"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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If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold? how can String theory state that there are only $11$ dimensions?

I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant ...
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1answer
13 views

Three cylinders on two inclined planes, each inclined at an angle alpha.

Two smooth uniform right circular cylinders, each of mass $m$ and radius $a$, are placed symmetrically in contact with each other and with 2 planes, each inclined at an angle $\alpha$ to the ...
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22 views

Short pedagogical introduction to Young-tableaux and weight diagrams?

I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them, which contains many detailled and worked out examples of how these methods ...
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1answer
48 views

Insight about compact groups

I'm quite familiar with the general notion of compactness in math but I have some troubles with its extension to group theory. I'm not talking about definitions or theorems: I would like to have some ...
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23 views

What does it mean to take the Laplace transform of a non-periodic position function?

What I'm trying to get through my head here is how taking the Laplace transform of a system with a position function like $X(t)=t$ is possible. To my current (admittedly incomplete) ...
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1answer
45 views

Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
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1answer
81 views

Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
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2answers
42 views

Three cyclists Raman, Mohan and Nitin ride around a circular course

Three cyclists Raman, Mohan and Nitin ride around a circular course 85 km around at the rate of 8, 12 and 20 km an hour. Raman and Mohan ride in the same direction and Nitin in the opposite direction. ...
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1answer
42 views

Affinity of lorentz transformations

Lorentz transformations are often defined to be linear. But suppose instead we only consider transformations that preserve the spacetime interval. Is it possible to prove that those transformations ...
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1answer
51 views
+50

Given a transformation, find the generating function

There's a mapping $(x,y) \mapsto(u,v)$ given by $u= x\cos\theta-y\sin\theta$ $v =x\sin\theta + y\cos\theta$ I'd like to find a generating function $G(x,y)$ for this mapping, which I understand to ...
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1answer
37 views

Intuition behind surface integrals

While line integrals derive their intuition from , and are analogous to, the concept of Work in physics, what intuition is there to back up the notion of surface integrals? In the texts I've been ...
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27 views

Thermodynamics based proofs

What are some mathematical inequalities and theorems that follow using thermodynamics "proofs" (rigorous or just intuitive)? Any suggested books on the matter? For example, AM-GM inequality follows ...
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17 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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2answers
32 views

Physics- projectile motion. Given values: time of flight and horizontal component velocity. Ball launched at an angle returning to the same height.

A person throws a baseball with a horizontal component velocity of 25m/s. It takes 3 seconds to come back to its original height. Calculate its horizontal range, its initial vertical component ...
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41 views

What is elnekiti's triangle? (edited) [closed]

Elementary ceĺular automata shows amazing complex systems such as pascal's triangle is similar to " wolfram rule 90 " , so i looked over youtube searching for extra content and i found this video Here ...
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12 views

Modeling smoke cloud as expanding Gaussian / ellipse

I am making a simplified model of smoke coming from a train's smokestack. You can imagine that if you want an accurate model you have to think in 3D and use computational fluid dynamics and stochastic ...
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1answer
28 views

The literature on Chern-Simons theory

Can any one give some literature on Chern-Simsons theory? I can not find any book introducing this theory. Thanks.
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1answer
41 views

What is $\mathrm{dim}(\mathrm{Sym}(\mathrm{Herm}(H)^{\otimes N})$?

The totally symmetric subspace of $(H^k)^{\otimes N}$, with $H^K$ a $k$-dimensional Hilbert space, has dimension $\binom{N+k-1}{k-1}$. But I now want to know the dimension of the totally symmetric ...
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1answer
28 views

Couple stress tensor reference.

Can someone give me a good mathematical reference for couple stress tensor in its most basic form. Thank you.
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14 views

Heat equation with $x\in [0,+\infty[$ and non-homogeneous initial and boundary condition

The IVBP that i need to solve is the follow: \begin{equation} \begin{cases} u_t=au_{xx} & x>0,t>0,a\in\mathbb{R}^+\\ u(x,0)=B_0e^{-kx}\cos(kx) & x\geq ...
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1answer
19 views

How to calculate $e^{ad\hat{A}}\hat{B}$?

For $\hat{a}$ and $\hat{a}^{\dagger}$ is annihilation and creation operators which, $[\hat{a},\hat{a}^{\dagger}]=1$. Could you please show me the way to calculate, ...
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107 views

What comes under Mathematical Physics. [closed]

Suppose Math has 5 sub parts: Analysis: Analysis, Complex Analysis, Measure theory and integration,Functional Analysis Algebra: Group Theory, Vectors space-rings-modules, Galois, Algebra Number ...
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1answer
24 views

Hamiltonian vector field and symplectic geometry

I want to show the following theorem: For any Hamilton function $H : M \rightarrow \mathbb{R}$ on some symplectic manifold $M$ and symplectomorphism $f : M \rightarrow M$ we have $X_{H \circ f} = ...
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1answer
34 views

Defining a partial derivative with respect to an antisymmetric tensor/matrix

I'm looking at some nonlinear electrodynamics, and have been following a textbook which contains a primer on some of the stuff I'm interested in following up. However, I seem to have fallen at the ...
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1answer
30 views

Impact of two bodies problem

A body of mass $M$ moving with a velocity $u$ collides with another of mass $m$ which rests on a table. Both the balls are perfectly elastic and smooth and the the body of $m$ is driven in a ...
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33 views

Stability of ground state under positive (not relatively bounded) perturbations

This is about positive perturbations that are not necessarily relatively bounded, but where the perturbed operator is known (by some independent proof) to be self-adjoint. Is this a known result (or ...
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2answers
105 views

What is the most general notion of “Fourier transform?”

I know the definition of a classical Fourier transform that maps a function f(x) on the real line X to a function F(p) on a dual space (here another real line and borrowing some physics notation) P. ...
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3answers
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What's the name of this theorem?

It happens very often in physics that we find relations like: $$\int_V f(x) dx = \int_V g(x) dx$$ for an arbitrary volume $V$. From this we usually say "Since the volume is arbitrary, the integrands ...
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39 views

Vector Calculus - Polar Co-ords

I am having a lot of difficulty finding an approach to solving the following question: A dyon is a particle with both electric and magnetic charge; in suitable units $$\mathbf{E} = ...
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27 views

Conformal field theories and critical points

I apologize in advance if this question belongs to physics.stackexchange. I've been trying to learn CFT following Zee for QFT background (approximately first and second chapter,) and then Di ...
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1answer
21 views

What set of straight ramps exist such that a ball sliding down any one of them would reach the base at the same time?

I'm looking for a set of straight ramps, which, under idealised conditions (uniform gravitational acceleration and no friction) would have a point-like body slide down them from the top to the base ...
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1answer
50 views

Is there an analytic or at least a numerical solution to an eqaution of the form $\sqrt{k_1\sqrt{x}+k_2}\;\Big(k_3x+k_4\sqrt{x}+k_5\Big)+k_6=0$?

So the problem comes from cosmology and I want to solve for the unknown function $a(t)$, which is the scale factor for the universe. So I have an integral involving $a$: $$ ...
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1answer
26 views

Deflection - irradiation direction.

Compute the beam deflection $\delta$ (see figure) through a prism with angle $\epsilon$ as in the figure and refractive index $n_p$. When is the deflection $\delta$ minimal compared to the original ...
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1answer
13 views

Glass prism - refractive index

Light falls perpendicular on one side of a glass prism with refractive index n. The light is totally reflected on the right side. 1.1: Determine the angle of incidence using the figure ...
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1answer
57 views

Stuck on computing distance travelled from velocity and yaw rate.

I am somewhat stymied on what appears to be a simple formula. Here is the problem statement: Assume that a rigid body is traveling with constant velocity $v$, and is rotating with a constant yaw rate ...
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7answers
775 views

Why does the “separation of variables” method for DEs work? [duplicate]

Heyho, I am using the separation-of-variables method for quite a while now, but what was always bothering me a bit, is why is it possible to do those operations. I'll give a concrete example (source ...
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2answers
105 views

Is the following PDE boundary value problem well-posed?

My Question Is the following Poisson boundary value problem well-posed, as stated? If so, how could I go about solving it? If not, what would it need to be well-posed? Does it satisfy the ...
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1answer
20 views

Inclined plane - euler-lagrange

A mass point of mass m moves frictionlessly down an inclide slope under influence of gravity. Solve the equations of motion and determine the constraint with the use of the lagrange equation of ...
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42 views

Euler-Lagrange - circle cone

A mass point moves on the wall of a hollow circle cone under influence of the homogeneous gravitational field of earth. Use spherical coordinates to solve this problem. a)Set up the ...
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0answers
21 views

Possibly use a Fourier Transform to perform deconvolution

I need to determine the function g(B) in order to determine its prefactors $a_n$. Here's what I have: h(f,B) is a Gaussian function g(B) is the unknown function i(f) is an inhomogeneous function and a ...
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1answer
59 views

Equation of motion - curve - particle

A particle of mass m moves frictionlessly under the influence of gravity on a curve defined by: $x=a(\phi+\sin\phi)$ and $y=a(1-\cos\phi)$. a) Set up the terms for the kinetic and ...
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0answers
27 views

trace calculation of an operator valued matrix

Heyho, i've got problems understanding a certain calculation of the trace of an operator valued matrix right now. We've got the Matrix $T(\lambda)= \begin{pmatrix} A(\lambda) && B(\lambda) ...
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37 views

Determining the group generated by a set of roots?

I have a set of 45 roots and I want to know which group is generated by the corresponding generators. In the set are 5 diagonal (=Cartan) generators $$ (0, 0, 0, 0, 0, 0)_1,(0, 0, 0, 0, 0, 0)_2,(0, ...
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1answer
88 views

$A^tA-AA^t$ in Mathematical Physics

In very different contexts of mathematical physics (rigid body mechanics, fluidodynamics, general relativity, quantum field theory,...) I have come across the following expression: $$ A^tA-AA^t, $$ ...
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1answer
19 views

Complex refractive index.

A linearly polarized wave with the vacuum wavelength $\lambda_0$ falls perpendicular on a material with the complex refractive index $\bar{n}=1.5+i\cdot 0.15$ with that wavelength. After what ...
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3answers
63 views

Figure out the component of a value in X and Y coordinates using trigonometry.

Alright. It's been long that I studied trigonometry and did Laws of Motion and Free Body Diagrams, and I was decent good at them, but somehow I am having trouble in understanding the following. Note ...
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Refractive index - medium

To determine the refractive index of a medium two identical em-waves are generated. One wave goes through unknown medium whereas the other one stays in vacuum. Determine the refractive index ...
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50 views

A problem on Constrained Motion

Q. A particle is moving in a smooth curve under gravity and its velocity varies as the actual distance from the highest point. Prove that the curve is a cycloid. Attempt: The eq. of motion is ...
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1answer
29 views

solving equation in terms of $w_1$ and $w_2$

I have a a physics problem involves the following equation $$\tan(\alpha) = \frac{(w_1 + w_2)^{1/2}}{w_3}$$ from a certain set of equations that I use I derive the following equation: ...
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2answers
61 views

Practical use for negative $dt.$

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would ...