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
6
votes
0answers
101 views
Help computing an integral
I need to calculate the following integral:
$$
\int_0^1 \int_0^1 \frac{1-\cos(2 \pi k_1 x) \cos(2 \pi k_2 y)}{4 \sin(\pi k_1)^2 + 4 \sin( \pi k_2)^2} dk_1 dk_2
$$
I have tried to use some contour ...
6
votes
0answers
192 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, ...
6
votes
0answers
429 views
Eigenfunctions of the Helmholtz equation in Toroidal geometry
the Helmholtz equation
$$\Delta \psi + k^2 \psi = 0$$
has a lot of fundamental applications in physics since it is a form of the wave equation $\Delta\phi - c^{-2}\partial_{tt}\phi = 0$ with an ...
5
votes
0answers
87 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 ...
5
votes
0answers
241 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 ...
5
votes
0answers
174 views
is any hamiltonian system with just one degree of freedom completely integrable?
An hamiltonian system with $n$ degree of freedom is said to be completely integrable when there exists an system $f_1,\ldots,f_n$ of first integrals mutually Poisson-commuting, such that ...
4
votes
0answers
108 views
Integral of a gaussian function of trigonometric functions
I need help with the analytical solution of this integral:
...
4
votes
0answers
83 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 ...
3
votes
0answers
198 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
votes
0answers
73 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
votes
0answers
65 views
Simplifying an integral arising in Physical Chemistry
I am struggling to understand the following transition (encountered in a paper on Physical Chemistry).
Let
$$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
3
votes
0answers
141 views
Equation of motion for a wobbling disc
While looking at a frisbee the other day, I suddenly had a question.
Suppose (in free space) you set a disc-shaped object spinning, and then you impart a sudden force perpendicular to the spinning ...
3
votes
0answers
52 views
Why are linear functions the natural analogue of exponential functions in a tropical semiring?
I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says
It's not obvious how to interpret the ...
3
votes
0answers
83 views
Finding $\mathbf r(t)$ for the parameterized two-body equations of motion
I'm trying to understand the equations of two-body motion. Namely, given the position, velocity and mass of two orbiting bodies at time $t$, how can I explicitly find their position and velocity for ...
3
votes
0answers
156 views
What can we say about $n\times n$ invertible matrices, all elements of which are $+1$, $-1$, or $0$?
Basically, the title says it all, except for why I am asking. I'm studying a paper that I can't do justice to in a few words here. (It is not freely available on the Web, as far as I know, but I can ...
2
votes
0answers
41 views
Circular Motion
A car is driven in a flat circular curve of radius $r$ m. The car’s engine supplies a
constant tangential driving force. The car experiences a friction heading towards the centre of the circle.
By ...
2
votes
0answers
61 views
Solving Generalized Eigenvalue Problem perturbatively
Let me formulate the problem to convey my notation.
I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation
$$ U_A A\,\,U_A^{-1} = A_{diag}$$
Now the matrix is changed, ...
2
votes
0answers
162 views
Publication date for Michael Spivak - Physics for Mathematicians II?
I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
2
votes
0answers
38 views
Molecular vibrations and a generalisation of Wigner's rule for (non-finite) compact groups
years student of mathematics and write my script for my bachelor. The topic is "Representations of groups and applications in physics". I understand the representations very good but now i want to ...
2
votes
0answers
72 views
Mathematical significance of the “Dirac conjugate”
Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ ...
2
votes
0answers
77 views
How to convert a hologram into an image?
Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...
2
votes
0answers
179 views
Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?
I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly.
I know
$$\nabla\times E= ...
2
votes
0answers
129 views
On the geometric arguments used in Newton's *Principia Mathematica Naturalis Philosophae*
When one reads Newton's Principia Mathematica, one is immediately aware of the complexity of the synthetic geometry that he uses to prove his propositions. This I understand because all of the ...
2
votes
0answers
99 views
Decomposing products of spinor representations into anti-symmetric tensors
There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
2
votes
0answers
96 views
Check my solution - Modelling of a spring with Differential Equation
I am doing some work with differential equations. I have solved the following problem but am uncertain if I'm doing it correctly. Could someone look over it for me and check if I'm doing something ...
2
votes
0answers
66 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 ...
2
votes
0answers
49 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 ...
2
votes
0answers
40 views
Heuristics for definitions of open and closed quantum dynamics
I've been reading some of the literature on "open quantum systems" and it looks like the following physical interpretations are made:
Reversible dynamics of a closed quantum system are represented ...
2
votes
0answers
68 views
A lower positive bound on the number of closed orbits with given energy for a mechanical system
Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the ...
2
votes
0answers
73 views
Extension of Uncertainty Relations to a specific potential in Schrödinger Equation
Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
2
votes
0answers
127 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 ...
2
votes
0answers
204 views
Hermitian operators and commutators
If I have three operators such that $[A,B] = C$, and I know that $A$ and $C$ are Hermitian, does it follow that $B$ is anti-hermitian? If $A$ and $B$ were Hermitian, $C$ would be anti-Hermitian, so ...
2
votes
0answers
163 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 ...
2
votes
0answers
129 views
How does physicist compute path integral in Chern-Simons theory
The space of connection has no measure right?
2
votes
0answers
136 views
What does “Gromov Witten potential” the “potential” mean
"Gromov Witten potential", when does "potential" mean here? What does the whole thing mean in physics? Thanks!
2
votes
0answers
90 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
0answers
28 views
Three body problem with point interactions
I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials).
Is ...
1
vote
0answers
40 views
Planar circular restricted 3-body problem
Hi and sorry for my bad English, it's not my first language. I'm trying to find the equations of motion of the planar circular restricted 3-body problem. I did the gravitational force, but I have some ...
1
vote
0answers
45 views
Hamiltonian of one and two unknots
Recently I calculated the Ising Hamiltonian of a Hopf link. First, I colored the Hopf link in a checker board pattern and drew the Seifert surface from it. Considering the shaded regions as vertices ...
1
vote
0answers
25 views
References for three body problems with Fermi statistic
I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?
1
vote
0answers
32 views
Regularization theory
In order to remove the collision singularity in the equations of motion of the three dimensional two body problem, one defines the coordinate transformation
$x_1=u_1^2-u_2^2-u_3^2+u_4^2$
$x_2=2(u_1 ...
1
vote
0answers
42 views
Simple equation misunderstanding
Im trying to use an equation on this page
http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics
The angle of lean, $\theta$, can easily be calculated using the laws of circular motion
$$ ...
1
vote
0answers
75 views
Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?
This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
1
vote
0answers
61 views
Where does the integral come from in this Spring formula describing displacement at a certain point?
I'm in the process of trying to combine some equations for a simulation.
One describes the position at a point along a hanging slinky using...
$$y(d) = (l_1 + {mg\over k})d - {mg\over 2k}d^2$$
...
1
vote
0answers
48 views
Convolutions of Path Integrals of Gaussian Functions
I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
1
vote
0answers
66 views
Wilson lines, boundary condisions, surface defects of TQFTs
I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
1
vote
0answers
45 views
Solving inverse square of visible scale
I'm not super-adept in mathematics, so I turn to you for help. As I read, the perceived scale of an object reduces by the inverse square as the viewed distance increases. In order to solve for this, ...
1
vote
0answers
92 views
damped harmonic oscillator driven by a stochastic momentum (not force)
Could you give references for solutions or solutions to the following problem:
Given: damped harmonic oscillator driven by stochastic force of very short duration (= stochastic momentum).
Find: ...
1
vote
0answers
57 views
Differential equations with different constants for different sub-domains
I remember that when I was studying differential equations, there was an example with solutions of the form $f(x) + C_1$ for $x>0$ and $f(x)+C_2$ for $x<0$ where $C_1$ and $C_2$ may be different ...
1
vote
0answers
122 views
Applications of mathematics to some kinds of sporting strategies
I am a rather newbie maths person. Haven't studied maths in a while and so not sure what things are called was hoping to get some information to push me in the right direction so I know what it is I ...
