Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

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4
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1answer
316 views

Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
1
vote
1answer
107 views

Does a constant of motion always imply hamiltonian?

If a dynamical system has a constant of motion, and is not already evidently Hamiltonian, is it always possible to use a change of variables and obtain a Hamiltonian system? Edit: the constant of ...
0
votes
1answer
138 views

Modelling a situation using a combination of functions

I am having trouble with a combined function task. It is a mass that is attached to a spring, and a spring to a wall (think, a doorstop). When it is pulled away from the wall it oscillates along the ...
9
votes
2answers
232 views

Is electrostatic energy positive definite?

This is a question coming from physics, but its nature is purely mathematical. Given some continuous distribution of charge $\rho$ (take it compactly supported, or "nice enough" depending on the ...
2
votes
0answers
107 views

Solving Generalized Eigenvalue Problem perturbatively

Let me formulate the problem to convey my notation. I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation $$ U_A A\,\,U_A^{-1} = A_{diag}$$ Now the matrix is changed, ...
5
votes
0answers
920 views

Publication date for Michael Spivak - Physics for Mathematicians II?

I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
2
votes
0answers
59 views

Molecular vibrations and a generalisation of Wigner's rule for (non-finite) compact groups

years student of mathematics and write my script for my bachelor. The topic is "Representations of groups and applications in physics". I understand the representations very good but now i want to ...
7
votes
2answers
5k views

What is dimensional consistency, mathematically?

When going through symbolic calculations involving physical measurements, it's common to check that the final result is dimensionally consistent. (If I'm calculating a frequency, I'd better get ...
1
vote
0answers
40 views

Regularization theory

In order to remove the collision singularity in the equations of motion of the three dimensional two body problem, one defines the coordinate transformation $x_1=u_1^2-u_2^2-u_3^2+u_4^2$ $x_2=2(u_1 ...
1
vote
2answers
31 views

Trigonometry in the DE

Just a simple undergrad physics student asking them mathematicians. I have a very simple 2nd order homogeneous DE of the form: $$y''=-a^2y$$ So, a solution will be of the form $$y(t)=c_1 ...
2
votes
1answer
99 views

Is $f''(f)df=f'(x)df'$ correct?

This question is about kinematics and $a , v, x$ stand respectively for acceleration, velocity, position. Supposing we have an expression of $a$ in function of $x$, we have the following theorem: ...
0
votes
1answer
433 views

Finding the slope for Euler's method of evaulating differential equations

The change in the belocity of a body falling at a relatively slow speed over a short distance is given by $\frac{\mathrm{d}v}{\mathrm{d}t}= g - kv$, where $g$ is the acceleration due to gravity and ...
3
votes
1answer
517 views

Mass Spring Oscillator Interpretation

I’m having some trouble predicting the behavior of ODE’s using the mass-spring analogy. For example, consider the second order IVP listed below: $$y’’ – \space 6y’ + 8y = 0, \space \space \space ...
5
votes
2answers
928 views

Volume charge density of hydrogen atom

The potential of the hydrogenatom is given by: $$\phi(r) = \frac{1}{4 \pi \epsilon_0} \frac{e}{r} ( 1 + \frac{r}{a_0}) e^{-\frac{2r}{a_0}}$$ I'm no supposed to find the volume charge density $\psi$ ...
2
votes
1answer
45 views

When does $\int_a^b \frac{dx}{1-x^2} \neq Ln(1 + x) - Ln(1 - x)?$

I have an integral Over a particular bound. $$\int_{V_o}^{V_f} \frac{dv}{1 - \frac{v^2}{v_t^2}}$$ Can this integral Be that of an Inverse Hyperbolic Tangent, even though it looks like it can be ...
2
votes
1answer
4k views

Phi vs Theta referring to angles of vectors

Even though this question arises from doing simple vector math from a physics textbook I figured it's much more general than that application. The problem is as follows: A spelunker is surveying a ...
2
votes
1answer
1k views

Differential equation for propotional absorbtion of light depending on intensity at a point

The rate (per foot) at which light is absorbed as it passes through water is proportional to the intensity, or brightness, at that point. a. Find the intensity as a function of the distance ...
2
votes
1answer
513 views

How to calculate the position of a turning object, based on its rotation?

I'm working on a program that periodically updates the position of an object. The object is able to move in straight lines, as well as turn gradually. In order to check that my object is turning ...
1
vote
1answer
219 views

Vector Function Magnitude

I was wondering, when you take the magnitude of the vector function $r(t)$, what does it represent geometrically? Does it represent the magnitude of the displacement vector, whose initial point is ...
0
votes
1answer
71 views

Help with a differential word problem

When the electromotive force (emf) is removed from a circuit containing inductance and resistance but no capacitors, the rate of decrease of current is proportional to the current. If the initial ...
1
vote
2answers
763 views

Dirac delta function

1)Prove that the dirac delta function property: $$ x\delta'(x)=-\delta(x)$$ 2)and : $$\int_{-\infty}^\infty \delta'(x)f(x)dx=-f'(0) \ $$
1
vote
1answer
188 views

Quaternions, torque, and impulse.

In a physics simulation I have a solid ball of mass $m$ and moment of interia $M$ (which is a diagonal matrix with all entries equal to ${2\over5}mr^2=i$). Its instantaneous rotation is given by a ...
3
votes
0answers
122 views

Who established the word “ Degree of freedom ” in statistics?

I wonder who is the first one that established and applied the word : "degree of freedom" in statistics? Why he/she need degree of freedom in the calculation of many statistical values?
0
votes
1answer
56 views

Corroboration Of Simple Algebra For A Physics Lab

I have a few equations that I need to solve for a specific variable, and I am wondering if anyone would care to look them over. The first equation is deriving Kepler's equation of orbital period, and ...
0
votes
2answers
320 views

Vector Analysis (Phasor-diagrams) of poly-phased circuits

I have a poly-phased circuit of $q$ phase ($q$ input voltage in equilibrium) such that $$1\le i \le q, \quad V_i= V_{max}\sin\left(\omega t - (i-1)\cfrac {2\pi}{q}\right) $$ How can I use vector ...
2
votes
1answer
314 views

Physics related question on Divergence Theorem for “general” smooth manifolds

I'm a physics major so bear with me here on the math. This is related to a problem from the textbook General Relativity - Wald. In classical electromagnetism say we have a vector field $V$ defined on ...
2
votes
1answer
241 views

Tensors in math and physics

I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me ...
6
votes
2answers
888 views

Mathematical significance of the “Dirac conjugate”

Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ ...
0
votes
4answers
4k views

Projectile Motion - Finding initial velocity

You need to send a bowling ball up exactly $205$ meters so someone at the top of the Washington Monument can take a picture of it "hanging" in space. The bowling ball will be shot from a mortar tube ...
3
votes
1answer
356 views

Is there a non-variational derivation of Snell's law from Fermat's principle?

Every proof I've seen of Snell's law from Fermat's principle uses some sort of variational argument, mostly involving variational calculus. Niven's wonderful book, Maxima and Minima Without Calculus ...
0
votes
1answer
296 views

Calculating Humidity, Wet Bulb Temp and Dew Point [closed]

I have been trying to simulate a climate system, but I have hit a wall. I am trying to calculate Humidity. But to calculate Humidity you need the Wet Bulb Temperature, but to calculate the Wet Bulb ...
2
votes
1answer
519 views

Solve integral by Mellin-Barnes representation

We start with the following integral $$I:=\int_0^\infty\mathrm{d}\alpha\,(a+\alpha)^{-\lambda}$$ which is easily evaluated to ${\frac {{a}^{-\lambda+1}}{\lambda-1}}$ by direct integration over ...
3
votes
2answers
996 views

Resolving vectors into components

The problem is to determine the components of $F_2$. Problem image Method 1 Method 2 My question is why do I receive different answers?
0
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2answers
2k views

Can somebody help and explain me with angular and linear Velocity textproblems?

So we got this as homework today, and I just don't understand it; how to start or how to do it. Explanations and/or setups would be great :) A wheel with ...
1
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0answers
148 views

Hermitian conjugation and representations of the Lorentzian Clifford algebras

The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma ...
7
votes
0answers
291 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
0
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2answers
1k views

Convert this figure to watts , where W = 1 J/s , and then estimate the average per capita energy consumption rate in watts. [closed]

The world consumes energy at the rate of about $466 EJ$ per year, where the joule$(J)$ is the SI energy unit. Convert this figure to watts$(W)$ , where $W= 1 J/s$, and then estimate the average per ...
2
votes
1answer
88 views

An integral for the Earth's insolation

Consider the function $$ [-\pi/2,\pi/2] \ni \theta \mapsto s_\beta(\theta) = \int_0^{2\pi} \sqrt{ 1 - \left(\cos \theta \sin \beta \cos \gamma - \sin \theta \cos \beta \right)^2} \, d \gamma $$ for ...
9
votes
1answer
255 views

Help computing an integral for Green's function of a discrete Laplacian on a square lattice

I need to calculate the following integral: $$ \int_0^1 \int_0^1 \frac{1-\cos(2 \pi k_1 x) \cos(2 \pi k_2 y)}{4 \sin(\pi k_1)^2 + 4 \sin( \pi k_2)^2} dk_1 dk_2 $$ I have tried to use some contour ...
1
vote
3answers
9k views

Find the component of $\vec{a}$ along $\vec{b}$?

I'm trying to do homework for my physics class, and it says I should find 'the component of $\vec{a}$ along the direction of $\vec{b}$'. The vectors are: $\vec{a}$ = 7.1i + 8.9j $\vec{b}$ = 5.8i + ...
4
votes
0answers
178 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
1
vote
2answers
348 views

How to get the derivative of a physical formula?

I've heard that if $U_{ind} \neq $ constant, you must use the following formula: $ U_{ind} = N \times \dfrac{d \phi}{dt}$ I however don't know how to get the derivative of this formula? In math I ...
2
votes
2answers
368 views

How to calculate the mass of a layer of an atmosphere?

In a problem I'm working on, I need to derive what fraction of molecules in the Earth's atmosphere are found in the troposphere (the bottom layer extending from the surface to approx. 12 km). I have ...
4
votes
0answers
136 views

Gaussian Integral with non-polynomial exponent

I am currently trying to evaluate this Integral: $$\int\limits_{u_0}^{u_1} \exp\left[-\angle(H(u),N)^2\right]du$$ Where ...
7
votes
4answers
312 views

Is length adimensional when space is not flat?

Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...
2
votes
1answer
455 views

Center of Mass of “Combinations” of 1-Dimensional Objects.

Center of mass for one-dimensional objects is given by $\displaystyle\frac{\int x \, dm}{M}$ or $\displaystyle\frac{\int x \rho \, dx}{M}$, where $\rho$ is density. Now, the center of mass of a rod ...
1
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1answer
600 views

Finding the initial velocity using calculus

I throw a stone at 20 degree, when the stone falls to the ground, it reaches 100m further. Using CALCULUS methods, find the initial velocity of the stone.
4
votes
1answer
203 views

Existence Energy of Wave Equation

I was just going trhough some properties of the wave equation, including the energy of the wave equation given by $E(t)=\int_{-\infty}^{\infty}u_t^2+c^2u_x^2 dx$, i.e the sum of kinetic and potential ...
7
votes
2answers
642 views

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding ...