"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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shifting integration variable and taking derivative seemingly giving problem

I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ...
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12 views

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra?

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra? https://en.m.wikipedia.org/wiki/Hodge_dual https://en.m.wikipedia.org/wiki/*-algebra This would seem to be a ...
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61 views

What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition?

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...
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22 views

Deriving E=mc^2 from hollow box with mass M and photon

I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the ...
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1answer
41 views

A Question Regarding Stone's Formula

Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$ A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an ...
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1answer
28 views

Can someone solve this in order to provide an example for solving central force problems?

Let $F(x)=x$ be a function that describes the magnitude and direction of a force that varies with distance from the origin. I understand that $m \frac {d^2} {dt^2} p(t) = F(p(t))$ is used to derive ...
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1answer
27 views

Central Force Problem

Given an equation $F(x)$ that represents the magnitude of some force $F$ that varies with distance from the origin, is it possible to derive the equation of motion $p(t)$ of a point particle $P$ ...
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1answer
20 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
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1answer
48 views

Proving an integral relation (isotropic function)

In Hansen-McDonald's book Theory of Simple Liquids the following relation is often used: We want to evaluate the integral $$\int_V f(\vec r_1, \vec r_2) d \vec r_1$$ We observe that if the function ...
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1answer
31 views

Gravitational attraction in different points in a uniformly dense sphere

Suppose we lived on the surface of a spherical planet, with uniform density. How would the gravitational force we experience change if we drilled a hole and descended towards the centre of the planet? ...
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1answer
22 views

Understanding elastic collision between two rocks with unknown masses

I have this problem here that goes like this: Two curling rocks of equal mass, one red and the other yellow, are involved in a perfectly elastic, glancing collision. The yellow rock is initially at ...
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1answer
35 views

Two blocks and a frictionless pulley problem

Block B ($m_{B}$=0.36 kg) is connected to a lightweight rope that passes over a lightweight, low-friction pulley.The other end of the rope is connected to Block A ($m_{A}$=0.72 kg), which is on a low-...
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61 views

Metric transformation

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
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1answer
45 views
+50

Schwarzschild metric, speed of ball as measured by observer who catches the ball, just before ball is caught?

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\...
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1answer
43 views

Hamiltonian Mechanics and the Symplectic Category

Are canonical transformations (in the sense of Hamiltonian mechanics) morphisms for a certain category? They seem to fit the archetypal description of morphisms being "structure-preserving maps". ...
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1answer
9 views

How to know whether a solution for a set of coupled non-linear differential equations are stable or not

I have 3-coupled ordinary non-linear differential equations, one second order in time and other two first order in time, with 3-dependant variables say x(t), y(t) and z(t). I have a particular ...
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13 views

Collision laws (formulas) for colliding discs with rotation [migrated]

Say you have two discs with fixed radii $r_{1}, r_{2}$, positions $q_{1}(t), q_{2}(t)$, momenta $p_{1}(t), p_{2}(t)$, which both depend on time $t$ and fixed masses $m_{1}, m_{2}$. They are assumed to ...
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1answer
24 views

Derivative of Lattice Laplacian

The lattice Laplacian is defined as, $$ \nabla_L^2x_j \equiv \frac{x_{j+1} - 2x_j + x_{j-1}}{a^2} $$ where the lattice spacing, $a$, is a constant. The derivative, with respect to $x_i$, then gives, ...
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2answers
41 views

“Inverse” Helmholtz Decomposition

So I am trying to write a report on the Helmholtz decomposition theorem on $\mathbb{R}^3$. The theorem states that under certain conditions, every vector field $\textbf{F}:U \subseteq \mathbb{R}^3 \to ...
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0answers
24 views

By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...
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1answer
72 views

A friend and I are trying to figure out who is doing more work benchpressing? Me? Him? Or is it the same amount? [closed]

Me and a friend have been working out for some time and have been bench pressing each week, increasing our weight. Once we started getting to heavier weight we noticed that he was able to get more ...
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1answer
15 views

Helmholtz Decomposition on $\mathbb{R}^3$ Proof

I am trying to prove the Helmholtz decomposition theorem which states that given a smooth vector field $\mathbf{F}$, there are a scalar field $\phi$ and a vector field $\mathbf{G}$ such that \begin{...
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2answers
21 views

Differentiating $v= \frac{d \theta}{dt}-r \sin \theta~i + \frac{d \theta}{dt} r \cos \theta~j$, with respect to time using the product rule

This is what I'm trying to differentiate with respect to time: The answer is supposed to be found using the product rule: However, I can't see how the product rule would be used here and what ...
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37 views

Space-time mathematics. Do I get this right?

Trying to understand general relativity like this, do I get this right? distance in $space \neq$ distance in $time$, if $space$ and $time$ = $2$ separate dimensions distance in $space =$ distance in ...
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35 views

Physical side of TQFT [migrated]

How would one go about understanding the physical side of TQFTs? What are the best introductory resources? I know Atiyah axioms but I don't know any QFT.
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1answer
68 views

Computing Curvature of a Connection (Dirac Monopole)

I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ...
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2answers
49 views

Numerical solutions of partial differential equations

I'm studying mathematical physics and working on numerical solutions of partial differential equations. I am having trouble understanding the way we solve partial dif. equations, e.g., $\frac{\...
2
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1answer
50 views

Gauge condition equivalent to condition that coordinate functions satisfy wave equation to first order

Let $\eta_{ab}$ be the metric of special relativity and let $x^\mu$ be global inertial coordinates of $\eta_{ab}$. Let $\gamma_{ab}$ be a small perturbation of $\eta_{ab}$. How do I see that the gauge ...
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1answer
16 views

Finding equation of a bent sufficiently flexible cardboard of length $l$ fitting into a gap of width $m<l$

I was thinking about how the walls of a barrel is made then I realized it is someone like fitting a piece of wood of length $l$ in between some "gap" of length $m<l$. This would cause the piece of ...
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0answers
18 views

Finding the steady-state temperature distribution in the half space $ z < 0 $

i thought about using cylindrical laplace equation. and stucked in the process, where i needed to use fourier-bessel series.
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1answer
25 views

How to interpret summation convention?

In Landau and Lifschitz Mechanics, p. 99, we have (implicit) the equality $$\Omega_i^2 x_i^2 = \Omega_i \Omega_k \delta_{ik} x_{\ell}^2 $$ written with Einstein summation convention. The left hand ...
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39 views

Special Relativity-Book

I would a good book to study the Special Relativity. In my course the professor has treated the following topics: $(1)$ Lagrangian and hamiltonian dynamic of a charged particle; $(2)$ Relaticistic ...
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25 views

Tensors as mathematical objects

Continuing my journey to understand Tensors, Maxwell's equations. Here is my current understanding. Is it correct? Tensors are mathematical objects, i.e., an entity in mathematical reality or a ...
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17 views

Ease out elastic function with equivalent start and end values?

I have an elastic ease out function: http://easings.net/#easeOutElastic formula in code: ...
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0answers
11 views

Direction of launch given tangential velocity vector

Angular velocity of the object = $<12.7848,7.38128,5.37313>$ Position of object in space = $<-0.1154,0.5867,-0.531658>$ Tangential velocity of the object = $<-7.07679,6.17696,8.3529>...
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1answer
24 views

Poisson brackets of angular momentum

So I'm trying to simplify this Poisson bracket of angular momentum vectors: {$L_1,L_2$} Where $L=r \times p$ I know that $L_1=r_2p_3-r_3p_2$ and $L_2=r_3p_1-r_1p_3$ (I can easily derive this from ...
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21 views

Vector diagram: forces

Using a vector diagram,explain why it is easier to do chin ups when your hands are 30cm apart instead of 90 cm apart.(Assume that force exerted by your arms is the same in both cases). If someone ...
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1answer
43 views

Difference between position vector and distance vector? [closed]

Its similarities and dissimilarities between position vector and distance vector?
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1answer
31 views

How to show that this integral operator is bounded?

Consider the integral operator $T : C([0,1])\to C([0,1])$ given by $$Tf(t)=\int_0^1 K(t,\tau)f(\tau)d\tau.$$ I'm solving one exercise which is to show this operator is bounded. The exercise is from ...
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1answer
40 views

Making ease-out-bounce formula have a linear start

I'm using a bounce ease out formula, the code for it: https://github.com/jesusgollonet/processing-penner-easing/blob/master/src/Bounce.java#L9. The function is copied here: ...
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0answers
7 views

$L^2$ and $C$ solutions of an initial-boundary value problem for 4$^th$ order equation

I study the initial-boundary value problem \begin{equation} \alpha^2\frac{\partial^4 w^0}{\partial x^4}+\frac{\partial^2 w^0}{\partial t^2}=P(t)\delta\left(x-\xi\right),~~ 0<x,\xi<1,~ t>0, \...
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0answers
6 views

Elastic Body Simple Deformation

In continuum mechanics we can consider a reference frame $B = [0,1]$ along with a homogeneous deformation $F$ where $x = Fp$ for $x \in \mathbb{R}$ and $p \in [0,1]$ and $F = 2$ so $F[B] = [0,2]$. ...
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0answers
31 views

How does the step in the picture transition to step 2?

:) I have a math question regarding this picture. The problem is that I do not understand how the first equation turns into the the second. Where did the integral come from?? (the dv and dt) Update: ...
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20 views

Physical Object to Pseudo-Riemannian Manifold

It is well known that Lorentzian mainfold is studied in general relativity. So this raises my curiosity about How about the classical mechanics? Does it correspond to the manifold $\mathbb{R}\times ...
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71 views

How low will a given string hang? [closed]

If I have a piece of string that is n meters long, attached at two points m meters apart, how low will the string hang? The two ...
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0answers
37 views

Riemannian metric as an operator

In the article http://www.sciencedirect.com/science/article/pii/0370269379905896 authors consider principal bundle $P(M, G)$ and then define induced metric $g$ on $\eta = Sp(A)/G$, where $Sp(A)$ - ...
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1answer
39 views

Scalar property of $ C(\Omega)=\sum_{|\alpha|\leq m}\color{blue}{\big|\Omega\big|^{\dfrac{2|\alpha|-n}{n}}} \int_{\Omega}|D^\alpha f|^2\ dx $

This is closely related to a previous question: Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$ This question focuses on the direct calculation (by change of ...
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35 views

Srednicki's QFT - chapter 2 - understanding from a mathematician's point of view

I am reading the first chapters of Srednicki's Quantum Field Theory book, trying to understand them from a mathematician's point of view. In particular, I'm interested to what happens when you try to ...