Coordinates Definition and 1000 Threads

In a textbook I own a formula is given for the integration of natural coordinates over an element. In this case it is a 1 dimensional element (i.e. a line segment) with coordinates ##x_i## and ##x_j##. The coordinate ##x## over the element is written as: $$ x = L_1(x) x_i + L_2(x) x_j $$ with...

It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.

7:03 what is second component of a(theta)? this -> 2 * r' * (theta)' I understand everything except that.

In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away) Starting from a five-dimensional embedded hyperboloid (as in carroll pg. 324) ## -u^2 + x^2 + y^2 + z^2 + w^2...

Hi PF! I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?

Hello, I am trying to solve the following problem: If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...

I calculate that \mbox{curl}(\vec{e}_{\varphi})=\frac{1}{\rho}\vec{e}_z, where ##\vec{e}_{\rho}##, ##\vec{e}_{\varphi}##, ##\vec{e}_z## are unit vectors of cylindrical coordinate system. Is there any method to spot immediately that ##\mbox{curl}(\vec{e}_{\varphi}) \neq 0 ## without employing...

I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following - Center point is known - area is known - Point A is known - Points B, C, and D are UNKNOWN I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...

The wavefunction of ##|\psi\rangle## is given by the bra ket ##\psi (x,y,z)= \langle r| \psi\rangle## I can convert the wavefunction from Cartesian to polar and have the wavefunction as ## \psi (r,\theta,\phi)## What bra should act on the ket ##|\psi\rangle## to give me the wavefunction as ##...

Good day! I am currently struggling with a very trivial question. During my studies, I operated with a parameter called "geometrical buckling" for neutrons and determined it in cylindrical coordinates. But thing is that we usually do not consider buckling's dependence on angle so its angular...

I'm making a program that generates lines in 3D space. One feature that I need is to have an incrementally increasing angle on a line (a bending line / curve). The problem is simple if the line exists in the xy-plane, then it would be a case of stepping say 1m, increase the azimuthal angle φ...

Say we are working in a 2D plane, with a camera and a ball flying past as shown. Camera at bottom, ball flying from left to right Given that I have the X/Y coordinates of the camera, as well as the coordinates of the ball at any given time during the 'flight', how am I supposed to calculate the...

Hello all, so I’ve been reading Jennifer Coopersmith’s The Lazy Universe: An Introduction to the Principle of Least Action, and on page 72 it says: If I understand it right, she’s saying that in our Euler-Lagrange equation ## \frac {\partial L} {\partial q} - \frac {d} {dt} \frac {\partial L}...

Not sure when to use Rindler coordinates to analyze dynamics in a constant accelerating reference system. Rindler coordinates seem messy because they are always changing. Wouldn't it be easier to invoke the principle of equivalence and treat the environment of an accelerating system as a...

Hello everybody, Currently I am doing my master's thesis and I've encountered a physics problem which is very difficult for me to solve. The problem I have is finding equations for the magnetic scalar potential inside and outside a ferromagnetic wire for specific boundary conditions...

Greetings! here is the solution which I undertand very well: my question is: if we go the spherical coordinates shouldn't we use the jacobian r^2*sinv? thank you!

I think I undeerstand Lagrangian mechanics but I have a question that will help to clarify some concepts. Imagine I throw a pencil. For that I have 5 generalised coordinates (x,y,z and 2 rotational). When I express Kinetic Energy (T) as: $$T = 1/2m\dot{x^{2}}+1/2m\dot{y^{2}}+1/2m\dot{z^{2}} +...

In my job, I was given the task of calculating a force that operates an ultrasound transmitter on a receiver. The calculation is made by assuming that each point on the transmitter is a small transmitter and integration should be made on the surface of the transmitter. Since the transmitter is...

I'd like to get some help on checking my understanding of special relativity, specifically I'm trying to clarify the idea of coordinates. Any comment is really appreciated! The spacetime is an affine space ##M^4##, which is associated with a 4 dimensional real vector space ##\mathbb{R}^4##...

I have a function in polar coordinates: t (rho, phi) = H^2 / (H^2 + rho^2) (1) I have moved the center to the right and want to get the new formulae. I use cartesian coordinates to simplify the transformation (L =...

I have sometimes seen the claim that one advantage of Lagrangian mechanics is that it works in any frame of reference, instead of like Newtonian mechanics which will hold only in the inertial frame of reference. However isn't this gain only at the sacrifice that the Lagrangian will need to take...

As a part of my self study, I am trying to derive the Laplacian in spherical coordinates to gain a deeper understanding of the mathematics of quantum mechanics. For reference, this the sphere I am using, where ##r## is constant and ##\theta = \theta (x,y, z), \phi = \phi(x,y)##. Given the...

For reference, the wikipedia entries are adequate for this discussion: https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates (henceforth, GP coordinates) https://en.wikipedia.org/wiki/Lemaître_coordinates (henceforth, LM coordinates) Both of these coordinates are based on a foliation...

I should use the cross product but I don´t know how. I tried to calculate it but it didn´t work out as expected. Please can you give me one example how to do it ?

This is the picture of the problem. My solution is: I'm not sure if the limit is 0 to 2 or 0 to 4...

Hi, starting from this post Basic introduction to gravitation as curved spacetime I would ask for a clarification about Rindler coordinates. From this wiki entry Rindler coordinates I understand that the following transformation (to take it simple drop ##y,z##) $$T = x\sinh{(\alpha t)} ...

(Goldstein 3rd edition pg 72) After reducing two body problem to one body problem >We now restrict ourselves to conservative central forces, where the potential is ##V(r)## function of ##r## only, so that the force is always along ##\mathbf{r}##. By the results of the preceding section, I've...

Hi, I am trying to find open-form solutions to the integrals attached below. Lambda and Eta are positive, known constants, smaller than 10 (if it helps). I would appreciate any help! Thank you!

Let the point P(2,8) be a point in xy-plane and line m: y = -0.75*x+3.25 be a line in the xy-plan. The distance from a point P to a point B is 7 unites. Where the x coordinate of B is negative. Find the acute angle between PB and m. To find B I then construct a circle of radius 7 with center...

Hi If i have 2 general vectors written in Cartesian coordinates then the scalar product a.b can be written as aibi because the basis vectors are an orthonormal basis. In Hamiltonian mechanics i have seen the Hamiltonian written as H = pivi - L where L is the lagrangian and v is the time...

(This is not about independence of ##q##, ##\dot q##) A system has some holonomic constraints. Using them we can have a set of coordinates ##{q_i}##. Since any values for these coordinates is possible we say that these are independent coordinates. However the system will trace a path in the...

Here is the initial problem and my attempt at getting Laplace solution. I get lost near the end and after some research, ended up with the Bessel equation and function. I don't completely understand what this is or even if this i the direction I go in. This is a supplemental thing that I want to...

I consider a scalar massless field obeying the Klein-Gordon equation ##\Box \psi=0 ##. First, in Minkowski spacetime, a solution of the equation is $$ u_{\mathbf k}(x^\mu)=((2\pi)^3 2 \omega)^{-1/2} e^{ik_\mu x^\mu}$$ where ##\mathbf k=(\omega, \vec k)##. So this mode has a frequency of...

Hello, I'm having some trouble understanding this solution provided in Landau's book on mechanics. I'd like to understand how they arrived at the infinitesimal displacement for the particles m1. I appreciate any kind of help regarding this problem, thank you!

Greetings! I have the following integral and here is the solution of the book (which I understand perfectly) I have an altenative method I want to apply that does not seems to gives me the final resultMy method which doesn't give me the final results! where is my mistake? thank you!

A person wrote that ##L=\frac{1}{2} m (\dot{r}^2+r^2 \dot{\phi}^2 +\dot{z}^2)## But, how to find equation of that coordinate system?

I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...

Are cartesian coordinates the only coordinates with a holonomic basis that's orthonormal everywhere?

I solve (1). But to solve (2), What should be the suitable separation constants? I am so confused... E=2/(m*(a+b)) * (a*(dWa/da)^2+b*(dWb/db)^2-k)+l^2/(2mab) where l(constant) is pc since c is cyclic. What should I do to solve the problem?

Hi If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b. Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...

I am confused. Why sometimes perturbation ##V'=\alpha xy## we can write as ##V'=\alpha x \otimes y##. I am confused because ##\otimes## is a tensor product and ##x## and ##y## are not matrices in coordinate representation. Can someone explain this?

how they got that value for the scale factors h?

For car 1, the parametric equations are x = 1 + 0.8t and y=t. For car 2, the parametric equations are x=0.6s and y=2+s. (Let t and s represent time). Solving the system of equations, when the x values are equated are the y values are equated, I get s = -13 and t = -11. I assume that the 2 cars...

Hi I was always under the impression that i could write a2 = a.a = a2 Equation 1 where a⋅ is a vector and a is its modulus but when it comes to the kinetic energy term for a particle in plane polar coordinates I'm confused ( i apologise here as i don't know how to write time derivative with...

So for this problem the solution used Cartesian coordinates but I was wondering wouldn’t it be easier to use Normal and tangential coordinate because the bar is undergoing centripetal motion? Also on the right diagram shouldn’t the acceleration be down and not up. The reason I think that is...

I have a question that might be considered vague or even downright idiotic but just wanted to know that once we find out the velocity & acceleration of a body in angular motion in plane polar coordinates, and are asked to integrate the expressions in order to find position at some specified time...

Good day here is the solution J just don't understand why the solution r=√2 has been omitted?? many thanks in advance best regards!

I am trying to derive the tangential acceleration of a particle. We have tangential velocity, radius and angular velocity. $$v_{tangential}= \omega r$$ then by multiplication rule, $$\dot v_{tangential} = a_{tangential} = \dot \omega r + \omega \dot r$$ and $$a_{tangential} = \ddot \theta r +...