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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9
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226 views

Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, ...
8
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0answers
228 views

What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
7
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0answers
210 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
6
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62 views

Examples of useful, insightful and interesting hand-waving

I am really amused by the answers to this question on "Most harmful heuristic" posed on MathOverflow, from which I've benefited a lot. However, it seems to me that some hand-waving may be really ...
6
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0answers
67 views

Symbolic manipulation inside integral

I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic ...
6
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84 views

What is the generalization of Gauss's Theorem to a manifold?

In a (pseudo-)Riemannian manifold with constant basis vectors, one certainly has that the integral of the divergence of a tensor field $T$ over a submanifold $\Omega$ is equal to the integral over the ...
6
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204 views

Renormalization for mathematicians

Can someone explain to me the processes of renormalization and regularization used in quantum field theory and similar fields in a way that a pure mathematician might make sense of it? Is there a ...
6
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230 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
6
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0answers
365 views

Studying quantum mechanics without physics background

I am a first year PhD math student, and I am wondering if I should study quantum mechanics even though I don't have an undergrad background in physics. I posted this question in physics ...
6
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105 views

Analytic caustics for 3D objects

Is it possible to efficiently calculate caustics for a given 3D object, like a torus, or a cube? To be more precise: let's assume that we have a 3d torus, resting on a 2d plane and a single light ...
5
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267 views

Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
5
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82 views

Earnshaw's theorem

Proposition Suppose $U\colon\Omega\to\mathbb R$ is a non-constant harmonic function, i.e. $U\in\mathcal C^\omega$, i.e. analytic, and $\Delta U=0$, where $\Omega\subseteq\mathbb R^n$ is a region. ...
5
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100 views

Solving numerically the equation of motion of D7 brane perturbation

I want to solve this equation $$ \partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0 $$ numerically. I know that ...
5
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0answers
51 views

Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray ...
5
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0answers
88 views

Second law of thermodynamics as a theorem about state space evolution

I once saw a mathematical explanation of the second law of thermodynamics. The statement was something like this: there is a mapping $f$ from the set of thermodynamic states $S$ to itself, and a ...
4
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56 views

Sorting out some integrals from physics

I'm doing some physics for a change, and I'm trying to sort things out a bit. From the definitions of mass, torque, momentum and angular momentum I've come up with the following integrals: ...
4
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89 views

Rolling a ball into a cone; what should the forces overall be?

Suppose there is a cone, with the apex pointing down, and the top of the cone at height $h$, apex half-angle $\psi$, ball of mass $m$, and the initial velocity into the cone (completely horizontal, ...
4
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46 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
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41 views

C. Neumann passage in Latin from *Annali di Matematica Pura ed Applicata*

Neumann, Carl. “Theoria nova phaenomenis electricis applicanda.” Annali di Matematica Pura ed Applicata 2, no. 1 (August 1868): 120–128. doi:10.1007/BF02419606. p. 121: Nova introducitur ...
4
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84 views

A light beam enters a closed room. What is the maximal number of reflections?

I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and the light beam enters the ...
4
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164 views

How to write $SO(2n)$ characters in terms of rotation angles?

Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says ...
4
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120 views

Where do I go from Linear algebra past Calc III to try to learn complex physics (relativity and quantum group theory)?

I'm mainly a programmer, but I have a love for Mathematics that's been, well, insatiable. I've had my eye on learning Quantum Groups and Relativity, but I want to stay in something I can do with ...
4
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0answers
118 views

Gaussian Integral with non-polynomial exponent

I am currently trying to evaluate this Integral: $$\int\limits_{u_0}^{u_1} \exp\left[-\angle(H(u),N)^2\right]du$$ Where ...
4
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0answers
67 views

Why are linear functions the natural analogue of exponential functions in a tropical semiring?

I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says It's not obvious how to interpret the ...
3
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0answers
39 views

Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...
3
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31 views

Fourier transform of $\frac{1}{r}$ (Coulomb potential)

When calculating the Fourier transform of a function of the form $f(\vec{r}) = \frac{1}{4 \pi \left|\vec{r}\right|}$, one encounters the problem that the resulting integral does not converge, i.e. ...
3
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38 views

Mathematical interpretation of pressure (gradient)

I'm having some problem with the following. Usually, the pressure of some mass on an area A is defined by $$P=\frac{F}{A}$$ where $F$ is the force of the mass exerted due to gravity. However, here ...
3
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136 views

Developing solution for electrodynamics problem

Although it is a question related to physics, since the point it really matters is its mathematical aspect, I post this question on MSE. There's an additional exercise from Introduction to ...
3
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0answers
83 views

Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$ \int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t) $$ for $(-\frac{1}{2}T< t < \frac{1}{2}T)$ is useful in photon ...
3
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0answers
402 views

Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
3
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86 views

Who established the word “ Degree of freedom ” in statistics?

I wonder who is the first one that established and applied the word : "degree of freedom" in statistics? Why he/she need degree of freedom in the calculation of many statistical values?
3
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143 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
3
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0answers
326 views

Equation of motion for a wobbling disc

While looking at a frisbee the other day, I suddenly had a question. Suppose (in free space) you set a disc-shaped object spinning, and then you impart a sudden force perpendicular to the spinning ...
3
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0answers
108 views

Stability of Orbits in Schwarzschild Spacetime

I'm looking at geodesics in the Schwarzschild geometry, and have come up against something I cannot prove. I've shown that for a particle moving on a geodesic with $r$ constant and $\theta=\pi/2$ we ...
3
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0answers
99 views

Finding $\mathbf r(t)$ for the parameterized two-body equations of motion

I'm trying to understand the equations of two-body motion. Namely, given the position, velocity and mass of two orbiting bodies at time $t$, how can I explicitly find their position and velocity for ...
3
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0answers
174 views

What can we say about $n\times n$ invertible matrices, all elements of which are $+1$, $-1$, or $0$?

Basically, the title says it all, except for why I am asking. I'm studying a paper that I can't do justice to in a few words here. (It is not freely available on the Web, as far as I know, but I can ...
3
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0answers
210 views

Designing a mathematical physics class

Surprisingly, the university (a major tech school) I attended does not offer a mathematical physics class. Consequently, I often get asked by my physics friends what are some good math classes to take ...
3
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0answers
112 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 ...
2
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0answers
41 views

Period of a pendulum

Consider the pendulum problem $\frac{d^2x}{dt^2}+\sin(x)=0$ $\frac{dx}{dt}(0)=v_0=0$ $x(0)=x_0$ Show that the period ...
2
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0answers
40 views

Launch angle required to hit coordinate (x,y) with air resistance

Finding Angle of Elevation to hit X, Y and Wikipedia Angle required to hit coordinate work, but don't calculate air resistance. Is there a way to find the launch angle of a projectile required to hit ...
2
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0answers
59 views

Forced oscillation in a pendulum and resonances

In a pendulum without the small angles approximation the equation describing the motion of the mass is: $$\ddot{\phi}(t)=-\dfrac{g}{l}\sin\left(\phi(t)\right)$$ Applying a sinusoidal force ...
2
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0answers
35 views

Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
2
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0answers
81 views

Why is Grassman integration so weird?

Why are Grassman integration and differentiation equivalent? The only justification of this definition I have ever scene is "Well, how else could it work?" Indeed, I don't have any other suggestions, ...
2
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0answers
55 views

ODE particular solution (physics)

I have to do this exercise: ($Z(t)=I(t)$, it's printed wrong). I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega ...
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97 views

Are there any legitimate examples of applications of set theory to Physics?

Sets are of course ubiquitous, so in this sense set theory is applicable to physics. But I'm thinking more like techniques like forcing, or constructs like cardinals or ordinals. Intuitively it ...
2
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0answers
32 views

Dhar's Burning Test - Confusion about Abelian Sandpile Model

Dhar's Burning test is a bijection between the spanning trees of a certain graph and the recurrent states Abelian sandpile model. I would like some help working out this bijection in different cases: ...
2
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0answers
80 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
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0answers
71 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
2
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0answers
62 views

Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
2
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0answers
75 views

How to maximize speed of rest position approach of nonlinearly damped spring oscillator?

Inspired by comments to answer for this question: Suppose we have a system which is described by the equation $$\ddot x=-x+g(\dot x),$$ with initial conditions $x(0)=1$, $\dot x(0)=0$. If ...