"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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In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
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0answers
38 views

Equivalent descriptions of “flat space with non Euclidean metric” and “curved space with (local) Euclidean metric”: the case of Minkowski space.

FIRST: I start with the guiding idea: 1. we have the surface of a paraboloid (z = x2 + y2); its metric, in an infinitesimal neighbourhood of one of its points is (we can choose it) EUCLIDEAN; now, ...
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1answer
38 views

When do isometries commute with the compatible derivative operator on a semi-Riemannian manifold?

Let $M$ and $\tilde{M}$ be smooth manifolds, each with a metric $g_{ab}$ and $\tilde{g}_{ab}$, assumed here to be smooth symmetric invertible tensor fields, which are non-degenerate but not ...
3
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1answer
40 views

Math formulas on Clustering

I am currently studying Clustering in Machine Learning. I have found a document regarding guessing the right number of clusters. I am reading the first part of it, having difficulties in understanding ...
0
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1answer
34 views

Rope question - integration

A 50-lb bucket is at the bottom of a 100-ft well. A 200 lb rope (also 100 ft long) is tied securely to the bucket. We will use rope to lift this bucket out of the wall, at a rate of 1 foot every ...
2
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1answer
64 views

Using metric to raise and lower indices

Everything I read on tensors makes it clear that using the metric matrix $g_{ab}$ and its inverse $g^{ab}$ to respectively lower and raise indices of a tensor is very important. As far as I know (and ...
3
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1answer
54 views

Is kinetic energy a positive definite quadratic form?

Recall (Arnold, Mathematical methods of classical mechanis, 4.19, B) Definition. Let $M$ be a riemannian manifold. The quadratic form on each tangent space $$ T = \frac{1}{2} \langle v, v ...
10
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2answers
180 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
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0answers
20 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
3
votes
2answers
102 views

Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
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0answers
25 views

Conformal group in two dimensions

In Conformal field theory, physicist says, the conformal group in two dimensions is infinite dimensional, so the associated with the infinity of generators and infinity conserved charges provided. Is ...
2
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1answer
47 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
3
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2answers
129 views

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
4
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2answers
94 views

Identifying the algebra

In order to solve an obscure (physics) problem I have been considering whose details are not important, I am looking for elements (I am thinking in terms of matrices and their products but this may ...
6
votes
1answer
77 views

How did Le Verrier calculate Neptune's position?

In the Wikipdia article on Neptune the discovery is described as a mathematical achievement: Subsequent observations revealed substantial deviations from the tables, leading Bouvard to ...
2
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0answers
31 views

Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
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0answers
29 views

Properties of functional integration

this question comes from theoretical Physics, the issue being the so called Path Integral. The measure of this thing is something written as $[d\phi]=\prod_x d\phi(x)$ And this should be the limit ...
3
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1answer
42 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
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0answers
34 views

Trigonometry, find distance of arc movement

Imagine I have the setup as follows: I want to compute the height x in State 2, depending on how much the blue axis have moved. That is, the distance ...
1
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1answer
47 views

The definition of scalar and vector concomitant of a metric

I'm reading Defrise-Carter's paper Conformal Groups and Conformally Equivalent Isometry Groups. One might find the paper at the following link: ...
3
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1answer
78 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
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0answers
21 views

Mathieu equation solution with non-periodic boundary conditions

I need to solve the Mathieu equation: $y''(x)+(a-2q \cos(2x)) y(x) = 0$ but with the unusal boundary condition: $y(x+\pi) = e^{i \alpha}y(x) \quad , \quad \alpha \in R$ if $\alpha = 0$ than the ...
0
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0answers
39 views

derivation of an equation involving the Fourier transform of the square modulus of a wave function

A textbook on electron optics states that, ignoring a factor of 2 for convenience, the result $\mathscr{F}(I(\vec{r}))=\mathscr{F}(\phi(\vec{r}))\cdot{}A(\vec{k})\cdot{}\sin[\gamma(\vec{k})]$ can be ...
0
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1answer
23 views

Stopping point of a sliding particle.

A particle with given mass > 0 and given coefficient of friction > 0 and given initial downward speed > 0 starts at (0,1) on the graph of y = exp(-x). The coefficient of friction applies only to those ...
0
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2answers
49 views

Complete ONS and pure point spectrum

In all that follows all operators are taken to be densely defined on a Hilbert space $H$. Some textbooks state that an operator $A$ on $H$ has pure point spectrum if $H$ admits a complete ONS (Hilbert ...
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3answers
62 views

integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
0
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0answers
54 views

How do you solve this differential equation? $\tfrac{dx}{dz} = i (M x)$

How do you solve this differential equation : $\tfrac{dx}{ dz} = i (M x)$ where $M$ is a tridiagonal matrix with elements $100$. That is, $M$ is an array with $100$ elements in triagonal form, ...
6
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1answer
181 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
1
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1answer
36 views

Function spaces for the 1-dim heat equation.

Consider the standard 1-dim heat equation: $u_t(x,t)-\alpha u_{xx}(x,t)=0$, where $u:\mathbb{R}\times\mathbb{R_+}\rightarrow \mathbb{R}$, with initial conditions $u(x,0)=g(x), x\in\mathbb{R}$ and ...
1
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1answer
63 views

Formulation VS Interpretation

I'm reading a book on Mathematical Physics and at some point the author says that we must distinguish between a "formulation" and an "interpretation" of a theory, although it's not easy to point what ...
0
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0answers
61 views

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
2
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0answers
85 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
3
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0answers
39 views

Functional Extremum

Let a functional $H[\phi]$ of a map $\phi\in\mathbb{R}^{\mathbb{R}^4}$ be given by: $$ H(x^0) = \int_{\mathbb{R}^3} ...
2
votes
2answers
50 views

Initial value of Newton Raphson Method

I am currently studying Newton-Raphson Method. I feel that I understand the concept of it. Somehow, I am facing some question in my head about how to actually apply it. The questions that I have are ...
3
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0answers
50 views

Why is entropy = the Legendre transform?

Can someone give me a mathematician's explanation (and not a physicist's) as to why $$\int_{\Omega}\Psi^*(b(u(t))$$ is called the entropy where $\Psi^*$ is the Legengre transform of ...
1
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0answers
50 views

Integral calculation - Gravity - Free Fall

I have read this article http://physics.stackexchange.com/questions/3534/dont-heavier-objects-actually-fall-faster-because-they-exert-their-own-gravity. In the best answer by David Z - there are some ...
0
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0answers
32 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
1
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1answer
47 views

An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
0
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0answers
36 views

Proof of Arnold-Liouville's Thm: movement in angular coordinates conditionally periodic

I'm reading Arnold's book Mathematical Methods in Classical Mechanics and got stuck on the proof of Liouville's theorem on integrable systems. The proof finishes with Problem 11: Show that the motion ...
0
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0answers
26 views

Homomorphism between $SL(2,\mathbb{C})$ and the Lorentz group and choice of metric

Just a quick question, I'm hoping someone can clarify how this probably small issue can be resolved. It is said that a Lorentz transformation $\Lambda$ is a linear tranformation of $\mathbb{R}^3_1$ ...
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0answers
66 views

Resources for learning Relativity

I´m looking for books to the study of Relativity. I know that this is math stack schange and not physics stack schage, but I believe that some of the users here are interesed in physical-mathematical ...
10
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3answers
225 views

Areas of contemporary Mathematical Physics

I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc have had a significant impact on pure Mathematics especially geometry ...
1
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1answer
71 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
2
votes
2answers
56 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
0
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0answers
49 views

Where does this unit vector come from?

Although this is a physics problem, the problem is in regard to vectors. There is the relevant reference to this image here: What I need to find is the unit vector of force $F_1$. The book gives ...
0
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1answer
46 views

Centre of Mass and Moment of Inertia of a sphere - spherical cap

I have been given a sphere of radius a, from this sphere a cap of hight h is cut off. 1) What is the centre of mass of the rest of the sphere? 2) What is the moment of inertia regarding the axis of ...
3
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1answer
34 views

Physical reflections of prime-number distribution

Not a purely mathematical question: I have read somewhere that Atomic Orbital is closely related to the distribution of prime numbers, but I am unable to find any reference to that. Can someone ...
2
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0answers
52 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
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2answers
76 views

is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
2
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0answers
110 views

If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? [closed]

If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? For instance, would the dynamics of a sea of virtual particles of cardinality $\aleph_1$ would ...