"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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Sturm-Liouville and Bessel function identity

Given S-L equation $\dfrac{1}{x}[\dfrac{d}{dx}(xy')+(\dfrac{-m^2}{x})y]=-\lambda y$ Say $\mathcal{L}$ is the Sturm-Liouville operator, $y_k$ is eigenfunction $J_m(j_{mk}x)$ where $J_m$ is Bessel ...
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11 views

How to obtain the velocity sigma

Hi i have a set of results below but i don't understand how the value 4.4646x10^-4 is obtained. This is the part I'm specifically talking about. i hope someone can help. thank you
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95 views

Reflection of light from function graph

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to \infty}f(x)=0$. Let the point light source be placed at $P(x_0,y_0)$ with $y_0>0,\,y_0 ...
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30 views

Non-dimensionalization using Buckingham-$\pi$: example of the “bead on a rotating hoop, with viscous damping” problem

I am trying to find a way to non-dimensionalize known equations, using the Buckingham-$\pi$ theorem. Consider the "bead on a rotating hoop, with viscous damping" problem---if you are interested in ...
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15 views

Finding the expectation of momentum using density matrices.

I've been given the task of showing that $$ \bar{<p>}=\int\nabla_r\rho(r,r')|_{r=r'} dr$$ using the defition that the expectation of an operator $O$ is given by: $$ \bar{<O>}=\int\int ...
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Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
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20 views

Question about DE related to physics; includes Hooke's Law and Newton's Second Law as well as system of DE equations and solutions, and a phase plane.

I mainly need help with part A, and a little bit on B and C. Thank you in advance for your answer or any comment or edit that helps!!! A second-order DE can be sometimes solved with clever ...
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60 views

Force between parallel conductors using amperes law

Two parallel conductors are 0.3m long, and 0.15m apart. They each carry 2.5A of current in the same direction. Calculate the force between them. I did (2 * 10^-7) (2.5^2) / (0.15) and got the ...
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44 views

Minkowski metric on a surface

Do closed surfaces admit a metric with lorentzian signature? Any reference?
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1answer
29 views

PDE with distributional coefficient

I'm a undergraduate major in physics and when learning mathematical aspects of quantum physics I run into for example this problem: for a 1-dim system with delta potential, $-\frac{d^2}{dx^2}\phi + ...
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25 views

Finding the Legendre transform of an “entropy type” functional

I want to find the Legendre Transform of $$T(f) = \int_{\mathbb{R}^2} f \log \left(\frac{f}g{}\right) \, dx$$ on a set $H_M = \{ f: f \ge 0 \text{ and } \int_{\mathbb{R}^2} f =M\}$, where g is some ...
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Why do we use the word “energy” for this integral?

I came across the following and I would like to know the motivation or idea behind using the word "energy". Suppose $u(x)$ is a $C^1$ function on some domain $\Omega \subset R^3$. Then, we can think ...
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2answers
82 views

Volume integral of electric field (hemisphere solid)

Let $S$ be a hemisphere of radius $R$, and let $\sigma$ be the constant charge density at each point $(x',y',z')$ in $S$. The electric field generated by the hemisphere is a vector function: ...
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1answer
33 views

The second law for rotations

Is Newton's second law for rotations derived from his three laws of motion, or is it an independent axiom of physics?
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48 views

A simple pendulum

The idealized simple pendulum model (see the following figure) assumes that at every point of time the string to which the bob is attached exerts an equal and opposite force on the bob as does the ...
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1answer
35 views

Center of mass of a composite body

Find the coordinates for the center of mass to the shaded out shape. How does one tackle these problems? I have tried a bunch of stuff... like considering the small halfcircle hole on the left under ...
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21 views

Is compatibility with a gauge sufficient to turn a parallel transport into a connection?

As the title suggests, is compatibility with a gauge sufficient to turn a parallel transport into a connection?
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38 views

How is SO(2) compact according to this definiton?

According to MathWorld, a compact Lie group is a group whose parameters vary over a closed interval. I'm not sure if this definition is rigorous enough. I've also seen a similar definition here: ...
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42 views

how do you find position vector?

So at time zero a particle is at x= 4 m and y= 3 m and has a velocity of $${ v= \left( 2.0 \ \boldsymbol{\hat{\imath}}-9.0 \ \boldsymbol{\hat{\jmath}} \right) \text{m/s}}$$ The acceleration of the ...
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2answers
43 views

Help in Differential Equations - find velocity at t seconds…

Hello Everyone I am stuck where I am , Would like to know if I'm going the write path, and any hints on how to proceed would be greatly appreciated !! Question: A boat carrying 7 people is being ...
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116 views

The meaning of variables and derivations in Souriau's book

As far as I see, Souriau is using unconventional notions in his book "Structure of Dynamical Systems". He explains these notions in §2. of Chapter I, but it is a puzzle for me. Mainly because he ...
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52 views

Harmonic Oscillation using Gaussian Quadrature [closed]

Assume that the potential is symmetric with respect to zero and the system has amplitude $a$ suppose that the potential $V(x)=x^4$ and the mass of the particle is $m=1.$ Write a java function that ...
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1answer
44 views

Where should the fire man place the hose for the water to reach its maximum height?

Water leaves a fireman’s hose (held near the ground) with an initial velocity ${v_0 = 22.5 m/s}$ at an angle θ = 35° above horizontal. Assume the water acts as a projectile that moves without air ...
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27 views

Compute the specific heat capacity of ideal gas under constant $V$ and $p$

Compute the specific heat capacities at constant volume and constant pressure for air at standard temperature and pressure, assuming it is diatomic ideal gas and a molecular mass of 28u. I have ...
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18 views

How do you calculate a percent of precision?

What does it mean when "calculating a percent of precision can be found from the ratio of your mass sensitivity to the equilibrant mass for each case." My ratio of sensitivity is .010 kg. My ...
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86 views

Non-commutative symplectic geometry

How is non-commutative symplectic geometry defined? How does it differ from symplectic geometry? Does Darboux's theorem apply also there?
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123 views

Moment of inertia of semi circular disc with hole

Suppose we have semi circular disc of radius $R$ with semi circular hole of radius $R/2$ , how can I find moment of inertia from axis thru $H$ I fail to see any symmetry so I cannot integrate , ...
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50 views

Angle at which a body bounces off a sphere

Suppose a solid body approaches a sphere of radius $R = 1$ and height $z$, how do I calculate the angle $\theta$ at which the body bounces off the sphere? I am writing a java code where the ...
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90 views

Green's function in a moving frame for a constant heat source

I am looking for the Green's function of the problem in two dimensions $r =(x,z)$, \begin{equation} \nabla^2g + \frac{v}{D}\frac{\partial g}{\partial z} = -\delta (r-r_0) \end{equation} Which ...
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180 views

Hawking's and Ellis' derivation of the form of Einstein's field equations

On pages 72-73 of the book "The large scale structure of space-time" Hawking and Ellis show while determining the form of the field equations of general relativity that there is a relation of the form ...
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19 views

Non-linear perturbation definition

What exactly is the definition of a nonlinear perturbation when applied to a background spacetime metric? I have seen so called "linear perturbations" which look like $$ds^2 = -(1+2\Phi)dt^2 ...
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87 views

A bit challenging integration. (at least for me its challenging)

Hello everybody I am trying to solve this integral. I show you how far I 've gone. $\displaystyle\int^{\infty}_{-\infty} \frac ...
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1answer
44 views

Locally Hamiltonian vector fields

Consider the following definitions (taken from [1]) Definition. Let $E$ be a Banach space and $B: E \times E \to \mathbb R$ a continuous bilinear mapping. Then $B$ induces a map $B^\natural: E \to ...
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23 views

If $\mathbb v$ denotes the velocity of a point $s$ in $\mathbb R^3$, what is $\mathbb R^3\{\mathbf v\}$? (notation)

I started reading 'Mathematical Aspects of Classical and Celestial Mechanics' by Vladimir Arnold (Third Edition). In page $2$ they say that the euclidean space is denoted by $E^3$; moreover: The ...
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65 views

Perturbation of Laplacian

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...
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35 views

Determining a matrix representation

Determine a $2\times 2$ matrix $\mathbb{S}$ that can be used to transform a column vector representing a photon polarization state using the linear polarization vectors $|x\rangle$ and $|y\rangle$ ...
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1answer
51 views

System of equations ac = ax + by and ac^2 = ax^2 + by^2

I have these two equations: $$ ac = ax + by$$ $$ac^2 = ax^2 + by^2$$ I have to figure out $\mathcal x$ and $\mathcal y$ using $\mathcal a, \mathcal b, \mathcal c$ which are variables but not set ...
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3answers
202 views

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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2answers
35 views

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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1answer
41 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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2answers
100 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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2answers
104 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
143 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
33 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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2answers
107 views

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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1answer
27 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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212 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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1answer
36 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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1answer
104 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
47 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 ...