"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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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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Anyone know how I would linearize this data

I think it might be a cubic function,however I am unsure of how I would linearize the data in order to find the regression equation. hi ian.
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26 views

how to linearize a cubic function [on hold]

So I have been trying to linearize this data for almost a month now, teacher and I have no clue what to do. Help would be greatly appreciated
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1answer
18 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
47 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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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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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
31 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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33 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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53 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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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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35 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
26 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
22 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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28 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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33 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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16 views

Trajectory of an object under gravity

Is there an equation (cartesian/polar)depicting the trajectory of the motion of an object relative to another (in a two body system) under gravity?
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1answer
30 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
49 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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26 views

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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19 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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25 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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35 views

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
27 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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33 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
46 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 ...
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1answer
35 views

Page 72 of Courant and Hilbert's Methods of Mathematical Physics, Vol 1.

We have the following identities: $$ \beta_\nu = b_\nu -\frac{1}{2}(b_{\nu-1}+b_{\nu+1}),\ \ \ \ (\nu=2,3,4,\ldots)\\ \beta_1=b_1-1/2 b_2 $$ $$s_n(x)=\sum_{\nu=1}^n b_\nu \sin(\nu x) \\ ...
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Integration of the product of Hermite Polynomial and exponential function

how to proceed with these two integration.. $$\int^0_{−∞}e^{−ax2}H_{2k}(x)dx=?$$ $$\int^∞_{0}e^{−ax2}H_{2k}(x)dx=?$$ where $$H_n(x)$$ is the Hermite Polynomial (physicist's convention).
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Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
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1answer
27 views

A proof in Hilbert & Courant vol 1 of Weierstrass theorem.

My question is regarding a derivation of an inequality on page 67 of Methods of Mathematical Physics. Here's a scan of the book: ...
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1answer
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Dividing before and after integration give different results

I'm having a physics exercise, but the question is more of math. Assuming I have the following constants: $m_1, m_2, \alpha, V_0$ and two variables: $v, t$. (v as velocity). I reach the following ...
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1answer
18 views

Difference between 'principal of indifference' vs 'the assumption of equal a priori probabilities'?

Is there a difference between the "principal of indifference" and "the assumption of priori probabilities" and if so what? If there is no difference why the use of two different terms? EDIT I have ...
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1answer
39 views

(Distributional) Fourier transform

I need to calculate the (distributional) Fourier transform of $$ f(x) = \frac{x^2}{x^2+1}. $$ I unsuccessfully tried to find a differential equation for $f$ in order to solve the Fourier-transformed ...
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Reference request: 2D conformal field theory and the honeycomb lattice

Would anyone know what is meant by "conformally invariant" functions defined on the plaquettes of the honeycomb lattice (ie the function is defined on the vertices of the dual tringular lattice)? ...
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How to compute Casimir elements of $g \otimes g$?

Let $g$ be a Lie algebra. How to compute Casimir elements of $g \otimes g$? I am asking this question because in the book a guide to quantum groups, page 80, there is an equation $r_{12} + r_{21}=t$, ...
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1answer
17 views

What is the difference between a first order compartment and diffusion

Biologists use Compartment models to represent the flow and storage of fluids in an animals body. In tissue (like muscle) the diffusion of blood is more accurately represented by the diffusion ...
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What is the mathematical understanding behind what physicists call a gauge fixing?

I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an ...
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1answer
53 views

Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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1answer
40 views

How to use Runge-Kutta 4th order method without direct dependence between variables

Following equation shall be solved using Runge-Kutta method of 4th order: $$ \frac{\partial E(z,t)}{\partial z} = \frac{\partial P(t)}{\partial t} $$ $P(t)$ is given as an array, so that the ...
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Density matrix respect to Hilbert space

If there a two dimensional Hilbert sapce $H$ with the basis, $\{ e_1, e_2 \}$ and state $\psi = \frac{e_1 - e_2}{\sqrt{2}}$. How could we express it as a density matrix ?
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Rewrite a Lagrange function to Euler-Lagrange equation in polar coordinate

If we have a Lagrange function in the form $L(p, q) = \frac{p^2}{2} + q^2$, how could it be re-written as a form of Euler-Lagrange equation in polar coordinates ?
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1answer
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Transformation of Graph

Hello all, I tried to solve this transformation and my answer was $-(x+3)^3+2$ my reason for thinking: reflect cubic power, shift to the left $3$ units, move up $2$ units. $-(x+3)^3+2$ However, ...
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29 views

Conversion of polar equations when you change the position of the origin

I'm working on a physics problem that is described as follows: "I am standing on the ground beside a perfectly flat horizontal turntable, rotating with constant angular velocity w. I lean over and ...
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2answers
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Simple, stable $n$-body orbits in the plane with some fixed bodies allowed

I'm working on a visual simulator for the $n$-body problem in the plane (here). The goal is to show how complex behavior can arise from the simple inverse-square law of gravity. To that end, I want ...
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Examples of self-adjoint operators on $L^2(\mu)$

I'd like to come up with a number of simple examples of (formally) self-adjoint operators on $L^2(\mu)$, where $L^2(\mu)$ denotes $L^2(\mathbb R)$ with respect to the Gaussian measure $d\mu$ ...