"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
0answers
15 views

Identity in continuum mechanics

For a problem in the textbook I am reading, I need to prove that $\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS$, where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a ...
3
votes
3answers
70 views

Falling objects - finding the speed [closed]

I am trying to work out how fast water will be falling by the time the water hits the ground. If it starts 100m high how fast would it be travelling and why? With the acceleration because of gravity ...
0
votes
0answers
176 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 ...
3
votes
1answer
92 views

Introductory book on probability for physicists

I'm a physics student looking to start learning more about probability. Is there some introductory book on measure theoretical probability theory that includes sections on quantum probability? To ...
4
votes
3answers
43 views

Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
0
votes
1answer
68 views

Motion of a pendulum with air resistance

I am trying to model the motion of a pendulum with air resistance. I have resolved perpendicular to the direction of motion to get this equation where $m$, $g$, $p$, $C_D$ and $A$ are constants: ...
1
vote
0answers
41 views

How can I solve this partial differential equation?

I'm modeling the dynamic localization, after solving the Helmholtz equation I obtained this partial differential equation, if anybody can give me a guideline I would be truly grateful. $$ ...
7
votes
1answer
208 views

In what sense does analyticity guarantee the following equality?

I was reading a paper$^1$ on particle physics, and at some point it is stated that, provided $f(x)$ is analitic, we have $$ f(x)-f(0)=\frac{x}{\pi}\int_0^\infty ...
0
votes
0answers
35 views

Wick renormalization of stochastic integral

I am trying to understand a paper that summarizes some results concerning Wick renormalization of some stochastic integral. In the last few lines of the paper the authors say: In Euclidean ...
3
votes
2answers
38 views

What is the speed of the car given the time taken to receive an echo?

I am trying to solve this question- The driver of an engine produced a whistle sound from a distance $800m$ away a hill to which the engine was approaching.The driver heard the echo after ...
1
vote
1answer
29 views

The inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$.

I am reading the book. On page 80, there is a concept the inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$. Here $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra ...
0
votes
0answers
18 views

Notation in Belavin-Drinfeld's classification of solutions to classical Yang-Baxter equations.

I am reading the paper, on page 6, equation (3.5), there is a notation $(1 \otimes \alpha)r_0$, where $r_0 \in g \otimes g$, $g$ is a semisimple Lie algebra, $\alpha$ is a root. For example, suppose ...
2
votes
1answer
24 views

Magnitude of velocity and acceleration around a track?

A car travels around a circular track with a radius of $r=250m$. When it is at point $A$ then $V_a=5m/s$ which increases at a rate of $\dot{v}=(0.06t)m/s$. Determine the magnitude of its velocity and ...
0
votes
0answers
13 views

What is the relation between solutions of classical Yang-Baxter equations and solutions of modified Yang-Baxter equations.

Let $g$ be a Lie algebra. The classical Yang-Baxter equation (CYBE) is: $$ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. $$ The modified classical Yang-Baxter equation (MCYBE) is: $$ ...
0
votes
0answers
18 views

Associated Laguerre Polynomials negative indices?

For the associated Laguerre polynomials/functions, it is taken (specifically when solving for the eigenstates of Hydrogen in QM) that the associated Laguerre functions with negative indices (and also ...
0
votes
1answer
27 views

Double integral multivariable calculus

Consider the following integral $$\int_0^1 dx_1 \int_0^{1-x_1} dx_2 \, (1-x_1-x_2)^{-\epsilon-1} (-sx_2 - x_1p_1^2)^{-\epsilon-1}$$ where $s$ and $p_1^2$ are to be treated as constants throughout the ...
1
vote
1answer
38 views

Interpretation of a reaction diffusion equation

I have a reaction-diffusion equation in 1-dimensions of the typical form: $$\frac{\partial }{\partial t} u(x,t)= \frac{\partial^2 }{\partial x^2} u(x,t)+ \alpha(x) u(x,t), \,\qquad (x,t)\in ...
0
votes
1answer
44 views

Limit of probability density function as random variable approaches +/- infinity

Consider a complex-valued function $\Psi(x,t)$ such that $|\Psi|^2$ is a probability density function for $x$ (for any time $t$). In his introductory Quantum Mechanics book, David J Griffiths writes ...
0
votes
0answers
19 views

The sum of two r-matrices.

Let $g$ be a Lie algebra. Suppose that $r \in g \wedge g$ satisfy the condition: $[[r, r]] = [r_{12}, r_{13}]+[r_{12}, r_{23}] + [r_{13}, r_{23}]$ is a non-zero unique, up to scalar multiple, ...
2
votes
1answer
96 views

Origin of delta

Why does delta mean change? What is the origin of delta? I understand that upper-cased delta is used in this way and that delta is the fourth letter of the Greek alphabet. I also read that delta is ...
3
votes
3answers
69 views

Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
1
vote
0answers
43 views

Why is $J_n$ not symmetric, for $n\notin\mathbb Z$, while Bessel's equation is still symmetric?

Bessel's equation, $$x^2y''+xy'+(x^2-n^2)y=0,$$ has even parity, regardless of the value of $n$. So a solution of this equation must be even or odd. However, the Bessel functions $J_n$, which are ...
1
vote
1answer
15 views

Double random number from a gaussian, how to evaluate the skewness

I have a question for an application in physics. So my description will be really concrete, sorry. It's about the estimation of a systematic error from a calibration system. I have a LED with an ...
0
votes
0answers
35 views

Damping in a fluid

I am trying to understand damping in a fluid.Take, for instance, water flowing down a surface. I know a damping term $-\alpha\mathbf{u}$, where $\mathbf{u}$ is the velocity field, is added to the sum ...
1
vote
1answer
27 views

How does the evaluation of $\dot q_i$ at $q_1+\mathrm dq_1$ yield $\dot q_i +\dfrac{\partial \dot q_i}{\partial q_1}\mathrm dq_1 \;?$

I've been following Reif's Fundamentals of Statistical and Thermal Physics; there I came before the derivation of Liouville's theorem: There I couldn't understood few things. I could conceive the ...
7
votes
0answers
155 views

A conjecture on Schatten 1-norm

I have a conjecture on Schatten 1-norm. Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite ...
1
vote
1answer
47 views

I've found two different definitions of a cylindrical Brownian motion and don't understand why they are consistent

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space ...
1
vote
0answers
18 views

Infinite total variation of complex measure in Feynman path integral

I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...
0
votes
1answer
38 views

Right Propagating Wave complex

Does it make sense to think of $e^{ikx}\equiv $cos$(kx)+i$sin$(kx)$ as a right propagating wave? I am rather confused by the imaginary term here. Context: \begin{equation} \phi(x)=\begin{cases} ...
0
votes
0answers
37 views

Average number of steps to return to the origin of a random walk on a 2-d lattice.

Suppose I have a random walker on a 2-d square lattice with periodic boundary conditions with equal probability of going in any of the four directions. The walk ends when the walker reaches the point ...
0
votes
0answers
25 views

Definite integrals from Feynman-Hibbs A.4, A.5 (complex exponential with reciprocal time)

I am reading the revised edition of Feynman & Hibbs "Quantum Mechanics and Path Integrals". At one point (scattering of electron by atomic potential) we need to use a definite integral from the ...
0
votes
1answer
24 views

Set of coupled partial differential equations

I've read that the Einstein equation is a set of 10 coupled partial differential equations. I know what a partial differential equation is, but I don't know what a set of coupled partial differential ...
1
vote
1answer
39 views

How to construct a complete set in $L^2(\mathbb{R}^3)$ starting with the Spherical Harmonics?

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty ...
0
votes
0answers
47 views

Writing 5-dimensional dynamical system as Hamiltonian system

I've got a 5-dimensional continuous dynamical system, i.e., $$ \dot{x}(t)=f(x,y,z,u,w)\\ \dot{y}(t)=g(x,y,z,u,w)\\ \dot{z}(t)=h(x,y,z,u,w)\\ \dot{u}(t)=q(x,y,z,u,w)\\ \dot{w}(t)=p(x,y,z,u,w) $$ Is ...
0
votes
2answers
38 views

Multiplying two Scalar dot products together

stackexchange community, I'm just wondering what the rules are for multiplying dot products together, such as: $$ (P_{3}\cdot{P_{4}})(P_{1}\cdot{P_{2}}) $$ How would this be expanded out to not ...
0
votes
0answers
33 views

virtual work and potential energy

I was just going through the thermal and elastic buckling of bars & plates ,I found some researchers use virtual work to derive the equations, another researchers use potential energy in other ...
1
vote
1answer
25 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
1
vote
1answer
54 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
0
votes
0answers
41 views

Having trouble interpreting the geometry of this setup.

A circular conductor, with cross section given by $(x-d)^2+y^2=b^2$, i.e. radius $b$ and centered on $x=d$, has a circular core, made up of the interior of the circle $x^2+y^2=a^2$, with ...
3
votes
4answers
42 views

Finding the formula for acceleration from $v=2s^3+5s$, where $s$ is the displacement at time $t$

This is the question: I first found $\frac{dv}{ds}=6s^2+5$, then I tried to find $\frac{ds}{dt}$ by messing about a little with implicit differentiation, but I had no luck and I therefore couldn't ...
1
vote
0answers
46 views

Acceleration of an air bubble under the sea

An air bubble arises from the bottom of the sea. Find its acceleration if the resistance force is proportional to $\rho$*A*$v$ where $\rho$ is density of water, A is cross section area and $v$ is ...
0
votes
1answer
35 views

First Order Differential Equation for a Harmonic Oscillator

A box with mass $m$ is attached to a spring with spring coefficient $k$. This system is then placed into a glass case filled with a liquid with drag coefficient $\alpha$. Now I have the following ...
0
votes
1answer
35 views

More equations than unknowns for maxwell equations?

I had one curiosity regarding maxwell equations in 3-D From the curl equations, you get 6 unknowns, with 6 equations. The divergence equations add 2 additional equations. When these are combined, we ...
2
votes
1answer
63 views

How do I determine if the equation is a conservation law?

We have the PDE $\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a ...
4
votes
1answer
70 views

Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a ...
2
votes
1answer
65 views

Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some ...
1
vote
3answers
27 views

Where does the extra $\omega$ come in velocity of Simple Harmonic Motion?

Position $x$ in a SHM is given by $x=A\space sin(\omega t+\phi)$. Where $A$,$\omega$ and $\phi$ are Amplitude,Angular frequency and phase constant and are three constants respectively. So,velocity ...
0
votes
1answer
31 views

Frobenius method to solve differential equations, different \alpha found

I am referring to Carl Bender's Advanced mathematics methods for scientists and Engineers. Well, actually I know how to solve it....However, if I choose to do a so called "powerful" method,which is ...
1
vote
1answer
14 views

Odd Vector Product Question

Here is a question that has me stumped: Use the geometric definition to find: $2 {\bf i} × ({\bf i}+{\bf j})$ Student solution manual says: By the definition of cross product, $2 {\bf i} × ({\bf ...
0
votes
0answers
35 views

How can we prove that the derivative of a generalized Hilbert space valued Brownian motion is a Gaussian white noise?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\lambda$ be the Lebesgue measure on $[0,\infty)$ $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of ...