"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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11
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3answers
675 views

Qualitatively, what is the difference between a matrix and a tensor?

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical ...
5
votes
1answer
75 views

Quantum Mechanics state space

In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm ...
0
votes
0answers
30 views

Does $\{f,g\}$ mean anything when neither $f,g$ are the hamiltonian of a system?

Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...
3
votes
0answers
120 views

Heat equation in 2d Circle polar coordinates

I was presented this problem in PDE class involving heat equation on unit circle in polar coordinates using separation of variables, giving the following heat equation problem: $ u_t = 9\Delta u ...
4
votes
1answer
70 views

An identity involving matrix norm and vector norm in relation to a density matrix in physics

This question is self-contained. For those who are interested, it arises from my study of the paper: K Lendi (1987), Evolution matrix in a coherence vector formulation for quantum Markovian master ...
-1
votes
2answers
79 views

Double integral of symmetric function

Can someone please derive and explain how the LHS is equal to the RHS using the fact that the function $f$ is symmetrical with respect to time variables $t_1$ and $t_2$. Here $t$ is some constant. ...
0
votes
1answer
123 views

Physics related initial value problem (horizontal spring mass system)

Consider the horizontal spring-mass system where the spring-force is the only force acting on the mass. Suppose that a mass is initially at $x=x_0$ with an initial velocity $v_0$. Show that the ...
0
votes
0answers
37 views

50ft cable weighs 1lb/ft lifts 100lb box. Find work done by lifting box 50ft

I've been trying to work on this old exam problem that states: A 50ft cable that weighs 1 lb/ft is used to lift a 100 lb box up 50 ft. How much work is done? The problem lists that the correct ...
1
vote
1answer
66 views

Gauss' law in differential form for a point charge

I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that $ \nabla\cdot E=\frac{1}{\epsilon 0}\rho$, but when ...
2
votes
1answer
29 views

Circular Banked Track Friction [closed]

I've tried answering this question by resolving forces, then finding an expression for friction and inserting the given data so I can prove $F = 0$. However, I never get an answer of $0$. How do I ...
0
votes
1answer
48 views

Can an object be in freefall if it is traveling upward? [closed]

Can an object be in freefall if it is traveling upward? I'm thinking the answer is no?
0
votes
1answer
52 views

solving for initial velocity using the position vector

I am having trouble wrapping my head around this problem. The big picture is that i have to calculate the initial velocity v= needed for a soccer ball to cross a goal line. this is a homework ...
2
votes
2answers
39 views

Are four vectors in Special Relativity considered to be tensors?

In particular, I would like to know if the four velocity and the four acceleration are tensors.
2
votes
4answers
791 views

Where do the maximum and minimum values of velocity occur in the elliptic orbit [closed]

Where do the maximum and minimum values of velocity occur in the elliptic orbit? Why? Find the velocities. I need a detailed answer of this problem. Please help me. I know the answer. Maximum and ...
0
votes
0answers
32 views

What is the form that this commutation take?

[AB + BA,CD] is it [AB,CD] + [BA,CD]? More specifically I'm trying to find [xp + px,p*p] where p is p_x. I have tried [xp,pp]+[px,pp] = x[p,p]p + xp[p,p] + pp[x,p] + p[x,p]p + p[x,p]p + pp[x,p] + ...
0
votes
2answers
29 views

Multiplying vectors w/i, j, k [closed]

Three vectors are given by $a= 3.0i+3.0j–2.0k$, $b= -1.0i-4.0j+2.0k$, and $c= 2.0i+2.0j+1.0k$. Find (a) $a·(b\times c)$, (b) $a·(b+c)$, (c) $a\times(b+c)$. I know that you can multiply vectors to ...
1
vote
2answers
44 views

Recurrence relation with reciprocal in a circuit

The motivation for this recurrence relation is to find the total resistance in this circuit: Assuming that the capacitor has no resistance, with only one loop of the circuit, (let us suppose) the ...
0
votes
1answer
70 views

How to express this equation in terms of v?

I read from my physics textbook that the magnitude of a ripple voltage decreases if the capacitance is increased in a rectifier circuit, but the textbook didn't specify what the exact mathematical ...
0
votes
2answers
108 views

What is the connection between a metric and a manifold?

I am in process of reading a paper which contains something called a "Shahshahani Metric" which has uses in mathematical biology ...
0
votes
0answers
13 views

Convexity of the space of isospectral density matricies

Given the manifold $M$ of all complex, square matricies $\rho$ s.t. $\rho$ positive semidefinite $Tr(\rho)=1$ $\rho^{\dagger} = \rho$ consider the submanifold $N(\rho) = \{U\rho U^{\dagger} \ \ ...
1
vote
0answers
71 views

Placement of protons and neutrons in the nucleus

So, I'm creating a program that would represent a given atom (also different isotopes) in 3d view. I'd need some kind of formula to calculate the position of protons and neutrons to form a nucleus. ...
1
vote
2answers
289 views

Conformal Killing vectors fields on Minkowski spacetime

As is well known Minkowski spacetime (which is four dimensional vector space with scalar product $\eta _{\mu \nu}$ of signature $-+++$) is maximally symmetric, which manifests itself in presence of ...
1
vote
2answers
60 views

Center of mass of the three pennies?

Three pennies each of radius R and mass M attached at their edges. How to find the center of mass of the three pennies?
0
votes
1answer
41 views

Hamiltonian systems. Canonical Transformation.Wikipedia

I have some questions regarding Canonical transformation and hamiltonian systems. I will upload an image with the text from wikipedia: How can I obtain same results... I have no idea how they ...
0
votes
1answer
31 views

Covariant derivative notation?

I was reading up on covariant derivatives and came across this document. On the second page it says: We define a procedure called parallel transport by defining a vector $\vec A (\lambda)$ along ...
10
votes
2answers
431 views

Arnold's theorem on action-angles.

I changed the question slightly in its form to make it more readable. I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this ...
2
votes
0answers
14 views

Covariance of nonlinear sde

My problem is to compute the covariance of the following Ito process $$ dX_t=AX_t+\sum_{k=1}^{n}B_kX_tdW_k, $$ where $A,B_k$ are nonlinear operators defined on a complex separable Hilbert space $H$. ...
1
vote
1answer
28 views

Are there some simple way to remember Braid equation and Yang-Baxter equation?

The Yang-Baxter equation is $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ and the Braided equation is $R_{12} R_{23} R_{12} = R_{23} R_{12} R_{23}$. The indices in the equations are complicated. Are ...
2
votes
1answer
80 views

Central orbit - Find eccentricity of the orbit

A particle moving in an ellipse under the action of a force towards the focus O, moves from greatest distance from O to an extremity of the minor axis in time t, and then to the least distance from O ...
0
votes
1answer
64 views

Radial & Cross-Radial Acceleration: A problem

A particle moves along $r=Ae^{\mu\theta}$ where $\theta=Bt$, prove that its acceleration is proportional to $r$ and makes a constant angle with the radius vector. Approach: $\dot{\theta}=B$ then ...
9
votes
1answer
308 views

What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like Fourier transform - it's actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
1
vote
2answers
55 views

Issue in first order differential equation

I've tried many times to reach the solution of a first order differential equation (of the last equation) but unfortunately I couldn't. Could you please help me to know how did he get this solution. ...
0
votes
0answers
14 views

Boundary conditions for a radiative heat transfer problem

Consider the heat equation $$ \frac{\partial T}{\partial t} - a\Delta T + \mathbf v \cdot \nabla T = S $$ where $S$ is a source term dependent of the radiation intensity $I$ and the temperature $T$. ...
2
votes
1answer
28 views

Evolution operator always in $SU(n)$?

Think about the evolution operator $U$ in Quantum Mechanics for finite dimensional systems. Then this operator satisfies an equation $$U'(t) = -iHU(t)..$$ Here, I assume that $H$ is ...
0
votes
0answers
47 views

How to scale a problem involving heat and the flow of a viscous fluid?

I recently got set this problem and I was wondering if anyone would be able to give me some help on the later parts. An incompressible thermal conducting fluid is contained between two infinite ...
2
votes
0answers
25 views

Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...
1
vote
1answer
46 views

Simple Harmonic Motion under Periodic disturbing force

A particle of mass $m$ is executing a SHM in a straight line under an acceleration $n^2 \times (distance)$. If a periodic force $mk \cos{pt}$ be introduced and the time period of forced vibration ...
1
vote
1answer
65 views

Introductory Reference for Mathematical Physics

I'm a senior undergraduate student studying differential geometry. I have experience with smooth manifolds, some elementary theory of Lie groups, and a little multi-linear algebra. I understand this ...
2
votes
1answer
151 views

Fourier-Bessel series of $f(x)=x^2$

I'm trying to calculate the expansion of $f : [0,1]\to\mathbb{R}$ given by $f(x)=x^2$ in a Fourier-Bessel series of zeroth order. In that case let $J_0$ be the $0$-th order Bessel function and ...
5
votes
1answer
2k views

What are super-translations?

There's been a lot of news lately about a possible solution to the black hole information paradox from a presentation given by Stephen Hawking to the KTH Royal Institute of Technology in Stockholm. ...
0
votes
0answers
42 views

Solving an integral that includes an exponential function and the error function

This question contains all the values needed to compute an equation. My question is, do you get the same result I get? Or do you get the result in the paper I've linked to? I'm trying to decipher ...
1
vote
1answer
71 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
1
vote
0answers
29 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
2
votes
2answers
42 views

Triangle of forces

Forces equal to $5P$, $12P$ and $13P$ acting on a particle are in equilibrium ;find ,by geometric construction and by calculation ,the angles between their directions? I have an problem that, With ...
0
votes
0answers
24 views

What is a conjugate weight?

The authors here write that the longest element of the Weyl group is $$w_{\max} = - id$$ except for $E_6$, $A_r$ and $D_r$ with $r$ even. There they write that $w_{\max}$ acts on a weight $\lambda$ ...
0
votes
0answers
45 views

Does complexification make a self-conjugate representation non-self-conjugate?

I recently learned that a non-self-conjugate representation is not the same as a complex representation. Given a real representation $\pi$, with highest weight $\mu$ $$\pi : \mathfrak{g} \rightarrow ...
1
vote
0answers
53 views

Scaling Two Equations

I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated. An incompressible thermal conducting fluid is contained between two infinite ...
2
votes
1answer
57 views

Einstein summation convention: Del operator and dot product

Now, I am aware of the summation convention for the dot product $$\mathbf{a} \cdot \mathbf{b} = a_i b_i$$ But I am unsure about how to represent $(\nabla \cdot \mathbf{a}) \mathbf{b}$ and ...
1
vote
1answer
78 views

Sketching a Graph of a Particle Trajectory

How can I sketch the trajectory of a particle of mass $m$ with a position vector $\mathbf{r} = \cos(\omega t)\,\hat{\mathbf{i}} + \sin(3\omega t)\,\hat{\mathbf{j}}$ ? Will this be a three ...
0
votes
0answers
31 views

Flow between two infinite horizontal plates

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...