"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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Associated Laguerre Polynomials negative indices?

For the associated Laguerre polynomials/functions, it is taken (specifically when solving for the eigenstates of Hydrogen in QM) that the associated Laguerre functions with negative indices (and also ...
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27 views

Double integral multivariable calculus

Consider the following integral $$\int_0^1 dx_1 \int_0^{1-x_1} dx_2 \, (1-x_1-x_2)^{-\epsilon-1} (-sx_2 - x_1p_1^2)^{-\epsilon-1}$$ where $s$ and $p_1^2$ are to be treated as constants throughout the ...
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38 views

Interpretation of a reaction diffusion equation

I have a reaction-diffusion equation in 1-dimensions of the typical form: $$\frac{\partial }{\partial t} u(x,t)= \frac{\partial^2 }{\partial x^2} u(x,t)+ \alpha(x) u(x,t), \,\qquad (x,t)\in ...
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49 views

Limit of probability density function as random variable approaches +/- infinity

Consider a complex-valued function $\Psi(x,t)$ such that $|\Psi|^2$ is a probability density function for $x$ (for any time $t$). In his introductory Quantum Mechanics book, David J Griffiths writes ...
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21 views

The sum of two r-matrices.

Let $g$ be a Lie algebra. Suppose that $r \in g \wedge g$ satisfy the condition: $[[r, r]] = [r_{12}, r_{13}]+[r_{12}, r_{23}] + [r_{13}, r_{23}]$ is a non-zero unique, up to scalar multiple, ...
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1answer
100 views

Origin of delta

Why does delta mean change? What is the origin of delta? I understand that upper-cased delta is used in this way and that delta is the fourth letter of the Greek alphabet. I also read that delta is ...
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3answers
71 views

Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
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Why is $J_n$ not symmetric, for $n\notin\mathbb Z$, while Bessel's equation is still symmetric?

Bessel's equation, $$x^2y''+xy'+(x^2-n^2)y=0,$$ has even parity, regardless of the value of $n$. So a solution of this equation must be even or odd. However, the Bessel functions $J_n$, which are ...
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1answer
16 views

Double random number from a gaussian, how to evaluate the skewness

I have a question for an application in physics. So my description will be really concrete, sorry. It's about the estimation of a systematic error from a calibration system. I have a LED with an ...
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35 views

Damping in a fluid

I am trying to understand damping in a fluid.Take, for instance, water flowing down a surface. I know a damping term $-\alpha\mathbf{u}$, where $\mathbf{u}$ is the velocity field, is added to the sum ...
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1answer
27 views

How does the evaluation of $\dot q_i$ at $q_1+\mathrm dq_1$ yield $\dot q_i +\dfrac{\partial \dot q_i}{\partial q_1}\mathrm dq_1 \;?$

I've been following Reif's Fundamentals of Statistical and Thermal Physics; there I came before the derivation of Liouville's theorem: There I couldn't understood few things. I could conceive the ...
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160 views

A conjecture on Schatten 1-norm

I have a conjecture on Schatten 1-norm. Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite ...
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1answer
64 views

I've found two different definitions of a cylindrical Brownian motion and don't understand why they are consistent

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space ...
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19 views

Infinite total variation of complex measure in Feynman path integral

I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...
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1answer
38 views

Right Propagating Wave complex

Does it make sense to think of $e^{ikx}\equiv $cos$(kx)+i$sin$(kx)$ as a right propagating wave? I am rather confused by the imaginary term here. Context: \begin{equation} \phi(x)=\begin{cases} ...
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38 views

Average number of steps to return to the origin of a random walk on a 2-d lattice.

Suppose I have a random walker on a 2-d square lattice with periodic boundary conditions with equal probability of going in any of the four directions. The walk ends when the walker reaches the point ...
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25 views

Definite integrals from Feynman-Hibbs A.4, A.5 (complex exponential with reciprocal time)

I am reading the revised edition of Feynman & Hibbs "Quantum Mechanics and Path Integrals". At one point (scattering of electron by atomic potential) we need to use a definite integral from the ...
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1answer
27 views

Set of coupled partial differential equations

I've read that the Einstein equation is a set of 10 coupled partial differential equations. I know what a partial differential equation is, but I don't know what a set of coupled partial differential ...
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1answer
40 views

How to construct a complete set in $L^2(\mathbb{R}^3)$ starting with the Spherical Harmonics?

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty ...
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48 views

Writing 5-dimensional dynamical system as Hamiltonian system

I've got a 5-dimensional continuous dynamical system, i.e., $$ \dot{x}(t)=f(x,y,z,u,w)\\ \dot{y}(t)=g(x,y,z,u,w)\\ \dot{z}(t)=h(x,y,z,u,w)\\ \dot{u}(t)=q(x,y,z,u,w)\\ \dot{w}(t)=p(x,y,z,u,w) $$ Is ...
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2answers
39 views

Multiplying two Scalar dot products together

stackexchange community, I'm just wondering what the rules are for multiplying dot products together, such as: $$ (P_{3}\cdot{P_{4}})(P_{1}\cdot{P_{2}}) $$ How would this be expanded out to not ...
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34 views

virtual work and potential energy

I was just going through the thermal and elastic buckling of bars & plates ,I found some researchers use virtual work to derive the equations, another researchers use potential energy in other ...
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1answer
30 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
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1answer
55 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
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41 views

Having trouble interpreting the geometry of this setup.

A circular conductor, with cross section given by $(x-d)^2+y^2=b^2$, i.e. radius $b$ and centered on $x=d$, has a circular core, made up of the interior of the circle $x^2+y^2=a^2$, with ...
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4answers
42 views

Finding the formula for acceleration from $v=2s^3+5s$, where $s$ is the displacement at time $t$

This is the question: I first found $\frac{dv}{ds}=6s^2+5$, then I tried to find $\frac{ds}{dt}$ by messing about a little with implicit differentiation, but I had no luck and I therefore couldn't ...
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47 views

Acceleration of an air bubble under the sea

An air bubble arises from the bottom of the sea. Find its acceleration if the resistance force is proportional to $\rho$*A*$v$ where $\rho$ is density of water, A is cross section area and $v$ is ...
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1answer
35 views

First Order Differential Equation for a Harmonic Oscillator

A box with mass $m$ is attached to a spring with spring coefficient $k$. This system is then placed into a glass case filled with a liquid with drag coefficient $\alpha$. Now I have the following ...
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1answer
36 views

More equations than unknowns for maxwell equations?

I had one curiosity regarding maxwell equations in 3-D From the curl equations, you get 6 unknowns, with 6 equations. The divergence equations add 2 additional equations. When these are combined, we ...
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1answer
64 views

How do I determine if the equation is a conservation law?

We have the PDE $\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a ...
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1answer
72 views

Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a ...
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1answer
65 views

Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some ...
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3answers
27 views

Where does the extra $\omega$ come in velocity of Simple Harmonic Motion?

Position $x$ in a SHM is given by $x=A\space sin(\omega t+\phi)$. Where $A$,$\omega$ and $\phi$ are Amplitude,Angular frequency and phase constant and are three constants respectively. So,velocity ...
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1answer
31 views

Frobenius method to solve differential equations, different \alpha found

I am referring to Carl Bender's Advanced mathematics methods for scientists and Engineers. Well, actually I know how to solve it....However, if I choose to do a so called "powerful" method,which is ...
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1answer
14 views

Odd Vector Product Question

Here is a question that has me stumped: Use the geometric definition to find: $2 {\bf i} × ({\bf i}+{\bf j})$ Student solution manual says: By the definition of cross product, $2 {\bf i} × ({\bf ...
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38 views

How can we prove that the derivative of a generalized Hilbert space valued Brownian motion is a Gaussian white noise?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\lambda$ be the Lebesgue measure on $[0,\infty)$ $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of ...
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36 views

Is this partial differential equation solvable?

Ok so I am asked to set up a partial differential equation and then motivate why it is solvable. I'm only 2 weeks into my course so we are not asked to solve anything yet. However, if someone would ...
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1answer
34 views

find angle given point the trajcetory passes through and inital velocity

I'm currently studying M1 for A level maths and we've derived the equation to prove that the trajectory is a parabola. $y=x\tan\theta - \sec^2\theta \dfrac{gx^2}{2u^2}$ I am curious as to how to ...
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1answer
60 views

Solving differential equation describing motion in a pendulum

I've been looking at Simple Harmonic Motion in particularly the period of a pendulum. This may seem like physics but my question is tailored towards mathematics. The differential equation is: ...
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37 views

Explanation Request for an alternative expression of a Gaussian integral over complex variables

We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...
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48 views

Runge-Kutta 4 in polar coordinates

How is the Runge-Kutta method implemented on this differential equation: $$ \frac{d^2 \theta}{dt} = -\frac{g}{l} \theta $$ (pendulum motion) which is in polar coordinates? Let: $c = \frac{g}{l}$ ...
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1answer
47 views

Velocity Verlet method: How to calculate acceleration

The velocity Verlet method algorithm is as follows: Calculate: $$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t)\, \Delta t+\tfrac12 \,\vec{a}(t)\,\Delta t^2$$ Derive: $\vec{a}(t + \Delta t)$ from ...
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Hamiltonian Elliptical Path

For a Hamiltonian of the form, $$ H = \frac{1}{2} p_i p^i - \frac{k}{\sqrt{q_iq^i}} $$ which is a Hamiltonian for a gravity system or something similar. These systems are know to have paths that ...
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Definition of Global Information and Local Information (CS)

I am a research student of computer science, I always feel like there are some thing missed when I am trying to define some concept mathematically. For example, I would like to define two concepts: ...
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Semi-infinite forms?

I am reading Vafa's paper 'Topological Mirros and Quantum Strings'(arXiv:hep-th/9111017). In this paper, the author says the Hilbert Space of a fermionic string theory corresponds to the space of ...
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Cyclic permutation

How did the author do the cyclic permutation? $\Gamma^k_{ij}g_{kl}+\Gamma^k_{lj}g_{ki}=\partial_jg_{il}$ We can cyclically permute these indices to generate two more equations: ...
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1answer
32 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$?

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where ...
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35 views

A computation from an article in computational neurosciences (from physical review) which doesn't fit

I am reading this article (with this erratum) in computational neuroscience, and there is a computation there that simply doesn't fit.. Maybe one of you can see something that I am missing? For the ...
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Simple Harmonic Motion; Tension in Elastic rope

I'm struggling to model this question out correctly. A glider and its pilot have total mass $230$ kg. The glider lands on a horizontal airstrip and when its speed is $16$ m/s it hooks on to the ...
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2answers
59 views

Calculating segment length on circle

I'm building a physical machine and I'm trying to figure out a geometrical problem. The machine is composed by a cylinder, and the wall of this cylinder is composed by many wooden boards, each of ...