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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4
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1answer
305 views

Fourier (Hankel?) transform of a discrete set of radial points (question from a chemist!)

I'm sorry because I'm not a mathematician so that my question may look a little bit messy. I have tabulated values [1] of a 3 dimensional radial function $f(r)$: ...
3
votes
0answers
235 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 ...
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2answers
881 views

How to configure an LED to emit white light of a certain color temperature?

I'm working on an open source hardware project for a video/photo light, and it involves a fair bit of color math. I am trying to find my way from Color Temperature (CT) in Kelvin to current values ...
3
votes
2answers
433 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 ...
2
votes
1answer
1k 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 ...
18
votes
2answers
609 views

Atiyah's definitions of Topological Quantum Field Theory

According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces. How does this definition relate with the physics of quantum mechanics? What does the ...
1
vote
3answers
366 views

How long would it take for light to go through a human eyeball and back to the other side

I couldn't find any TAGS that fit my question... I don't know if I'm doing this correctly. I want to know how long a "twinkling of an eye" would be. A twinkling of an eye is the time it takes from ...
8
votes
1answer
855 views

The mathematics behind Clebsch-Gordan Coefficients

In quantum physics we have to work a lot with Clebsch-Gordan coefficients and generalizations like the Wigner 3j,6j, and 9j symbols. In our coursework we are taught that the coefficients are coupling ...
1
vote
1answer
67 views

find point of object in the future given direction and velocity

I have a point A, and another point B. I know the distance from point A to B. I also know the velocity with which B is moving and its direction. I also know the angle of B with respect to A. I would ...
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2answers
138 views

Integration of piece-wise arguments

Hallo all, I am from electronics background. I have a simple mathematical problem which seems daunting for me. The general equations for calculating charge is given below. It integrates the ...
5
votes
3answers
924 views

Parabolic shape in Bow (not arrow!)

This is what I am thinking for some days. And I think here are some experts who can answer this question. If I bend any stick made with material that uniform density and its shape is cylindrical ...
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 ...
10
votes
2answers
865 views

What is the relationship between the Boltzmann distribution and information theory?

I'm reading a paper on Boltzmann machines (a type of neural network in Machine Learning), and it mentions that "The Boltzmann distribution has some beautiful mathematical properties and it is ...
0
votes
2answers
110 views

Touch up on Trig

I have forgotten a few things about trigonometry and angles. I have this trig equation, $\sin \theta = \frac{200\text{ dyn}}{224\text{ dyn}}$ What exactly are the steps of getting the angle, ...
0
votes
1answer
716 views

Intuition behind using dot product to find component of vector in direction of another

I'm reading Chris Hecker's third article on rigid body dynamics http://chrishecker.com/Rigid_body_dynamics Quoting... "More importantly, if our collision detector supplies us with a 'normal vector' ...
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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 ...
2
votes
4answers
159 views

The motion of a system as a level set of the energy

Suppose we have a mechanical system with 1 degree of freedom, i.e. an ODE $$(1)\quad \ddot{q}+V^\prime(q)=0, $$ where $V \colon \mathbb{R} \to \mathbb{R}$ is some smooth function (potential ...
5
votes
2answers
825 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, ...
3
votes
2answers
504 views

Diffraction and Computer Generated Holography Calculations

I've tried this through Mathematica, and hit my own limit in math ability trying to do this, both to no avail. I'm assuming there is no way to do so, as a simple solution to this problem would be a ...
3
votes
1answer
167 views

Fourier coefficients in oscillation problem with viscosity

On the oscillation problem of a rope with fixed extremities, $$\left\{\begin{matrix} \left.\begin{matrix}\left.\begin{matrix} u_{tt}(t,x) = a^2u_{xx}(t,x)\\ u(0,x) = \varphi(x)\\ u_t(0,x) = ...
3
votes
1answer
1k 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 ...
2
votes
1answer
439 views

Uniqueness of Helmholtz decomposition

Helmholtz theorem states that given a smooth vector field $\mathbf{H}$, there are a scalar field $\phi$ and a vector field $\mathbf{G}$ such that $\mathbf{H}=\nabla \phi +\nabla \times \mathbf{G}$ ...
4
votes
1answer
1k views

How to find angle of motion in frictionless space based on vx and vy

This might, perhaps, be more suited to Stack Overflow or the Gaming section, but since it's math that is above my head, I thought I'd come straight here for it. I'm making a space game. So far I ...
1
vote
1answer
1k views

Subtract gravity from 3D object in different orientations

I'm currently working on a project that involves calculating a sensor module's distance and orientation. The problem I'm running into is the fact that once the sensor is, for example, rotated around ...
12
votes
2answers
4k views

Energy norm. Why is it called that way?

Let $\Omega$ be an open subset of $\mathbb{R}^n$. The following $$\lVert u \rVert_{1, 2}^2=\int_{\Omega} \lvert u(x)\rvert^2\, dx + \int_{\Omega} \lvert \nabla u(x)\rvert^2\, dx$$ defines a norm on ...
0
votes
1answer
57 views

How to determine the function of a data list?

I'll be the first to admit, I am not a math wiz. That said, please forgive my ignorance. I need to write a script for work that will take a given temperature/pressure and output the other ...
7
votes
3answers
1k views

Energy functional in Poisson's equation: what physical interpretation?

Let's consider this boundary-value problem: $$\begin{cases} -\Delta V = \rho & \rm{in}\ \Omega \\ V=0 & \rm{on}\ \partial \Omega \end{cases}.$$ We know that this problem has a ...
1
vote
1answer
4k views

How do I apply an angular velocity vector[3] to a unit quaternion orientation?

I have an angular velocity vector[3] in three dimensions and a unit quaternion (magnitude of 1) representing an orientation in three dimensions. I need to apply the angular velocity to the quaternion ...
4
votes
1answer
978 views

Incandescent light bulb lifetime - exponential distribution?

It's been said that incandescent light bulbs lifetime has exponential distribution. As I understand, this means a 10,000 hours time-to-failure has the same probability no matter how long the light ...
2
votes
0answers
446 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 ...
16
votes
3answers
1k views

Quantum mechanics for mathematicians

I'm looking for books about quantum mechanics (or related fields) that are written for mathematicians or are more mathematically inclined. Of course, the field is very big so I'm in particular ...
2
votes
4answers
2k views

Calculating acceleration based on a starting and ending distance

I have been trying to figure this problem out and just can't. The answer is supposed to be (A): 1.0 m/s/s. Can anyone tell me how they got that answer? I thought this problem was based on the ...
0
votes
1answer
12k views

Damping Constant Equation?

I've been looking through my textbooks and I've found a number of different equations - so I wanted to confirm with you which it is. What is the equation which determines the damping constant (gamma) ...
1
vote
1answer
433 views

Matsubara sum and complex analysis

Let $\omega_n=e^{\mathrm{i} \pi (2n+1)/N}$, for n=0,...,N-1, be the N-th roots of (-1). So that the sum for a suitable analytic functions $f(z)$ can be turned into a contour integral $$S=\frac{1}{N} ...
6
votes
1answer
280 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
1answer
886 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 ...
4
votes
2answers
2k views

Integrating Legendre Polynomials over half range

Solving for the potential of a conducting sphere with hemispheres at opposite potentials, (not using Green's function) I am stuck at this point: $$I_l = V_1 \int_0^1 P_l(x)dx+V_2 \int_{-1}^0 ...
0
votes
1answer
87 views

Deccelerate and Accelerate to achieve timed journey

I have a vehicle traveling at initial speed 100m/s. Expected destination speed is also $100m/s$. Maximum deceleration is $3$. Maximum acceleration is $10$. Distance to the target is $2000m$. The ...
2
votes
2answers
452 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 ...
55
votes
10answers
7k views

Is learning (theoretical) physics useful/important for a mathematician?

I'm starting to read The Princeton Companion to Mathematics, at the beginning it says: A proper appreciation of pure mathematics requires some knowledge of applied mathematics and theoretical ...
1
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0answers
227 views

Complete vs General Integral of first order PDE

The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics: ... we should recall the fact that every first-order partial differential equation has a solution depending ...
5
votes
1answer
285 views

Cartan 3-form on a Lie group G

Does anyone have a reference to learn more about the Cartan $3$-form on a group manifold $G$? I have read that the WZW Lagrangian is nothing more than the integral of the pullback of the Cartan ...
1
vote
1answer
148 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 ...
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vote
0answers
394 views

(Geometry-Physics) Given a radius and velocity calculate position of an aircraft banking to make a turn in three dimensional space

I have a radius, R, for an aircraft traveling at velocity, V. If we start at point, (X,Y,Z), what is the position of the point at time, t in terms of coordinates(X1,Y1,Z1)? For example: The aircraft ...
3
votes
1answer
187 views

Trouble deriving DE for fourier transform from DE of function

I am trying to derive an equation which is a standard result in physics (the momentum space Schrödinger equation). (Background: The wavefunction is a complex valued function of position coordinates ...
8
votes
4answers
540 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 ...
2
votes
1answer
494 views

wave equation and superposition

If I have this equation: $$\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial t^2}$$ And this general solution: $$u(x,t)=\sum^\infty_{n=-\infty}\cos k_nx(C_n\cos k_nt+D_n\sin k_nt)$$ ...
8
votes
2answers
3k views

What are “Super Numbers”?

I'm reading Hyperspace by Michio Kaku and in the chapter on SuperGravity "Super Numbers" are mentioned and are described as a number system where for any super number $a$, $a*a=-a*a$. I was wondering ...
3
votes
3answers
873 views

The relationship between mean and variance in the context of system energy and the partition function

I'm looking at a specific derivation on wikipedia relevant to statistical mechanics and I don't understand a step. $$ Z = \sum_s{e^{-\beta E_s}} $$ $Z$ (the partition function) encodes information ...
1
vote
2answers
393 views

What are the differences between classical Yang-Baxter Equation and quantum Yang-Baxter Equation?

what are the differences between classical Yang-Baxter Equation and quantum Yang-Baxter Equation? Thank you very much.