"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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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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23 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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20 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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33 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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33 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
41 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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105 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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45 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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34 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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19 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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15 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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80 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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88 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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35 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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81 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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169 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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14 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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84 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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34 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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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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60 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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33 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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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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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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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
25 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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90 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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56 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
122 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
29 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
68 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
26 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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195 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
32 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
72 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
40 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
23 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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27 views

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
125 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
25 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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20 views

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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45 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
93 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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31 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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1answer
48 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
73 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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Is the calculation of Green's function correct?

I am not sure if all the calculations are correct.Could you check for me please ? ...
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71 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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70 views

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: ...