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Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elemetary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.

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What is wrong with this product of measures in the $m\to \infty$ limit?

This question is about a passage in Reed & Simon concerning the Wiener measure and path integrals. To give the context, the authors consider the Trotter product formula $$e^{-t(H_0+V)}f=\lim_{m\...
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1answer
38 views

What kind of math should I learn before I tackle policy search PEGASUS research paper by Andrew Ng?

I provided the link below https://ai.stanford.edu/~ang/papers/uai00-pegasus.pdf the paper was referenced in the AI: Modern Approach book, and I would like to dive in depth into it. But my math is ...
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0answers
32 views

Oscillations of an Energy Eigenstate

Energy eigenstates of a 1-dimensional particle are given by solutions to differential equations of the form $$ \left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi(x) = E\psi(x) $$ where $V$ ...
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21 views

convolution of generalized functions

Good Evening! How to calculate the convolution of generalized functions in $D'(R^2)$ $$\theta(t-|x|)*\theta(t-|x|)? $$ in $D'(R^1)$ I've done it. And what should I do in $D'(R^2)$? In my opinion, ...
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15 views

Euler Lagrange Equation with co-/ contravariant Tensors

I am currently trying to show the following equation which, appears from the Euler Lagrange equation: $$\frac{\partial}{\partial \partial_{\nu} A^{\mu} } \left( \frac{1}{4} F_{\mu \nu} F^{\mu \nu} \...
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52 views
+100

Is Leibniz integral rule (basic form) allowed in this (physics) improper integral? Why?

Electric potential at a point inside the charge distribution is: $\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ ...
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0answers
20 views

Clausius paper “ On the motive power of heat & on the laws which can be deduced from it for the theory of heat”

My question is simple. Clausius expesses his fundamental proposition as follows: "In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to ...
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1answer
59 views

Equivalent of Pauli matrices in 4 dimensions

I would like to decompose the following 4x4 matrix: $$ \mathrm{H} = \begin{pmatrix} a & b & b & 0 \\ b & 0 & 0 & b \\ b & 0 & 0 & b \\ 0 & ...
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0answers
12 views

Difference for even and odd values for $n$ in the equation system $u_{tt}=a^2u_{xx}$ and $u|_{x=0}=0$ and $u|_{x=l}=\sin\frac{n\pi a}lt$

This is a follow-up question of What would happen if the boundary value for $u_{tt}=a^2u_{xx}$ is that $u|_{x=0}=0$ and $u|_{x=l}=\sin\frac{n\pi a}lt$. In the following one-dimensional wave equation ...
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1answer
35 views

The reason for symplectomorphism to conserve the canonical form of the Hamilton equations.

If I have $(M,\omega)$ with Hamiltonian a symplectic manifold, let $(q_1,p_1,...,q_n,p_n)$ be the Darboux coordinates. With these coordinates, the integral curves of the Hamiltonian vector field ...
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1answer
23 views

Why the relation $\mathbf{E}=-\nabla \psi$ holds at points $P \in V'$?

Let there be a continuous charge distribution in space having volume $V'$ and density $\rho$. Let: $\displaystyle \mathbf{E}=\int_{V'} \rho\ \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3}...
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0answers
21 views

Sum of the residue at poles of a meromorphic differential

I have the following rational function meromorphic differential $$w_{11}:=1/12\,{\frac {5\,{{\it z_1}}^{8}-8\,{{\it z_1}}^{6}+18\,{{\it z_1}}^{4}-8 \,{{\it z_1}}^{2}+5}{{\it z_1}\, \left( {\it z_1}-1 \...
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0answers
58 views

Notation Atiyah-Singer Index theorem

I am trying to have a go at the Atiyah-Singer Index theorem and my question is very basic. Any help is appreciated! In Nakahara's book Geometry,Topology and Physics the theorem is stated as follows : ...
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1answer
42 views

What would happen if the boundary value for $u_{tt}=a^2u_{xx}$ is that $u|_{x=0}=0$ and $u|_{x=l}=\sin\frac{n\pi a}lt$

During the discussion of non-homogenous boundary values for the one-dimensional wave equation $$ u=u(x,t),\;\frac{\partial ^2u}{\partial t^2}=a^2 \frac{\partial ^2u}{\partial x^2} $$ where the ...
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20 views

When the curvature of a connection is changed is the metric associated to the connection also changed?

For example in physics metric describes curvature of spacetime and electromagnetic field strenght is the curvature if the connection if the u(1) complex line bundle.
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1answer
35 views

Integral Solution Help

Could you help me solve this integral \begin{equation} \int_{-\pi/2}^{\pi/2} \sin{\varphi} \ e^{jka\sin{\varphi}} \ d\varphi \end{equation} Thank you so much!
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1answer
35 views

Show invariance under linear transformation [closed]

A simple question but I'm currently stuck. Let $\kappa\in\mathcal{R}$, and let $\sigma = \kappa I$ and $\pi$ be real $2\times2$ matrices, where $I$ is the $2\times2$ unit matrix. Define the map $R: (\...
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0answers
32 views

Probability of chain length of $\infty$-dimensional Noether space

(My motivation of asking it is about the modeling of learning space (i told it).How can we learn about something? This says you use a finite step to learn something. But in this world, there are ...
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1answer
47 views

What exactly is a potential?

From my basic physics days I understand that potential energy is the capacity of an object to do work. Now I am studying fluid mechanics and I am coming across things such as "velocity potentials" ...
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1answer
24 views

Symplectomorphism, intuitive interpretation.

What is an intuitive way to think about symplectomorphisms? A symplectomorphism between two symplectic spaces is a map $(M_1, \omega_1)\xrightarrow{\phi} (M_2,\omega_2)$ such that for the pullback $\...
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1answer
70 views

Is there a name for the set defined by the Minkowski sum of circles in orthogonal planes?

Recently, I started thinking about the set of points defined by the Minkowski sum of 1D circles in orthogonal planes. The reason for this is to extend the well known result that 1D linear/harmonic ...
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1answer
19 views

Determine the magnitude and direction of the electrical field

Consider a positively charged infinite slab with uniform charge density $\rho_1$ and thickness $4a.$ This slab is oriented such that the two faces of the slab are located on the planes $x=-2a$ and $x=...
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0answers
45 views

Why it is contradicting?

Let: $\nabla$ denote dell operator with respect to field coordinate (origin) $\nabla'$ denote dell operator with respect to source coordinates The electric field at origin due to an electric dipole ...
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1answer
24 views

d’Alembert Lagrangian for second rank tensors

Consider the Lorentz scalar Lagrange density $$\mathcal{L}=\eta^{\mu\nu}\partial_\mu T^{\alpha\beta}\partial_\nu T_{\alpha\beta}$$ for a second rank tensor whose contravariant and covariant ...
2
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1answer
76 views

Conservation laws and the heat equation

I studied equations of the form $u_t+f(u)_x=0$ are called conservation laws. Recently, somewhere I read that heat equation is a parabolic conservation law(I know that heat equation is a parabolic pde)....
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0answers
32 views

Hamiltonian mechanics is good because of the symplectic structure of Hamiltonian systems.

I was reading the wiki on Hamiltonian mechanics, and I stumbled across the moto : The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic ...
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2answers
35 views

Commuting Operators Have the Same Eigenvectors, but not Eigenvalues.

The following problem is out of my quantum mechanics textbook. Assume that two operators $H$ and $\Gamma$ commute. Show that if $|\psi\rangle$ is a non-degenerate eigenvector of $H$, that is, $H|\...
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1answer
29 views

If $\frac{\partial u}{\partial \phi} = - \frac{\partial v}{\partial \lambda}$, is $u$ = $-v$

Under the continuity equation in atmospheric dynamics: $\frac{\partial u}{\partial \phi} + \frac{\partial v}{\partial \lambda} = 0$ Therefore, making: $\frac{\partial u}{\partial \phi} = - \frac{\...
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1answer
21 views

Find the reciprocal lattice vectors for a triangular lattice with primitive lattice vectors $\vec a_1=(d, 0)$ and $\vec a_2= (d/2, \sqrt{3}d/2)$

Find the reciprocal lattice vectors for a triangular lattice with primitive lattice vectors $\vec a_1=(d, 0)$, $\vec a_2= (d/2, \sqrt{3}d/2)$ Using the condition that the reciprocal lattice vectors, ...
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1answer
23 views

Physics: Biot_Savort Law Bent wire

An infinitely long wire carrying a current I is bent at right angle: \begin{align*} dB&=\frac{\mu_0}{4\pi}I\frac{d\mathbf{s} \times \mathbf{\hat{r}}}{r^2} \\ \int dB&= \frac{\mu_0}{4\pi}I\...
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0answers
16 views

How can I proof or demonstrate mathematically that solid angle integral from a point outside a closed surface is zero?

I would like to know how can I proof or demonstrate mathematically that solid angle integral from a point outside a closed surface is zero. How can this be done? I have already demonstrated the value ...
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1answer
54 views

Rigorous proof of quantum electrodynamics renormalization

In most physics books they give proofs of renormalization of quantum electrodynamics that are not mathematically rigorous. Is there any book or article that give a formal proof of quantum ...
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1answer
40 views

Where does this formular for rotating a vector in 3D space around another 3D vector comes from?

I found this formular: $\mathbf{R}_{\vec{n}}(\alpha)\vec{x}=\vec{n}(\vec{n}\cdot\vec{x})+\cos(\alpha)(\vec{n}\times\vec{x})\times\vec{n}+\sin(\alpha)(\vec{n}\times\vec{x})$ here: https://de.wikipedia....
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1answer
25 views

Particle traveling on a helix

In this problem, we consider a particle moving on a helix: a) Suppose the position vector of a particle as a function of time $t$ is given by $r(t) = ( \cos t, \sin t, 3t )$. Find the speed of the ...
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1answer
63 views

ADM Formulation in General Relativity

In the ADM(Arnowitt – Deser – Misner) formulation, we can foliate a globally hyperbolic spacetime by spacelike hyper-surface(Cauchy surface) $\Sigma_{t}$, which parametrised by global time function $t$...
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2answers
20 views

Calculating the velocity of a particle on a spinning disc

The particle is moving towards the center as the disc is spinning, the position of the particle is described by the following expression: $r = r(t) cos(ωt)i + r(t) sin(ωt)j$ How do I calculate the ...
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1answer
29 views

Reference Request - Applied Math books and resources

I am second-year undergrad from Physics. I wanted to move to applied mathematics because I like the topics they study, like differential equations, fluid dynamics and related stuff. So I am looking ...
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0answers
29 views

for arbitrary vector $v,u$, is there the matrix X which satisfy the relation exp$[X]\,v=u$?

Nowadays, I'm studying for exponential map of Lie group. my question is, To make the form of exp$\begin{pmatrix}x_{11}&x_{12}&\cdots \\x_{21}&\ddots \\ \vdots\end{pmatrix}$, I have to ...
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1answer
66 views

Is Mathematical Biology analogous to Mathematical Physics? [closed]

Mathematical Physics seeks to create mathematical tools and methods to solve physics problems. Is mathematical biology roughly analogous? Is mathematical biology the discipline that seeks to create ...
4
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1answer
138 views

Getting representations of the Lie group out of representations of its Lie algebra

This is something that is usually done in QFT and that bothers me a lot because it seems to be done without much caution. In QFT when classifying fields one looks for the irreducible representations ...
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1answer
28 views

Kuramoto model coupled equations query

Short query about the 'Kuramoto Model', which is a mathematical model of synchronized coupled oscillators. If we consider the $N=2$ case then the governing equations are $$\frac{d \theta_1}{dt} = \...
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1answer
37 views

How can we justify $\vec F = \frac{d \vec p}{dt}$ by just vector subtraction? [closed]

I want to justify Newton's second law for the linear momentum of a particle: $$\vec F = \frac{d \vec p}{dt}$$ using really basic linear algebra. Basically, just with vector subtraction. This is ...
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0answers
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What is the matrix of inertia of a thin rectangular plate? [closed]

I only want to get sure how the matrix looks like, Is it : $\begin{bmatrix} \frac{Mb^2}{3} &0 &0 \\ 0&\frac{Ma^2}{3} & 0\\ 0& 0 & \frac{M(a^2+b^2)}{3} \end{bmatrix}$ Or ...
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1answer
23 views

Line integral of multi dimensional Dirac Delta

How do we compute the line integral of a Dirac Delta? e.g. $\int_a^b \delta(x-x(t),y-y(t))dt$ ? Consequently, is $\int_{-\infty}^{\infty}\delta(x-t)\delta(y-t)=\delta(x-y)$ or $\delta((x-y)/\sqrt2)$? ...
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0answers
26 views

What is the operational way of discovering scale invariance of differential equations?

Context The answer here by @Keenan Pepper gives an instance for what it means for an algebraic or trigonometric formula to be scale invariant. For quick reference, I quote his answer here but with a ...
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2answers
65 views

How to turn into conditional statements the equations of physics? Looking for basic examples ( at the high school physics, college physics level).

My question is related to philosophy, but I do not ask for a philosophical answer. I would be interested in a technical answer from a logician's / mathematician's point of view In basic philosophy ...
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3answers
120 views

Robust & Linear Control Systems

The characteristic polynomial of a control system is the following uncertain polynomial: $$s^3 + a_2 s^2 + a_1 s + 3.5 $$ Where $a_1 \in [1.5,4.2],a_2 ∈ [1.2,4.25]$ and $ 4.2 ≤ a_1 + a_2 ≤ 6.3$ . Is ...
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1answer
102 views

Why is symplectic geometry the natural setting for classical mechanics?

I was reading this very nice document, to understand why symplectic geometry is the natural setting for classical mechanics. I more or less understood why there is naturally a 2-form that arises. ...
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0answers
19 views

Contradiction in potential theory

Let $\mathbf{I}$ be a continuous vector valued function in a volume $V'$ having boundary $S'$ Let $\mathbf{r}$ be the distance vector from points in $V'$ to a point $P$ Then the potential at point $...
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1answer
16 views

Differential in Polar coordinates (velocity)

The velocity of circular motion in polar coordinates is like this; $$\vec v(t) = \frac{d}{dt}\vec r(t) = \frac{dR}{dt} \hat u_R (t)\ +\ R \frac{d\hat u_R}{dt} $$ where $R$ is the radius, and $\hat ...