"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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Projection operator is Hermitian

Use Dirac notation (the properties of kets, bras and inner products) directly to establish that the projection operator $\mathbb{\hat P}_+$ is Hermitian. Use the fact that $\mathbb{\hat ...
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is it possible to intergrate this function to get x(t) and y(t)?

say you have a function as below; $d^2V(t)/dt = -B^2V(t)$ B is a constant Initial conditions $V_x(0) = V$, $V_y(0) = 0$ I can't see how to integrate to get x(t) and y(t); I ended up with ...
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Significance and physical meaning of diagonalization of linear maps and bilinear forms, eigenvalues and eigenvectors

In linear algebra, I have studied the diagonalization of a linear map and of a bilinear form; and also the concepts of eigenvalues and eigenvectors. However, the importance of diagonalizing a linear ...
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+100

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
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2answers
71 views

Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
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1answer
16 views

an object is rolling down a circular curve

An object is dropped from the top of a circular curve with radius r and rolls down the curve until it reaches the bottom. What would be the equation that would give the velocity of the object at any ...
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Proving that the derivative of the LRL vector $=0$ [on hold]

How to prove that the derivative of the Laplace–Runge–Lenz vector $=0$? $$A=\dot{x}\times(x\times\dot{x})-\dfrac{k}{\mu}\cdot\dfrac{x}{||x||}$$ $$\dot{A}=0$$
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160 views

How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables. Can anyone offer any advice on how I could go about solving this? 3 days of staring at a wall has so far given me no ...
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33 views

Mathematical Puzzle: A Drag Race of Who Wins

I'm having a real difficult time understanding how this problem is solved: "Two drivers, Alison and Kevin, are participating in a drag race. Beginning from a standing start, they each proceed with a ...
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2answers
92 views

What exactly are pseudovectors and pseudoscalars? And where could I read about them?

I can't find good information on the internet. In my mathematical physics class the definition of a vector was given as: That object with magnitude and direction which doesn't change under ...
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1answer
23 views

Effective Acceleration for Non-Constant Acceleration Motion

This question uses the same symbols as "Effective Acceleration" is Distance-Averaged Acceleration?. One of the kinematics formulas for constant acceleration is: $\Delta x=v_0*\Delta ...
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1answer
25 views

Functional expansion

I am confused by this expansion in Landau and Lifshitz: First, they define $\textbf{v}' = \textbf{v} + \textbf{$\epsilon$}$. So for a function $L$, $$L(v'^2) = L(v^2 + ...
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25 views

Second order differential equation, physics.

I need your input on this exercise I'm doing: "A 2-kg mass is suspended from a string. The displacement of the spring-mass equilibrium from the spring equilibrium is measured to be 50 cm. If the mass ...
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0answers
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How can i Plot Laguerre-Gaussian mode vortex by Mathematica? [closed]

I've been using Mathematica for a few years. I couldn't plot the Laguerre-Gaussian vortex vawefron by using the Mathematica. I know i should use the Laguerre-Gaussian function in coding, but i ...
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2answers
556 views

What is the exact and precise definition of an ANGLE?

On wikipedea I found a definition of an Angle as such: "In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of ...
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1answer
34 views

Physically, what meaning have Taylor series which have their lower order terms equal to zero, but their higher order terms non zero?

Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the ...
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1answer
45 views

“Effective Acceleration” is Distance-Averaged Acceleration?

My question involves simple math, but to be precise on what I'm asking, I need to write a lengthy description. Let us define the following symbols: $t$: time $x(t)$: distance as a function of time ...
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1answer
13 views

Shifting Velocity and Position functions

I'm given a function $A(t)$ that defines the acceleration of an object w.r.t. time $t$ and am tasked with finding the position function and velocity function for that object. Finding the functions ...
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Computing the angular momentum in spherical coordinates [migrated]

How to compute the angular momentum of a particle in spherical coordinates? It's given by: $$x_1=r\cdot\cos(\phi)\cdot\sin(\theta)$$ $$x_2=r\cdot\sin(\phi)\cdot\sin(\theta)$$ ...
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Angular momentum in Cylindrical Coordinates

How to calculate the angular momentum of a particle in a cylindrical coordinates system $$x_1 = r \cos{\theta}$$ $$x_2 = r \sin{\theta}$$ $$x_3 = z$$ Thanks.
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2answers
103 views

Why $F(\mathbf q,\dot{\mathbf q},t)$ and not $F(\mathbf q,t)$?

In beginner classical mechanics, which I've just started learning, a particle with coordinates $\mathbf q\in\mathbb R^n$ has its equation of motion specified by $F(\mathbf q,\dot{\mathbf ...
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1answer
18 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
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2answers
51 views

Electric field of semi-sphere

I have to find the electrical field in the center (of the base) of a semi spherical shell of radius R. The total charge Q (Q > 0) is uniform on the intern surface of the semi sphere. Here's a scheme: ...
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1answer
39 views

The Hodge dual and the Moyal product related or just notation?

The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an inner product space $V$ of dimension $n$. So we can we write; \begin{equation} \lambda\in ...
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How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$?

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$? I was reading a papar and found out that the desity matrix in $d$-dimensional Hilbert Space can be expressed ...
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38 views

General Relativity perturbation

Could anyone explain to me what I have misunderstood/missed when trying to understand this paper on GR perturbation? The paper is http://arxiv.org/pdf/0704.0299v1.pdf In equation 25 for $R_{00}$, ...
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2answers
366 views

Calculating the linear output force a threaded rod and nut would produce based on the input torque

I would like to calculate the amount of linear thrust a threaded rod combined with a rotationally static nut would be able to produce based on the rotary force applied to the rod. The information ...
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0answers
25 views

Commutators of Schur polynomials of Lie algebra elements

Question: Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
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61 views

Existence of classical solution for equation $\Delta u + v(x) u = 0$

It is known that if $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$ then the Dirichlet problem $$ \begin{array}{rl} \Delta u & = 0 \quad \text{in $D$}, ...
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1answer
46 views

How to prove $\mathcal{L}e^{-U}g = e^{-U}\mathcal{L}^*g$ for Fokker-Planck operators

Let $U$ be sufficiently smooth, $\beta$ a constant and $$ \mathcal{L}p = \frac{1}{\beta}\Delta p + \nabla\cdot(p\nabla U)\\ \mathcal{L}^*g = \frac{1}{\beta}\Delta g - \nabla g \cdot\nabla U. $$ Now ...
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1answer
53 views

The Underlying Manifolds of the Special Unitary Lie Groups SU(n)

I want to find the underlying manifolds of Lie Groups $\mbox{SU}(n)$ for general $n$. $$ \quad $$ My lecturer told us last year that the only n-spheres that admit a Lie group structure are ...
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m-th Dertivative hermitian for even m, and antihermitian for odd m

How to show that the $\left(\frac{d}{dx}\right)^m$ operator, is anti-hermitian for odd $m$ and hermitian for even $m$. I can use mathematical induction to show this, but I need a more formal proof.
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Stability of Lax-Wendroff Approach for Advection Equation

The Problem: I am attempting to solve the following problem in 1D over a periodic region: "In one dimension, the mass density $\rho$ is advected with velocity $v$, so that it follows the equation: ...
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1answer
30 views

well-posedness of a mathematical model

what is the meaning of Well-posedness of a mathematical model of a physical phenomena for example stokes equation in fluid dynamics ? what is the necessity to prove that a model is well-posed? how ...
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Solve the following IVP for the wave equation in 2D with “Hadamard’s method of descent”

I must resolve the following IVP for wave equation in 2D: $$\begin{cases} u_{tt}=u_{xx}+u_{yy} & (x,y,t)\in\mathbb{R}^2\times(0,+\infty)\\ u(x,y,0)=f(x,y)= \frac{1}{1+x^2+y^2} & ...
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1answer
373 views

Cylinder-ray intersections equation

I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation: $$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$ In the end of the first page I quote: ...
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Derivatives : trouble to understand formulas

My teacher gave us some useful formulas, but honestly I don't know how to understand it. gradient of a scalar field : $d_{x}i{V^{i}}f(M)\varepsilon ^{i}$ gradient of a vector field : ...
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1answer
45 views

Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
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57 views

Algebraic treatment of Feynman potential wells?

This question refers to the discussion here: Chomsky, Feynman, Thom I will try the divide-and-conquer strategy to try to make some progress in the problem dealt with in the above-mentioned link. My ...
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Stochastic resonance

Having a look at the book Dynamical Cognitive Science, by Lawrence M. Ward, MIT PRESS, I encounter something which might be useful for my research, namely what he calls stochastic resonance, of great ...
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1answer
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Total thrust on the face of a vertical dam

"A vertical dam is a parabolic segment of width $12m$ and maximum depth $4m$ at the center. If the water reaches the top of the dam, find the total thrust on the face." Is it possible to answer this ...
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1answer
86 views

An $L^2$ bound for a function of the form $w(x)=|x|^{-1} - (g^2*|x|^{-1}*g^2)(x)$

It is stated in the paper [E. Lieb and H.-T. Yau, The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanics Commun. Math. Phys. 112, 147-174 (1987)] that this function is ...
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210 views

Can someone identify the following key equations in physics? [closed]

I need help identifying the following equations in physics. Most equations are related to quantum mechanics, a few is from relativity and electromagnetism. Thanks
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1answer
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Charge given to the electroscope

The question is: What is the charge given to the electroscope? Also each ball has a weight of 25g. Here's how I started. First I draw the scheme on the forces acting one one ball (I took the one ...
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29 views

Finite difference scheme for the continuity equation

I am currently trying to solve a system of PDE's numerically, one being the equation; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 $$ I have been reading up on ...
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Derivative of position is velocity and of velocity is acceleration?

This is more of a personal question (i.e. not school related), but how has it been proven that the derivative of position is velocity and derivative of velocity is acceleration? I did some Google ...
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1answer
66 views

An elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system

Arnold in his essay On teaching mathematics made the following statement: The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only ...
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1answer
32 views

Help for understanding a vectorial equation found in a paper.

Trying to implement a code for the algorithm described in this paper I found something not very clear to me that leads me to misunderstand the whole concept. To calculate the vector $\vec{b_{3d}}$ ...
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1answer
57 views

Basis for Clifford algebra $Cl^2 (W)$ and quotient space $Cl^3(W)/Cl^2(W)$

Consider a basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W=V \oplus V^*$. I need help to write the basis for Clifford ...