"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)

0
votes
0answers
172 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
2
votes
0answers
52 views

Relation between a linear second order differential equation and Riccati special differential equation

Consider the following differential equation \begin{equation} \frac{d}{dx}\left[N(x)\frac{dw}{dx}\right]+\sigma^2\rho(x)w=f(x,\sigma),~~ 0<x<l, \end{equation} $0<N\in C^1(0,l)$, $0<\rho\in ...
0
votes
0answers
22 views

how do I resolve equations that are both dependant on each other

I'm working on a project concerning the ideal power equation of aerodynamic bodies seen here: $$P = \frac{1}{2}C A D v^3 + \frac{W^2}{Db^2v}$$ where $P$ = power, $C$ = coefficient of drag, $A$ = ...
3
votes
1answer
32 views

Helmholtz decomposition of a vector field on surface

Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and ...
1
vote
1answer
19 views

Uniqueness for Dirichlet problem in exterior domain

I have the following problem: $\Delta u =0$ in $\Omega_e = \mathbb{R}^3 - \overline{\Omega}$, and with condiction $u=0$ on $\partial \Omega$ and $u=o(1)$, that is $\lim_{r \rightarrow 0} u(x) =0$. ...
1
vote
1answer
75 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
1
vote
0answers
16 views

How to determine when the Green's function do not exist?

I've been solving some problem which asks us to find the Green's functions for some problems when it exists. Now, there's a theorem which allows us to guarantee that it exists. The theorem is as ...
-2
votes
0answers
28 views

Math / Physics - Curcular motion [on hold]

What is the maximum speed a car can turn a 50m radius corner if the corner is banked at 15 degrees and has a coefficient of friction of 0.3?
5
votes
1answer
152 views

Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
5
votes
1answer
157 views

A trigonometric identity

If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind, $$\ln(2) + \sum _ {n=1} ^{\infty} ...
2
votes
1answer
149 views

Tough second order differential equation

I can't figure out this diff equation (in cylindrical coordinate). How can I solve it ? Any comments appreciated $$ \frac{1}{r}\frac{d}{dr}(r\frac{dE}{dr})+\frac{d^2E}{dz^2}+(\epsilon_0 ...
0
votes
0answers
19 views

Interpretation of Equations of Motions

I started a lecture on differential equation with following example. If a body is moving in a straight line in plane with constant speed, how can we describe this motion mathematically? To answer ...
2
votes
0answers
57 views

Cubes in cubes in cubes in… ad infinitum.

Suppose I have a cube with one open side (with a volume of let's say $1\ m^3$) for the sake of simplicity; the problem is scale invariant) made from a material that makes the cube just float in water ...
1
vote
1answer
23 views

finding curvature radius

given a projectory equation of the form $ y=y(x) $find the curvature radius as a function of $x.$ a projectory equation , hence $ x=x(t)$, input that in y and we get $y=y(x(t))$, which is what one ...
1
vote
0answers
29 views

Heat problem with an internal source of heat for which the maximum principle doesn't hold.

Heat problem with an internal source of heat for which the maximum principle doesn't hold. The problem is the following and honestly I don't know how to solve it... $$u_{t}=u_{tt}+2(t+1)+x(1-x) , ...
2
votes
0answers
22 views

The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
1
vote
1answer
36 views

Solve question with $E=mc^{2}$ [closed]

In the sun, energy is derived from the fusion of hydrogen into helium via the proton-proton chain. The reaction proceeds as follows: ${}^{1}$H$+{}^{1}$H$\to {}^{2}$H$+e^{+}$ ${}^{2}$H$+{}^{1}$H$\to ...
2
votes
1answer
27 views

Projectile motion: Proving:$ x^2 + 4 \left(y-\frac{v^2}{4g} \right)^2 = \frac{v^2}{4g^2} $

Question: Projectiles are fired with initial speed $v$ and variable launch angle $0< \alpha < \pi$. Choose a coordinate system with the firing position at the origin. For each ...
0
votes
1answer
21 views

Finding the reflection of a plane wave from a sphere

The physical problem I'm trying to solve is this: I would like to find the "reflection" of a harmonic plane sound wave in a liquid, from a spherical air bubble. I'm modeling the problem as follows: ...
2
votes
0answers
78 views

how to solve this integral ? It seems bounded and well defined integral but I don't know how to solve this

how to solve the following integral ? It seems well defined i.e. bound but I could not solve it. I tried by expanding series expansion of tanh[x] but after that I got a series as an answer, which I ...
0
votes
0answers
40 views

need help to solve the following integral [closed]

I need help to solve the following integral. It seems bounded and well defined but I don't know how to solve it. I used series expansion of tanh[x] but then I got answer as series which I could not ...
-1
votes
0answers
8 views

Numerical Solution of Matrix with Diagonal Elements of Highly Varying Order

I am trying to solve following set of equations: A(i,i-2)*u(i-2) + A(i,i-1)*u(i-1) + (A(i,i)+β(i) )*u(i) + A(i,i+1)*u(i+1) + A(i,i+2)*u(i+2)= B(i) + β(i) where i=1:1000000 If values of β ...
0
votes
0answers
20 views

Specifics on the Williamson normal form algorithm

I'm looking at the algorithm for Williamson normal form for symplectic diagonalization of positive-definite symmetric real matrices, given on pp.24 here: ...
-2
votes
0answers
19 views

Can crossing trajectories exist in 2D phase diagrams? [closed]

The other day my research team were talking about dynamical systems and chaos, and we were talking about the impossibility of the existence of crossing trajectories when the system has two variables ...
1
vote
3answers
32 views

Physics problem on derivatives and integrals

So there are a few basic formulas I'd like to start with, $W=\int_0^bFdx$, $F=ma$, and $a=\frac{d^2}{dt^2}x$. In words, Work $(W)$ is defined as the area under a Force versus Displacement $(F/x)$ ...
5
votes
0answers
72 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
1
vote
0answers
28 views

Dynamics of fluid

While reading on Wikipedia about the partial differential equations (https://en.wikipedia.org/wiki/Partial_differential_equation), I wondered how dynamics for the fluid occur in an ...
2
votes
2answers
30 views

Find an function that oscillates between a given upper and lower envelope

Suppose I'm given two real, continuous functions $f(x)$ and $g(x)$ such that $f(x)\ge g(x)$ for all real $x$. I'd like to determine an oscillating function $h(x)$ that has $f(x)$ as its upper-envelope ...
0
votes
0answers
17 views

vertical projection with gravity as acceleration

$\mathbf Question.$ A stone projected vertically upwards with initial speed of $u\ m/s$ rises $70\ m$ in the first $t$ seconds and another $50\ m$ in the next $t$ seconds. $\bullet$Find the value of ...
0
votes
0answers
9 views

What are the factors that will determine the complex impedance $Z_d$? In a simple series MOSFET PWM switcher. [closed]

Say I could model and measure the impedance of an inductive heater driver circuit, $Z_L$. Then what factors determine how $Z_d$ should be chosen knowing: $V_L = 24$ V as well. I'm not sure how to ...
1
vote
1answer
15 views

Find the equilibria

Consider the equation $\ddot s = s-s^3.$ Let $m=1.$ 1) Write this as a first order system. Let $\dot s=v.$ Then we get $\dot v=s-s^3.$ So first order system is $$\begin{pmatrix} \dot{s} \\ \dot{v} ...
0
votes
2answers
382 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 ...
0
votes
0answers
17 views

Find the potential energy and sketch it

I'm given the equation $\ddot s=s-s^3.$ I'm asked to compute the potential energy and sketch it. I'm also given $m=1.$ To do this I have done the following: $$F=ma=\ddot ...
1
vote
1answer
29 views

Book's for potential theory: single and double layer potential

Does anyone know recommend me some book about the theory of the potential, especially that concerning the layer potential. Besides the theoretical part in the higher dimension, if there are concrete ...
1
vote
0answers
8 views

The expectation number of collision to slow down below certain value

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward ...
0
votes
1answer
24 views

Units of $F(x) = x-x^3$

If we are given that $F(x) = x-x^3$ is a force function, which means it is in the units of $[M][L][T]^{-2}$, then how do we determine what kind of "unit units" participate in this function? Namely, ...
10
votes
1answer
107 views

Identity in general relativity, not sure if true or not

Let $(M, g_{ab})$ be a spacetime and define a new metric, $\tilde{g}_{ab}$, on $M$ by $\tilde{g}_{ab} = \Omega^2 g_{ab}$, where $\Omega$ is a smooth, positive function. Let $\nabla_a$ denote the ...
3
votes
1answer
25 views

Using linear algebra to find resonance frequency and normal oscillations and motion

I am stuck part way through the following and not sure how or if finding eigenvalues will help with finding modes of oscillations: Consider the system of three masses and two ideal elastic bands: ...
2
votes
2answers
102 views

Where should the Lorentz transformations fit into this?

I am trying to figure out how to "see" things in relativity via a toy model. With a pinhole camera I'd like to capture a relativistic scene consisting of a vertical marked stick which is moving ...
0
votes
1answer
18 views

another help with partial differential equation.

I am to solve next task. solve this PDE with boundary conditions. $\Delta u = \frac{64}{r^5}\sin\varphi, \quad 1<r<2,$ $u'_r|_{r=1} = 2\cos^2\frac{\varphi}{2}, \quad u'_r|_{r=2} = ...
0
votes
1answer
32 views

Is there a general way to prove this Fourier transform property?

We know that one of the important Fourier transform properties is that, the Fourier transform of a narrow function has a broad spectrum, and vice versa, We can easily see this in this example, the ...
0
votes
0answers
17 views

How to model the following scenario with an ODE if possible

Consider a cylinder, full of charged particles travelling through. From the perspective of looking through the tube, you would see a circle of particles and obviously this circle continues down the ...
3
votes
1answer
73 views

Diffraction and Fresnel Integrals

Migrated from Physics SE due to mathematical content I am trying to derive the intensity variation function for a single slit diffraction. Sorry for the poor diagram... So I decided to take the ...
0
votes
0answers
11 views

Modelling a charged particle flowing travelling through a conductive pipe.

I'm on an internship and have a project to model how a charged particle might be affected by a conductive surface either side of it. Here's how I approached it: I assumed the particle had some charge ...
4
votes
1answer
40 views

“Flow lines” of “dust” are geodesics?

The stress-energy tensor representing "dust" takes the form$$T_{ab} = \rho u_au_b$$where $u^a$ is a unit timelike vector field, i.e., $u^au_a = -1$. Does it necessarily follow that in any solution to ...
0
votes
1answer
16 views

MOI about a diagonal

If by taking a thin rod, and finding its Moment of Inertia about an axis, say through the mid point of its side, one can observe that stretching the rod uniformly along the axis of rotation will give ...
1
vote
3answers
32 views

Moment of Inertia (Square Laminas)

If I have a uniform square lamina of side length 2a and intend to find its Moment Of Inertia about a perpendicular axis to its plane, is there a general formula for this? If there isn't, I have tried ...
0
votes
1answer
36 views

Proving solutions to the anisotropic kepler system that meet certain constraints lie on the position axes of configuration space

The system is: \begin{equation*} x''=\frac{-\mu x}{(\mu x^2 + y^2)^{3/2}} \end{equation*} \begin{equation*} y''=\frac{-y}{(\mu x^2 + y^2)^{3/2}} \end{equation*} With $\mu>1$ a constant ...
1
vote
0answers
25 views

Equivariant Cohomology and Mayer Vietoris sequence [closed]

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
0
votes
1answer
19 views

Hamiltonian mechanics: constant energy hypersurfaces with $dH \neq 0$

I read substantially the following sentence in Frankel's "Geometry of physics": Look now at the level set $$V_{E}=\left\{(p,q)\in T^{*}M:H(p,q)=E\right\}$$ where $T^{*}M$ is the cotangent space, $p$ ...