A set of numbers used to quantify location in space.

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62 views

How coordinate system shifting is related to similarity transformations?

I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?
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4answers
203 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
3
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2answers
182 views

Can a curvature in time (and not space) cause acceleration?

I realize that the curvature of space-time causes acceleration (gravity). Is it possible to have a curvature only of space, or a curvature only of time? If so, would a curvature only of space, or a ...
5
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3answers
147 views

Integral in different coordinate systems

In Griffiths' electrodynamics book, he uses the equation, $$\nabla^2\mathbf{A}=-\mu_0 \mathbf{J},$$ to state that $$\mathbf{A}(\mathbf{r}) = ...
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0answers
256 views

Understanding and deriving ellipsoidal coordinates geometrically

If one were to read old texts on mathematical physics, like Maxwell, Morse & Feshbach, Hilbert and Courant, Jacobi, etc... they'd find ellipsoidal coordinates popping up, but the authors derive ...
7
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1answer
2k views

How to calculate roll, yaw and pitch angles from 3D co-ordinates (Euler Angles)

I have digitized a video of a flying fly in a 3-dimensional space. At all instants I know the x, y, and z co-oridinates of the following points on the fly's body --- The points are my choice, and ...
6
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2answers
505 views

Why isn't invariant notation common?

In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis ...
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1answer
78 views

Prove one of the following trajectories is circular

In his classical mechanics lecture, Prof Susskind gives a short exercise, which I "feel" is very simple, but don't know where to start with. The question is: "There is a coordinate system ...
6
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1answer
108 views

Curvilinear Coordinates and basis vectors

In these notes, $\frac{\partial \vec{r}} {\partial q_i}$ is stated to form a basis set for the vector space. How does this happen? Also, how does one justify this equation from Goldstein's ...
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1answer
84 views

How big or small is a reference frame in Relativity?

What exactly is a frame of reference? Does it have an objective existence and if so what is it? What's the size of a reference frame? Is a reference frame the same size for a stationary frame of ...
2
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1answer
278 views

How to transform material permittivity tensor from Cartesian coordinates to another orthogonal coordinate system?

I have a material specified by a permittivity tensor written in Cartesian coordiantes: $$\begin{pmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz}\\ \epsilon_{yx} &\epsilon_{yy} ...
1
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1answer
234 views

How can I find the motion equations of the 2-dim harmonic oscillator?

First of all: I am no physicist, so I am rather helpless. I need to find the moving equations of the 2-dim. harmonic oscillator. If it is possible it should be rather elementary, because, as I said, ...
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1answer
82 views

Which symmetry for which distance function

For evaluating the electric field of some charge distribution one can use $$\phi(r):= \frac{1}{4 \pi \varepsilon_0}\int_{\mathbb{R}^3} \frac{\rho(r')}{||r-r'||_2} dr'.$$ My question is: What symmetry ...
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1answer
53 views

Inconsistent integral and distance in spherical coordinates

I am currently studying this problem: 14 b) There you see an integral $$A(r) = \int f(\theta) (-\sin(\phi), \cos(\phi),0) d \Omega$$ where $f$ is the function containing all the rest of the integrand ...
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0answers
74 views

Jacobian of a transformation on Maxwell equations in cylindrical coordinates

In an area called transformation optics, they transform Maxwell equations from one space coordinate system to another, and then using the fact that Maxwell equations retain the same format under ...
2
votes
2answers
65 views

How do I modify a 3-D simulation grid to be 2-D?

I am creating a particle in cell simulation that models an electron plasma in a cylindrical container. Part of this process is assigning charge density to grid points based on the position of each ...
2
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1answer
675 views

Calculate the latitude / longitude coordinates of the location where the Sun is at the zenith

There are plenty of resources showing how to calculate zenith and azimuth of the Sun when the time and the location are given. However, I need to calculate the location where the sun is at the ...
10
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5answers
1k views

Minkowski Metric Signature

When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
0
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1answer
93 views

How do I find the generalized coordinates in a certain system?

I'm learning about constraints and I know the following: If there are $N$ particles in 3 dimensional space, I have $3N$ degrees of freedom. If I have $n_b$ holonomic constraints and I switch over to ...
3
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2answers
382 views

The wave equation in general relativity, special relativity, and Cartesian coordinates

The relativistic wave equation is $$\square\varphi=\rho$$ where $\varphi$ is the field, $\rho$ is the source, and $\square$ is the D'Alembert operator, defined by ...
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0answers
122 views

Center of mass coordinates in Lagrangians and Laplacians

Is there a quick nice and easy way to write Lagrangian's and the classical/quantum Laplacian operator in terms of center of mass coordinates? The algebra is so involved and it has me confused about ...
0
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3answers
143 views

Is this webcomic accurate?

I was considering this xkcd comic from 5/10/14, with the alt-text "Trains rotate the Earth around various axes while elevators shift its position in space." I'm wondering about its accuracy. I ...
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0answers
126 views

About the proof of the second Bianchi Identity

The second Bianchi Identity is $$ \nabla_{[a}R_{bc]de}=0 $$ As far as I know, the proof (say, Walfram Mathword) start by stating the representation of Riemann tensor in local inertial coordinates $$ ...
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0answers
122 views

Compute and plot satellite position on 2D Earth map

I would like to create a simple satellite tracking software, that will compute the position of a satellite at any time, and show its location on a 2D map of Earth. Let's assume it is possible to ...
2
votes
2answers
76 views

Show that two families of curves are orthogonal (without using orthogonal trajectories)

I'm reading through Hartle's General Relativity and came across this question: Consider the following coordinate transformation from rectangular coordinates $(x,y)$, labeling points in the plane ...
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0answers
179 views

How to solve the Laplace equation in ellipsoidal coordinates?

It seems that popular textbooks on electrodynamics do not discuss how to solve the Laplace equation in ellipsoidal coordinates. I could not find any reference, but there must be references about this. ...
2
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1answer
139 views

Curvilinear coordinate system around body of revolution

In Boundary-Layer Theory by Schlichting he gives the boundary-layer equations for a body of revolution according to the paper by Boltze$^1$. Unfortunately, this paper is in German. He apparently uses ...
3
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1answer
108 views

Is there a technical term for “meaningfulness” of mathematical operations?

Is there a technical term for "meaningfulness" of mathematical operations? For example, adding vectors that represent forces has a meaning regardless of the coordinate frame, but an elementwise ...
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0answers
76 views

Compatibility between solutions of explicit Maxwell equations vs. wave equation?

When trying to solve for the allowed propagation frequencies in a cylindrical waveguide, I approached the problem by solving the wave equation for all three components of $\bar{E}$, and subsequently ...
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1answer
107 views

Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
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1answer
457 views

When are we now? Time coordinate system [closed]

The Second and the metre are well defined. We have those units available. Of course units are meant to express measurement, tools to compare with precision, and interchange that information ...
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0answers
55 views

Active and passive transformations and the change in potential energy

Under active transformation, the particle moves. On the other hand, for a passive one, the coordinate is just relabel. I've read that the passive one will not affect the potential energy and the ...
0
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2answers
74 views

Which coordinate system confirms quantum-level experimental data?

We often use the Cartesian coordinate system, since it is the naive approach at macro level (placing a box just "next to" or "above" the other box). There are, however, many more such systems, incl. ...
2
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0answers
59 views

Solutions of PDEs in different coordinate systems

Suppose we have a PDE, for example the Helmholtz paraxial equation: $$ \nabla_\perp^2A+2ik\frac{\partial A}{\partial z}=0 $$ Solutions depend on the coordinate system we are using, i.e. we obtain ...
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0answers
69 views

How to prove the relations between the variables in bispherical coordinates?

When I was reading papers about the forces between two charged spherical conductors in a uniform electric field, I was confused with bispherical coordinates the author used to describe the spherical ...
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1answer
130 views

Coordinate-free derivation of the Lamé-Navier's elasticity equations

Linear static elasticity provides a local equation $-\mathrm{div}\sigma=f$, the constitutive law $\sigma=2\mu\epsilon+\lambda \mathrm{tr}(\epsilon)I$ as well as the strain-displacement relationship ...
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1answer
65 views

Electric field in a cylinder

We have electric charge density $\rho(r) = kr$ in a cylinder of infinite height and radius $a$. I'm asked to find the electric field. I'm doing it using two methods and I don't undesrtand why then ...
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1answer
77 views

Heliocentric Worldview [duplicate]

Isn't the whole historic Discussion of Heliocentric vs. Geocentric Worldview just about a Calculation-Technique. I mean I could also choose my coordinate-center to be in the middle of Earth and setup ...
5
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0answers
106 views

What coordinate system is used to describe planets positions in the universe?

How are planets positions described in the space and in respect to what? For example is Sun the origo and right now at this moment Earth has [coord_X, coord_Y, coord_Z]? or maybe [lng, lat]? ...
4
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1answer
459 views

failing to see the conundrum in the Einstein hole argument

I've been reading about the Einstein hole argument, and i fail to understand what makes active diffeomorphisms "special" compared to passive diffeomorphismsm also known as good old coordinate ...
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1answer
239 views

Optimal selection of generalized coordinates in Lagrangian system

EDITED: The number of bonds is actually 2, not 1 (look at edit history). Fixed for archiving purposes. Problem: The edge A of an homogeneous rod (of length $\ell$ and mass $m$) is performing a smooth ...
1
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1answer
688 views

Radians to Rotate Earth to Match ECI Lat/Lon with ECEF Lat/Lon

I am attempting to model GPS Satellite positions on the globe for a set of ephemerides. I have a verified set of ECI ( http://en.wikipedia.org/wiki/Earth-centered_inertial ) XYZ Coordinates and a ...
0
votes
1answer
62 views

Parametric equations of a hypersurface

In light-front QFT, in the Minkowski space, we define a hypersurface, $\Sigma_+ : x^3+ x^0 = 0 $. How can I write its parametric equations?
2
votes
1answer
132 views

Space-Time Continuum [duplicate]

In special relativity it is said that " Time and space cannot be defined separately from one another. Rather space and time are interwoven into a single continuum known as spacetime. " What is the ...
2
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4answers
95 views

Special Relativity moving in space

Given that time is 'just another' dimension and people get hung up on the fact that we cannot go back and forth in time like the other dimensions. Is there any proof that the corner of 8th Avenue and ...
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1answer
205 views

Symmetry, Transformations and non-linear transformations

I am a physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) or differences between coordinate transformation of two kinds : Rotation of ...
2
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1answer
76 views

Riemann normal chart and special relativity

When you pick Riemann normal coordinates at a point, then the Christoffel symbols vanish and the metric is flat, but the Riemann curvature tensor does not necessarily vanish. Since Einstein said that ...
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3answers
382 views

Does the Relativity Principle of Special Relativity imply homogeneity and isotropy of all the reference frames?

In Rindler's book: Relativity, Special, General and Cosmological, is stated on page 40 that the Relativity Principle (RP), when applied to just one Inertial Frame (IF), guarantees the homogeneity and ...
5
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1answer
140 views

Extent of coordinate freedom to set metric components along a spacetime path

If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ ...
5
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1answer
369 views

6 independent Einstein field equations?

I can't understand the comment on page 409, Gravitation, by Misner, Thorne, Wheeler It follows that the ten components $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ of the field equation must not ...