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Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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How to Construct Proper Spherical Coordinates in Minkowski Spacetime?

In $n$ dimensional Euclidean space, we only need one radial coordinate, and $n-1$ angular coordinates, where one ranges from $[0, 2π)$ and the rest range from $[0, π]$ Spherical Minkowski coordinates ...
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2answers
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Relativity Confusion

In the time dilation formula $\Delta t=\gamma\Delta t_0$, I am confused about what $\Delta t$ and $\Delta t_0$ measure. If we have two people, Tom and Bob, moving at constant velocity relative to ...
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1answer
29 views

Galilean transformation and differentiation

Given $x=x’-vt$ and $t=t’$, why is $\frac{\partial t}{\partial x’}=0$ instead of $1/v$? $t$ seems to depend on $x’$ because if $t$ changes, $x’$ changes. Also, in this problem, $dx=dx’$ as well, but I ...
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Trouble deriving the Lorentz transforms

After separately arriving at the two equations $x'=\gamma(v^2)(x-vt)$ and $t'=\phi(v^2)(t-\frac{vx}{c^2}$). Where $x'$ and $t'$ are the coordinates a moving observer ascribes to events. I had to show ...
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1answer
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Equation for $N$-body problem using Jacobi Coordinate

For reference on Jacobi Coordinate used for solving 2-Body problem, I referred Wikipedia Jacobi Coordinate, and on looking at those equation I can't get the meaning of the symbol q in the equation for ...
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Converting the deformation gradient from Cartesian to Cylindrical

Suppose I have a Cartesian deformation gradient tensor F for a domain $\Omega_0$. This tensor deforms $\Omega_0$ into a new domain $\Omega_1$. Also assume that I know the values for each entry of F at ...
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1answer
35 views

Canonical commutation relation for spherical coordinates?

What coordinates systems can the canonical commutation relation be generalised to? I also ask specifically for spherical coordinates. This is because I want to prove $\hat{\bf p}\cdot{\bf e}_r-{\bf e}...
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How to determine the best-fit magnetic dipole Gauss coefficient in a shifted coordinate system?

Introduction A planetary magnetic field $\vec{B}$ can be described outside of the planet using Gauss coefficients $g_n^m$ and $h_n^m$ and a spherical harmonic expansion: $$\vec{B} \;=\; -\vec{\...
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How does a tensor from cotangent and tangent spaces transform?

In Sean Carroll's Spacetime and Geometry An Introduction to General Relativity Chapter 2, there is an example of tensor transformation from $x,y$ coordinates to primed ones using $$(x',y') = (\frac{2x}...
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How is time “homogeneous”?

My book$^1$ states: Let's consider a clock moving freely over a curve such as: \begin{equation} \frac{dx^i}{dt}=\text{const} \tag{1.20} \end{equation} We define the proper time $\tau$ as the ...
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Interpolation on curvilinear grid [migrated]

How could someone interpolate spatial data on a curvilinear grid, like the one in the picture shown?
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35 views

Cylindrical coordinates of a ring

A classical problem for electromagnetism students is the calculation of the electric field on the central axis of a ring. It can be solved in many different ways, but I got stuck with the pure ...
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2answers
142 views

Intuition behind Manifold

As the majority of concepts in dynamical systems are based on Manifolds. How can one think/imagine about the concept of a manifolds intuitively? (A Lucid explanation is highly encouraged!!!)
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4answers
132 views

Does $A_\mu$ transform with the coordinates, or as a vector in a tangent space?

Does the vector potential $A_\mu$ transform when we merely relabel events in space-time (coordinate transformation), or does it transform with the basis vectors of a tangent space in which it lives? ...
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2answers
59 views

Null Kruskal Coordinates and affine parameterization

Let $(M,g)$ be the Schwarzschild spacetime. The usual metric tensor in Schwarzschild coordinates reads: $$g=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2\quad f(r)=1-\frac{2M}{r}.$$ Now consider radial null ...
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Azimuthal Symmetry is on a certain axes that is not simply the $z$-axes

let's say in a problem, the azimuthal symmetry is along the axis $\frac{1}{\sqrt6}(-1, -1, 2)$. If I create two new basis angles and get solved the LaPlace's equation with that, what should I do to ...
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1answer
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Why this constant is included in the tortoise coordinate?

In the Schwarzschild spacetime, the tortoise coordinate $r_\ast$ is defined by the property that $$\dfrac{dr_\ast}{dr}=\left(1-\dfrac{2M}{r}\right)^{-1}$$ Now, we cam integrate this. Multiply by $r$ ...
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2answers
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Intuition for formula of tangential component of acceleration in general curvillinear motion

In certain problems of plane motion, the position of the particle P is defined by its polar coordinates $r$ and $\theta$. It is then convenient to resolve the velocity and acceleration of the particle ...
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1answer
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Train and lightning bolts: why the time difference does not depend on the position of the moving person?

So imagine two lightning bolts hit the ground, simultaneously to a stationary observer. There is also a person on a train traveling to the right at a constant velocity. I know that if he started in ...
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1answer
87 views

When to use an orthonormal basis in GR?

I am working on a problem in the textbook Gravity: An Introduction to Einstein's General Relativity by Hartle. The problem is to show that an observer moving radially through the throat of the ...
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1answer
50 views

How to place the axis so that I can calculate the center of mass for the two instances?

I've got this: A wagon of mass $M$, initially at rest, can move horizontally along a frictionless track. When $t = 0$, a force $F$ is applied to the cart. During the acceleration of M by the force $F$...
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2answers
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Different results when using generalized coordinates?

I'm solving an imaginary double pendulum (that is, two pendulums whose motion doesn't affect each other). The two pendulums have a "normal" motion, but they are attached. Taking the point to wich the ...
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2answers
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What is meant by 'The components of a force along a given axis'

I'm new to mechanics, and I'm having trouble interpreting what this means. So there's a force, okay. Then it has its components on the xyz axis, okay. But what does it mean by along a given axis, I ...
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0answers
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Does the Choice of Coordinate System Matter for a Particle Falling Downwards in a Linear Medium

I have a question regarding John Taylor's treatment of a particle falling down in a medium that exerts a linear drag force (Chapter 2 of his book "Classical Mechanics").First some background, he ...
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1answer
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Coordinate transformations in general relativity

Let's assume a non-rotating point mass with mass $M$. A non-massive object travels with constant velocity $\mathbf{v}_t$, with respect to the point mass, in the vicinity of the point mass. A non-...
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Question about holonomic constraints

Goldstein says that when a system of $N$ particles is subject to $k$ holonomic constraints, the positions $\mathbf{r}_1, \dots, \mathbf{r}_N$ can be parameterized by $3N - k$ independent coordinates $...
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Tangent vector to a curve [migrated]

I am trying to relate things simply. If a curve is on a flat 2D space represented by the parameter $\lambda$. In polar coordinate system $(r,\theta)$ at any lambda the tangent vector components are $$...
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Does absolute motion/stillness exist?

I am not a physicist :) I (think I) understand the idea that motion can be relative in the sense that, I'm standing still inside a moving train in regards to the moving train itself, but I'm moving ...
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1answer
48 views

Can all canonical transformations be generated using a generating function?

In Classical Mechanics, a gauge transformation is of the form \begin{equation} L \to L' = L + \frac{dF(q,t)}{dt} \, . \end{equation} Any transformation of this kind leaves the Euler-Lagrange equation ...
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What is the definition of coordinates of an observer? [duplicate]

I am confused about coordinates of an observer. Suppose we have an observer $A$ whose world line $\gamma$ is a geodesic and another observer whose world-line $\gamma$ is not. Since coordinates is just ...
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0answers
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Deriving Canonical Transformation from Generating Function using Principle of Stationary Action

In Hamill's "A Student's Guide to Lagrangians and Hamiltonians", section 5.2, the equations for a canonical transformation $(q,p) \to (Q,P)$, induced by the generating function $F(q,Q,t)$ are derived ...
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What are the conditions coordinate transformations need to fulfil in general?

Clearly, the number of new coordinates cannot be smaller than the number of degrees of freedom within the system. But otherwise, there seem to be little restriction on maps between coordinate system....
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3answers
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What exactly does the observer's coordinate system mean in special relativity?

This seems like a very basic question but I'm having a hard time understanding it, more precisely hard time visualizing it. Let say there is an observer, and we wish to somewhat formalize his ...
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1answer
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How does the Equivalence Principle imply that derivatives of the metric vanish in a freely falling frame?

Why do the first derivatives of $g_{\mu\nu}$ vanish in a freely falling coordinate system? I would like to start from the Equivalence Principle that for any point in spacetime there exists a locally ...
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1answer
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Invariance of length [closed]

Invariance of interval in Minkowski space under coordinate transformation was proved by the postulates of special relativity. (https://physics.stackexchange.com/a/453536/213658 .see this answer) Is ...
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Chart(s) of space-time as a smooth manifold

So we all know that space-time in general relativity is modeled as a smooth (pseudoRiemannian) manifold. Each point (event) on space-time is labeled with a unique coordinate $(t,x,y,z)$ in a specific ...
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2answers
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Confusion about Change in Integration Variable [closed]

I'm working through example 3.2 of Zangwill's Modern Electrodynamics and have come across a change in integration variables that I just can't seem to get. The example has two different change ...
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31 views

Confusion about valid set of virtual displacement

The answer of the problem below makes me confused about my thinking about virtual displacement... Consider the system in the figure below, similar to mechanisms used for aerial filming of sporting ...
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5answers
177 views

Is Relativity of simultaneity just a flaw in perception?

So I went through a couple of different books on special relativity ( including Einstein's own book) and I just can't seem to accept relativity of simultaneity as a real thing. I'll explain my ...
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Rigid Body Equations in terms of Body Coordinates by Hamilton's Principle

I sought-for the equations of motion of an unrestrained rigid body. The equations of motion are readily available in the literature, but my concern is to derive them by Hamilton's principle. ...
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1answer
80 views

Partition function in spherical coordinates

Suppose I write the Hamiltonian/energy of my system in spherical coordinates ($r,\theta,\varphi$) with conjugated momentums($p_r,p_\theta,p_\varphi$). How do I calculate the partition function? If ...
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1answer
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How do frames of reference work in general relativity, and are they described by coordinate systems?

In both Newtonian gravity and special relativity, every frame of reference can be described by a coordinate system covering all of time and space. How does this work in general relativity? When an ...
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2answers
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What is the unit vector in electric field formula? [duplicate]

What is the $\hat{r}$ (vector) in the formula $\vec{E} = k\frac{q}{r^2} \hat{r}$ for the electric field ? Why we dont use the vectors $\vec{i},\ \vec{j},\ \vec{k}$? Also why this vector doesn't ...
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0answers
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Help with the proof of the independence of the form of Lagrange equation wrt. choice of coordinates

I am reading about Lagrange-Euler equation here. When they prove that the formula is independent of the choice of coordinate, there is this reasoning, but I could not understand (probably my calculus ...
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1answer
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Spacetime diagram of LCDM

I have a question concerning the LCDM spacetime diagram https://i.stack.imgur.com/Uzjtg.png published on the Physics Forum Stack Exchange website Can space expand with unlimited speed? How are the set ...
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1answer
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Electric field on the boundary of a continuous charge distribution

In Purcell and Morin's Electricity and Magnetism, 3rd Edition, the claim is made that the magnitude of the electric field on the boundary of a continuous charge distribution is finite (assuming the ...
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Why is my height negative if I'm pointing upwards?

I'm solving this typical problem of finding the height I have to aim my crossbow in order to get to a distance. The maths are OK but since gravity is negative my height is also negative. What am I ...
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0answers
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Acceleration interpretation in accelerated frames in general relativity

I would like to know whether my physical interpretation of some dynamics in accelerated frames is correct. In a frame with acceleration $a$ we have the metric $$ds^2 = (1+ax)^2 dt^2 - dx^2$$ The ...
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1answer
54 views

Accelerated frame approximation in Schwarzschild metric far from the horizon

It is clear to me that if I take the Schwarzschild metric $$ds^2 = \left(1-\frac{2M}{r}\right)dt^2 - \left(1-\frac{2M}{r}\right)^{-1} dr^2$$ and choose $\rho = 2\sqrt{\frac{r}{2M} -1}$ then I get the ...
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2answers
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Christoffel symbol

In flat space, it is possible to find a coordinate system in which the Christoffel symbols are equal to zero at every point on the flat manifold. However, I was wondering if it is possible to find a ...