Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
3,361 questions
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Visualise Hadamard gate as addition of $x$ and $z$ gate
I try to imagine the Hadamard gate on a Bloch sphere and it is obvious that it is the sum of the Pauli $x$ and $z$ gate matrices.
So that means we rotate the "vector" around the x axis and ...
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How does the concept of "coordinates" makes sense when considering large scales of the universe?
I will describe two instances, that led me to ask the question of relevancy or meaningfulness of coordinates when we are not local in our observation of the universe, but rather we consider a large ...
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Transforming Initial Cartesian Uncertainty into Delaunay Variables for Orbital Uncertainty Propagation [closed]
I am working on orbital uncertainty propagation where the initial uncertainty is given in Cartesian coordinates
(
𝑟
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𝑣
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(r,v), either as a covariance matrix or as bounded (non-statistical) ...
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A (probably wrong) proof that the fundamental Poisson brackets are independent of the special choice of the canonical variables
In Wolfgang Nolting's book Theoretical Physics 2 - Analytical Mechanics, the following theorem is stated:
While the formal definition of canonical trasformation is given only several pages later in ...
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Using conserved quantities as variables in Mechanics
I am currently taking a course in theoretical (classical) Mechanics, where I have learned about the Darboux theorem. My professor has also mentioned one can "reduce the system by symmetry", ...
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Invariance of the interval in Schutz - maths error?
Reading Schutz's book on GR: On page 9, there's a derivation that I don't follow.
$\newcommand{\d}{\Delta} \newcommand{\b}[1]{\bar{#1}}$ Then in the
expression for $\d\b{s}^2$, $$\d\b{s}^2 = -(\d\b{...
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Most general isotropic spatial metric from Schur's lemma
Consider a generic spatial metric $$\mathrm{d}\ell^2 = \gamma_{ij}(\vec{x}) \, \mathrm{d}x^i \mathrm{d}x^j.$$ Assume isotropy of the universe around the origin. Then the metric can only depend on the ...
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Nr. of constraints and entries when considering coordinate transformation of the metric in GR
For the metric $g_{\mu\nu}$, when considering coordinate transformations, one can write:
$$g_{\alpha'\beta'}=\frac{\partial x^\mu}{\partial x^{\alpha'}}\frac{\partial x^\nu}{\partial x^{\beta'}}g_{\mu\...
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How come the fact that momentum is a covector does not contradict its coordinate change law?
First of all, I am aware of existence of this question, as well as other relevant questions at MSE and PSE. However, all answers to those questions focus too much on why momentum can take a vector as ...
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Intuitive understanding of what coordinate basis elements of $T_p$ mean in relation to the manifold
I am trying to have a "visual" or intuitive understanding of the description that was made in my GR lecture about the object $\partial_\mu$. The following was said:
We have a manifold $M$, a ...
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Do values of the metric tensor elements depend on the coordinate system of our choosing, but stress-energy tensor elements don't?
If that's the case, how is the Einstein field equation satisfied for different coordinate systems of our choosing? I'm asking in context of my previous question.
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Why do we use spatially expanding metric to measure the size of the expanding universe? [closed]
Being material observers, we do not expand with the universe. Our ruler for measuring its increasing size does not expand either - its scale does not change. If I identify the ruler with a metric, ...
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Why can't only time dilation alone take place in special relativity, without length contraction? [closed]
An object moving relative to an observer experiences time dilation, as stated in the theory of Special Relativity.
But suppose only time dilation occurred while lengths remained unchanged.
Wouldn’t ...
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Centrifugal term in cylindrical coordinates
When solving an equation of motion with spherical symmetry using the WKB approximation, one usually encounters a centrifugal term that diverges as
$$
\sim \frac{l(l+1)}{r^2}.
$$
What is the ...
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What would the Negative $𝑟$-Coordinate Region of the Kerr Metric look like from afar?
In the Kerr Metric there's a Negative $𝑟$-Coordinate Region on the other side of the ring singularity.
If you went through the ring, continued on and then looked back, how would it appear? What would ...
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