayleigh-Lamb equations:
$$\frac{\tan (pd)}{\tan (qd)}=-\left[\frac{4k^2pq}{\left(k^2-q^2\right)^2}\right]^{\pm 1}$$
(two equations, one with the +1 exponent and the other with the -1 exponent) ...
3
votes
1answer
484 views
Prove Pythagoras theorem through dimensional analysis
I've recently become acquainted with Buckingham's Pi theorem for the first time . Then I've found an excercise that says:
Use dimensional analysis to prove the Pythagoras theorem. [Hint: Drop a ...
3
votes
3answers
748 views
Given a radius and velocity calculate position of an aircraft banking to make a turn
I have a radius, R, for an aircraft traveling at velocity, V. If we start at point, (X,Y), ...
2
votes
1answer
476 views
Solid body rotation around 2-axes
I am trying to understand how to describe the rotation of a solid body flying in 3D space. From physics forums, I understand that the rotation of any solid object in space, is around 2 axes ...
1
vote
1answer
222 views
Parametric curve of intersection - line integral with respect to arc length
This comes from Apostol's Calculus, Vol. II, Section 10.9 #14:
A uniform wire has the shape of that portion of the curve of intersecion of the two surfaces $x^2+y^2=z^2$ and $y^2=x$ connecting the ...
13
votes
2answers
202 views
“Curled-up dimensions”?
I'm a grad student in math, but I don't know as much about physics as I should. I've read a handful of pop expositions of string theory, and they often refer to "curled up dimensions" and I've always ...
6
votes
1answer
301 views
Representations of a non-compact group are labeled by its maximal compact subgroup?
I don't have much of any awareness about the representation theory of non-compact Lie groups but I bumped into it for my work.
Is there some idea that the representations of a non-compact group are ...
5
votes
3answers
156 views
Physics: Uniform Motion
Ok so I have this homework problem. I dont want to give out the information because i want to put in the values myself, but basically I have one object moving at said speed. Said time later, another ...
5
votes
2answers
437 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, ...
5
votes
3answers
225 views
Summing divergent series based on physics
Solving a heat equation with central symmetry i got the following result:
The problem is to find a sphere center temperature vs. time given that the surface of the sphere is kept constant at $$T_1$$ ...
4
votes
1answer
172 views
Is a twisted de Rham cohomology always the same as the untwisted one?
I am a physicist studying now some supersymmetric sigma models. My question can, however, be reformulated in a purely mathematical language:
A twisted de Rham complex involves $d_W = d + dW \wedge $ ...
4
votes
1answer
288 views
Dyson series and T product
One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand.
$\{H(t_i)\}$ are ...
3
votes
2answers
107 views
Help solving differential equation
I want to solve the following differential equation:
$y[t]$ : vertical position (height) of the object at time t
$y_c$ : height of the ceiling
$y_e$ : equilibrium point, the height at which the ...
3
votes
1answer
504 views
Why does acceleration = $v\frac{dv}{dx}$
If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$
However I also understand $a=v \frac{dv}{dx}$. Can ...
2
votes
2answers
371 views
Projection of a 3D spherical distribution function in to a 2D cartesian plane
Consider a 3D spherical Gaussian distribution function that depends on radius only,
$$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$
where $R_\mu$ is the radial offset of the distribution and ...
2
votes
2answers
104 views
What is the $x(t)$ function of $\dot{v} = a v² + bv + c$ to obtain $x(t)$
How to solve $$\frac{dv}{dt} = av^2 + bv + c$$ to obtain $x(t)$, where $a$, $b$ and $c$ are constants, $v$ is velocity, $t$ is time and $x$ is position. Boundaries for the first integral are $v_0$, ...
2
votes
2answers
165 views
Unitary Operator as a complex valued function
A book on Quantum Mechanics by Schwinger states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator."
Please give a hint on how to prove this assertion.
2
votes
2answers
249 views
Delta function in curvilinear coordinates
I have been looking everywhere but I am unable to prove $$\delta(\vec{x}-\vec{a}) = \frac{1}{fgh}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$$
Where $f,g,h$ are scale factors for an orthogonal ...
2
votes
0answers
91 views
Analysing an optics model in discrete and continuous forms
A discrete one-dimensional model of optical imaging looks like this:
$I(r) = \sum_i e_i P(r - r_i)$
Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread ...
1
vote
1answer
208 views
Energy of wave equation decreasing
I have problems checking that the energy $E(t)=\frac{1}{2}\int_I(u_t^2+c^2u_x^2)dx$ on an open interval $I\subset \mathbb R$, such that $u(0,x)=0$ and $u_t(0,x)=0$ for $x\in\mathbb R\setminus I$ is ...
1
vote
1answer
314 views
Can the differentiating and squaring process in the cochlea explain a reported dichotic stimulation experiment?
On this math.stackexchange
on url What is Octave Equivalence?
in an answer on the related ( octave equivalence ) question is stated:
Mathematically, this signifies that the mammalian cochlea ...
1
vote
1answer
116 views
Finding a derivative
I'm working through a book on relativity so this may end up being a physics question but I'm pretty sure that my problem is mathematical so I'm asking here. In deriving the "special Lorentz ...
1
vote
3answers
341 views
Fourier analysis for waves
I'm studying physics, so I'm sorry if I'll write some inexact things in this post. I wish you can understand me.
If we have 1D wave equation:
$$\frac{\partial^2 \psi}{\partial ...
1
vote
2answers
416 views
two-body problem circular orbits
I've been trying to google the answer to this question, but have had no luck, so I'm asking this question here. Let's say the origin is at (0, 0), body 1 with mass m1 is at (-r1, 0), and body 2 with ...
0
votes
0answers
120 views
sinusoidal word problem
in tidal waves the sea level drops first leaving the seabed exposed (normally 30 feet below sea level), then it rises a equal distance above sea level. waves hitting a city have a max height of 38.9 ...
