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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54
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10answers
6k 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
225 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
282 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
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1answer
146 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
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0answers
393 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
533 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
490 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: $$u(x,t)=\sum^\infty_{n=-\infty}\cos k_nx(C_n\cos k_nt+D_n\sin k_nt)$$ ...
7
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
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3answers
868 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
380 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.
4
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1answer
344 views

How to improve linear interpolation in 3D

in one physics problem, there is a cube. A computationally expensive function can be calculated inside the cube. But one needs to do the calculation faster and to know for a given point inside the ...
0
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0answers
174 views

The quantum harmonic oscillator [closed]

I want to ask how to solve the equation $$-\frac{{\hbar}^{2}}{2m}\frac{\partial}{\partial_{r}^{2}}u(r)=(E-\frac{1}{2}Kr^{2})u(r)$$ with $K$ being a constant.
2
votes
1answer
160 views

expansion of an expression

The Fokker-Planck equation for several variables is : $$\frac{\partial W}{\partial t} = L_{FP}W\qquad(1)$$ where $$L_{FP} = -\frac{\partial}{\partial x_i}D_i(\{x\})+\frac{\partial^2}{\partial x_i ...
0
votes
2answers
104 views

Summation for Average Speed

How would I write a summation for the average speed of an object, given $10$ speeds (which were calculated $r=\frac{d}{t}$)? Would it be: $$ \sum_{i = 1}^{10} \frac{d_i}{t_i}$$ or am I wrong? I'm ...
0
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1answer
1k views

Hydrostatic Force/Pressure

I'm having difficulty understanding how to find Hydrostatic Force on various shapes. One problem is a right triangle with a height of 3ft and a width of 7ft. Its submerged 1ft below the surface with ...
17
votes
6answers
855 views

Geometrical construction for Snell's law?

Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the ...
2
votes
2answers
147 views

A body falls through a medium

I'm hating these variable resistance questions. A body of mass $m$ falls from rest in a medium that produces a resistance of magnitude $m\cdot k \cdot v$. where $k$ is a constant, where the speed of ...
6
votes
1answer
501 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 ...
0
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1answer
69 views

Variable Resistance

I'm stuck on this question from a past paper. A parachutist jumps out of a hot air balloon. The resistance is proportional to the speed of the parachutist. The parachutist has a terminal velocity of ...
2
votes
2answers
137 views

why does $\frac{b\partial a}{a \partial b} = \frac{\partial \log a}{\partial \log b}$?

I am doing some background research on sensitivity and elasticity analysis, and I came across the following definitions of elasticity: $$e_{ij}=\frac{a_{ij}\partial \lambda}{\lambda \partial ...
0
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2answers
123 views

Thermal transfer calculation

Say I have a flat rectangular bar of aluminum with a known volume. Sitting directly on top of that piece of aluminum is a rectangular circuit board made of FR4 (standard circuit board material). I ...
4
votes
2answers
251 views

Is there a point of no hearing if two speakers are emitting the same sounds - Wave Superposition

I'm talking bout this phenomenon (The third one): I would like to know if with sound waves, happen the same as with (water) waves. So, If two speakers were emitting the same sound, would there ...
9
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0answers
237 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, ...
3
votes
1answer
303 views

Laser paradox - A dot restricted to moving across an infinite plane

A laser is suspended 1 metre above an infinite plane. The laser swivels on its aperture, which remains at point A. The aperture is directly above point B. The laser shines a infinitely small dot onto ...
5
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1answer
331 views

WANTED: Short proof for challenging identity

I have to prove the following without any use of further mathematical theories except basic calculus and linear algebra: Let $ \ell $ be a positive integer and $A$ a real, symmetric and positive ...
2
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2answers
1k views

Homework Help - Calculus III / Physics / Work-Force Problem #2

I'm in a similar scenario as with my other question. I have tried to answer the problem correctly, and I have found similar (and even an identical) problem in my textbook, yet I still can't seem to ...
2
votes
2answers
1k views

Homework Help - Calculus III / Physics / Force-Work Problem

At the moment, I can't find a similar example in my textbook or in my class notes, and I have looked for one and contemplated this problem for the past hour or so. Can anyone give me an idea of what ...
3
votes
1answer
317 views

Best Practices for Learning Mathematics (especially in the classroom)

What study practices have led you to the best success in learning mathematics (or applied math, or theoretical CS) in the classroom (undergraduate/lower graduate level), especially in courses that ...
1
vote
3answers
677 views

calculation to change a helicopter's vertical position, adjusted with gravity

This is my very first question, so don't be too hard on me :) I am a beginner programmer, and I am working on the typical helicopter game, working with a one-button push: When the button is pushed, ...
2
votes
3answers
245 views

Applying a linear transformation to time sequences to separate interfering oscillations

This is an applied problem, which arises from the problem of reorienting of a sensor axes according to particle displacement directions: Consider a sensor which is located inside the solid substance. ...
9
votes
1answer
787 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 ...
2
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1answer
1k views

What does the little d and d^2 mean in equations?

I'm reading a text on ray tracing. There is this section about radiometric quantities where radiance is defined as $L = \frac{d^2\Phi}{dA cos\Theta d\omega}$ $\Phi$ is the radiant flux $\Theta$ is ...
2
votes
3answers
552 views

Scale Operator $Uf(x)=f(kx)$

I am looking for an operator $U$, that can do this to a function: $$Uf(x)=f(2x).$$ In particular I am happy if there is an $U$ for the general case: $Uf(x)=f(kx)$. Does such an operator exist for ...
1
vote
1answer
125 views

Continuum limit of (DE) equation

I have a system described an equation, and I want to find an (DE) equation for z(x,t), in the limit as l->0. First some definitions to simplify it some: $Z1=z(x,t)-z(x+l,t)$ $Z2=z(x,t)-z(x-l,t)$ ...
0
votes
1answer
432 views

Is it possible to calculate an angle of movement from the horizontal from accelerometer and magnetometer data?

I have a home made device (think baseball) with an accelerometer and a magnetometer embedded within it and would like to capture some details of an attempted 'throw' of the device. All I need is the ...
15
votes
7answers
3k views

Books to learn physics, being a math major

What books would you recommend to learn physics, being a a Math major, from classical mechanics, electricity, etc. to modern physics?
3
votes
2answers
184 views

sum of N light sources

[sorry for the bad English] I am fond of astronomy and environment. I want to try to make a "light pollution map" but I haven't my satellites... so I use as approximation of light pollution the ...
3
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1answer
657 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 ...
0
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2answers
731 views

Newtonian Physics: ball A is dropped, ball B is thrown - where do they coincide?

I'm doing a physics course in my spare time and the problem set is: Ball A is dropped from rest from a building of height H exactly as ball B is thrown up vertically from the ground. When they ...
3
votes
2answers
197 views

Vector Force Fields and Their Physical Interpretations

The vector force field F=(yi,-xj) has a curl of -2. The acceleration of a particle in space is given by: ax=y/m ay=-x/m This vector field has a divergence of 0. Will particles in this vector FORCE ...
45
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 ...
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1answer
404 views

Complexifying representations

Let me try to split the question in a few parts, I would like to understand the claim that all non-degenerate bilinear symmetric forms are equivalent over the complex while for the reals they can be ...
6
votes
2answers
507 views

sl(2,C) and the harmonic oscillator

I've been studying the finite-dimensional representations of the lie algebra sl(2,C). I've read that these representations are related to the harmonic oscillator and the associated raising and ...
1
vote
3answers
531 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 ...
15
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4answers
2k views

Why does dust gather in corners?

I've noticed when sweeping the floor that dust gathers particularly in the corners. I assume there is a fluid mechanics reason for this. Does anyone know what it is? Edit: No, really, this is a ...
15
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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)
2
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1answer
264 views

Temporal aberration. Shape of telescope mirror large enough that light transit time is significant

Suppose a parabolic mirror of a telescope is large enough (or equivalently that the camera exposure time is short enough) that you need to take into account the transit time of the light signal from ...
3
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4answers
3k 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), ...
0
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1answer
377 views

Calculation of G force

I have a formula which is G-force = velocity*omega/9.8. Omega is the angular velocity. I've seen on the internet that G force is actually acceleration/9.8. I'm confused as to which formula is correct. ...