"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 ...

learn more… | top users | synonyms (1)

1
vote
1answer
52 views

Minkowski space is locally Euclidean?

The Minkowski spacetime $\mathbb{R}^{1,3}$ is said to be a manifold (isomorphic to $SO^{1,3}$. But according to the definition of a manifold it should be locally euclidean. However, this seems to be ...
0
votes
0answers
16 views

Description of distributions with support in a linear subspace

The following lemma is true: any distribution $\lambda$ on the real line with support included in $\{0\}$ can be written as $$ \lambda = \sum_{i = 0}^N a_i \partial^i(\delta_0)$$ with the $a_i$ being ...
1
vote
1answer
66 views

Longitude and latitude problem

I find this question challenging. I am trying to solve this question for my younger brother. So here it goes: An airplane leaves an airport $X$, 20.6$^0E$ and 36.8$^0N$, and flies due south along the ...
1
vote
0answers
49 views

Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?

By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots ...
3
votes
1answer
51 views

Complex charge in RLC circuit

Calculate the absolute value of the complex charge in the RLC circuit: $$Q(t)=\frac{V_0e^{i\omega t}}{-\omega^2L+i\omega R+\frac{1}{c}}.$$ Find the frequency where $|Q(t)|$ is maximum. This ...
0
votes
2answers
165 views

How to find a vector normal to a cylinder in cylindric coordinates?

I'm trying to solve a problem which demands to multiply a vector M and vector normal to a cylinder's surface in cylindric coordinates. Height of the cylinder is infinite and its radius is R. So how do ...
0
votes
2answers
25 views

Finding vertical displacement

I am being asked to find the distance a shuttle travels upward after a given amount of time. I know that time passed is 79s, the rate of acceleration is 6.244 m/s^2, and the speed at 79s is 493.276 ...
3
votes
1answer
61 views

How to mathematically determine if the magnitude of a cross product is up/down(positive/negative?)?

So, I'm a newbie at complex vector math. I'm working on a 2D physics engine, and my issue is, with angular acceleration from torque, is it supposed to be positive or negative? I understand the right ...
0
votes
2answers
56 views

Vectors, How to measure total force and direction.

I am currently looking for some math help that I am quite struggling with. The problem is: (Vectors) A fisherman use his pole and line to pull a fish out of the water. The line exerts a force on ...
1
vote
1answer
83 views

Is 1/x the “slowest” asymptotically falling off differentiable function?

As a physicist, I tend to think about $\sim 1/x$ as the "slowest" fall-off of a "reasonable" function. Let us state this formally: $${\rm lim}_{x \to \infty} f(x) = 0, f(x) \in Reas \implies \exists A ...
1
vote
2answers
86 views

Water Refraction and the depth of the water.

I'm not sure if this is the right place to ask my question! But I hope I will find some help!. Image distortion occurs by refraction of light at the boundary surface between air and water when a ...
2
votes
1answer
56 views

Why does a heating model work?

I am referring to: $T=T_0 e^{kt}$ where T=temperature,t=time and k=constant. It seems to work, I as just curios to why it works?
1
vote
1answer
76 views

Losing all races by the same margin of time

Suppose two cars are racing along a (straight) road at a constant speed $v_{0}$ m/s. At time $t = 0$, Car 2 is ahead of Car 1 by $d_{0}$ meters; or, one could say, Car 1 is losing by $d_{0} / v_{0}$ ...
1
vote
0answers
38 views

Formally evaluating integral to calculate electric or gravitational field.

I never understood how such integrals are calculated, formally. In a line is easy, just a line integral. In a surface, sometimes is easy, like in a disc. But, some surfaces, like sphere, it gets ...
2
votes
0answers
38 views

Resolvent and spectrum of a self-adjoint extension

In this paper, they give the resolvent, spectrum, and eigenfunctions of the self-adjoint extension of the Laplacian on a rectangle that corresponds to a delta potential at an arbitrary point (items ...
2
votes
2answers
43 views

Calculate initial speed to launch the cat at specific spot

BACKGROUND: I’m trying to create a game where cat jumps from platform to platform, but as any other cat this furry devil won’t do the things I’m asking for. I want the cat to jump and land at the ...
0
votes
2answers
68 views

Solving a differential equation numerically to plot particle path

I'm trying to plot the evolution of a particle in an accretion disk by solving the equation $$2X\frac{\partial X}{\partial\tau}=V_R(X,\tau)$$ where I have found $V_R$ numerically to be ...
1
vote
1answer
56 views

Angular momentum of an accretion disk

I need to plot the time evolution of the total angular momentum in an accretion disk. This confuses me because I thought this should be constant, since angular momentum has to be conserved? I'm given ...
6
votes
0answers
152 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
0
votes
0answers
27 views

Examples of quasilinear wave equations

Consider a quasilinear wave equation equation of the form $\sum g^{ij}(u, Du)\partial_i\partial_j u = F(u, Du)$ on $R \times R^n$ subject to initial data $u(0,x)=g, \; \partial_t u(0,x)=h.$ Given ...
5
votes
0answers
148 views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
0
votes
0answers
24 views

Conformal transformation

The problem is following. This is an Exerciese of Polchinski $2.6$ (explanation about conformal field transformation) Consider the flat Euclidean metric $\delta_{ab}$ in $d$ dimensions. An ...
1
vote
0answers
42 views

Lecture notes on holomorphic Yang-Mills theory

Some time ago I've found these lecture notes on the gauge theory. In particular, in these lecture notes the author introduces and studies the Yang-Mills equations in the case of real bundles and ...
1
vote
1answer
65 views

Simple Harmonic Motion DE

The simple harmonic motion DE is $x''(t)=-x(t)$ I solved it as a homogenous linear equation and after inserting the Euler's formula into the equation, my solution becomes ...
0
votes
1answer
112 views

Trigonometry in projectile motion

I initially posted this question on Physics SE but got no responses probably because it's more related to maths than physics. A plane surface makes an angle $\bf X$ with the horizontal. From the ...
3
votes
0answers
65 views

My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why? [closed]

I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it ...
0
votes
2answers
148 views

Finding the equation for a (inverted) cycloid given two points

If I have two points on a Cartesian plane, and I know that they are connected by a cycloid, then how do I find the equation for that cycloid? For background information, I have been playing around ...
1
vote
0answers
30 views

Gauge covariant derivative on principal bundle over $\mathbb R^d$

I try to understand the physical definition of covariant derivative in gauge theories in terms of the exterior covariant derivative of vector-valued forms defined as the horizontal projection wrt a ...
2
votes
2answers
141 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
7
votes
1answer
117 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
2
votes
2answers
60 views

Decomposition of a positive semidefinite self-adjoint operator?

If I have a positive semi-definite self-adjoint operator $H:D(H) \rightarrow L^2$, is it true that there is always a decomposition $H=A^* A$ available? If this is true, what can we say about the ...
0
votes
0answers
37 views

Subject to in equation

I have the formula below: $$\hat x_2=\arg\min\lVert x\rVert_2\quad\text{subject to}\quad A{x}=y.$$ But I didn't understand what was meant by "subject to" ? does $x$ is replaced by $x_2$? please can ...
2
votes
0answers
27 views

Compatibility between the Lax, Hamilton and St. Petersburg formalisms in terms of the conserved quantities.

In classical integrable models, in the discrete case we have the Lax equation discrete $\frac{dL_{n}}{dt} = M_{n+1}L_{n}-L_{n}M_{n}$ the Sklyanin algebra, $\lbrace T_{a}(u),T_{b}(v)\rbrace = ...
13
votes
5answers
837 views

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 ...
0
votes
0answers
55 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, ...
1
vote
1answer
47 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
votes
1answer
47 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
votes
1answer
58 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
votes
1answer
73 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
votes
1answer
88 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
votes
2answers
190 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 ...
1
vote
0answers
31 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
128 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 ...
1
vote
0answers
30 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
votes
1answer
57 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
votes
2answers
131 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
votes
2answers
99 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
102 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
votes
0answers
35 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 ...
0
votes
0answers
31 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 ...