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
82 views
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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2answers
40 views
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
17 views
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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0answers
12 views
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:
$$\vec{B} \;=\; -\vec{\...
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1answer
48 views
How does a tensor from cotangent and tangent spaces transform?
In Sean Carroll's Spacetime and Geometry An Introduction to General Relativity Chapter 2, there is an example of tensor transformation from $x,y$ coordinates to primed ones using $$(x',y') = (\frac{2x}...
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0answers
74 views
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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0answers
23 views
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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2answers
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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0answers
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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
38 views
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
34 views
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
41 views
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
44 views
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
27 views
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
85 views
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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4answers
172 views
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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0answers
41 views
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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0answers
50 views
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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0answers
20 views
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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1answer
40 views
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
108 views
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
46 views
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
36 views
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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1answer
52 views
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
46 views
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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0answers
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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0answers
49 views
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.
...
2
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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
119 views
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
99 views
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
60 views
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
27 views
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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2answers
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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
42 views
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$$
The ...
3
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1answer
54 views
Accelerated frame approximation in Schwarzschild metric far from the horizon
It is clear to me that if I take the Schwarzschild metric $$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
90 views
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 ...
