Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
1,878
questions
-1
votes
1answer
43 views
What does spacetime interval really mean? [duplicate]
Is there any simple way to intuitively understand spacetime interval, proper time and proper length?
1
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2answers
54 views
Why do the tangent vectors of coordinate curves correspond to the coordinate basis vectors?
In Schutz's General Relativity Chapter 5, after he defines vectors in the modern view as tangents to a particular curve, he states the relationship between bases of different coordinates as:
$$\vec{e}...
0
votes
1answer
38 views
How different observers measure time?
Suppose I have a coordinate system, call it $S$, in which an observer $O$ is not moving, and $O'$ is moving with constant velocity and another coordinate system $S'$ where $O'$ is not moving and $O$ ...
1
vote
1answer
62 views
Equation of motions for simple pendulum in cartesian coordinates instead of generalized coordinates [closed]
I am trying to write the equation of motions for a simple pendulum but instead of writing them in generalized coordinates ($\theta$), I want to write them in cartesian coordinates (x, y), as I will ...
0
votes
1answer
65 views
How to define the components of the Poincare group?
I know that the Poincare group/inhomogeneous Lorentz group can be defined as:
$$
x^\mu = (t,-x) \\
t \rightarrow t^\prime = \gamma x + \delta t + b^0 \\
x \rightarrow x^\prime = \alpha x + \beta t + b^...
0
votes
1answer
32 views
Oblate spheroid conductor [closed]
Oblate spheroid coordinates $(\xi, \eta, \theta)$ are related to
cylindrical coordinates $(r, \theta, z)$ as follows:
$r=\sqrt{\frac{(\xi+a^2)(\eta+a^2)}{a^2-b^2}}, \theta=\theta, z=\sqrt{\frac{(\xi+b^...
0
votes
1answer
38 views
Eigenfunctions and eigenvalues of particle in 2D box
A particle in a 2D potential box has two degrees of freedom. It is bound by the infinite potentials at the boundaries.
Our professor asked us to resolve this into its respective $x$ and $y$ components,...
5
votes
5answers
120 views
The effects of Lorentz transformation on shape
Imagine a solid 3D cube. Now imagine that this cube is traveling close to the speed of light. To what degree will the spatial geometric properties of this object (or in general of any 3D object) ...
0
votes
0answers
16 views
What do bending moments look like after coordinate frame rotation?
Suppose a 3D body had certain bending moments about the x, y, and z axes. What would the bending moments about a different coordinate system look like? In my case, I have restricted myself to a planar ...
1
vote
3answers
174 views
Why do the Christoffel symbols not transform as a tensor?
Let $(P, \pi, M)$ be a principal $G$ -bundle. Given $A \in L(G),$ we define the vector field $X^{A} \in \Gamma(T P)$ by
$$
\begin{aligned}
X_{p}^{A}: \mathcal{C}^{\infty}(P) & \stackrel{\sim}{\...
0
votes
1answer
52 views
Transformation of coordinate in Lagrangian
Lagrangian for a Central force problem is:
$$\mathcal{L} = \frac{1}{2}\mu(\dot{r} + r^{2}(\dot{\theta}^{2} + sin^{2}\theta\cdot \dot{\varphi}^{2})) - U(r)$$
We know that angular momentum is defined as:...
0
votes
0answers
56 views
Rotation Reference Change Transformation CRTBP Python
I have created the following script that will propagate the dynamics of the CRTBP set for the Earth-Moon system. I have the initial conditions set for a vertical Lyaponov orbit. The goal I am trying ...
0
votes
1answer
57 views
Time derivative of $\rm{atan2}$ when $x=0$
I want to take the time derivative of the $\rm{atan2}$ function to calculate an azimuth rate in spherical coordinates, given position and velocity in Cartesian $xyz$ coordinates.
$$\rm{atan2}(y, x) =
\...
2
votes
3answers
74 views
Does Schwarzschild metric in Kruskal-Szekeres coordinates admit asymptotic ($r \to +\infty$) timelike observers?
I thank in advance whoever will answer my question.
Schwarzschild metric in Schwarzschild coordinates in $\mathbb{R}^{1,3}$ is [1]:
$$ds^2=-\bigg(1-\displaystyle\frac{2M}{r}\bigg)dt^2+\bigg(1-\frac{2M}...
2
votes
2answers
100 views
$\frac{0}{0}$ from Curvilinear Dirac Delta
The definition of the Dirac Delta in an arbitrary curvilinear coordinate:
$$\delta(\vec{r})=\frac{\delta(x^1-x^1_0)\delta(x^2-x^2_0)\cdot \cdot\cdot \delta(x^N-x^N_0)}{h_1h_2\cdot\cdot\cdot h_N}$$
...
0
votes
1answer
63 views
Lorentz transformation in system of unit [closed]
In a system of units in which the velocity of light $c =1$,
$$x' = 1.25x - 0.75t$$
$$y'= y$$
$$z' = z $$
$$t'=1.25t -0.75x$$
is a Lorentz transformation. Why is this so? That is, how can we ...
3
votes
0answers
62 views
Mathematical description of systems of reference - classical mechanics vs special relativity
Notation: In the following, $E^n$ denotes an euclidean space of dimension $n$ (an affine space with inner product $\langle\,\cdot\,,\,\cdot\,\rangle$ on the translation space).
The answer to this ...
0
votes
1answer
23 views
Evaluation of vertex function with Feynman Parameters
On page 191 of Peskin & Schroeder, they show that after using Feynman parameters $x, y, z$, the denominator of the integrand of the vertex function is:
$$D = k^2 + 2k(yq - zp) + yq^2 + zp^2 - (x+y)...
2
votes
1answer
34 views
Coordinate transformation of boundary condition
Let us suppose a heat transfer problem inside a cylinder of radius $r_a$. If we neglect changes along $z$ and $\theta$ directions, i.e. only a cross section of the cylinder, the problem can be ...
1
vote
2answers
96 views
Where does this formula come from? [closed]
I am doing revision for my module stellar & galactic astrophysics and have come upon this formula which I cannot seem to derive. Could someone please explain where it comes from?
"For an ...
0
votes
0answers
32 views
Rotation of crystal structure to match with match another structure of same compound/polymorph
Say, I have two crystal structures of a particular organic molecule, the crystal structures are basically identical, apart from a rotation and redefinition of the lattice vectors/angles, a simple ...
1
vote
1answer
71 views
$F=ma$ if $ a < 0$
The equation $F= m\cdot a$ is known, but what if $a < 0$? Is the force exerted negative?
1
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0answers
52 views
Difference between covariant derivative, gradient, Lie derivative, Lie bracket, Poisson bracket, Christoffel symbols. With examples and cases
I have some knowledge but when I started to examine these notions I was confused. I have a sess in my mind. Do someone clear up the difference between covariant derivative, gradient, Lie derivative, ...
1
vote
1answer
100 views
Dirac delta in spherical coordinates. What I'm doing wrong?
I must show that the integral
$$\frac{1}{(2\pi)^{3}}\int_{\vec{k}}d^{3}k\frac{\cos(\vec{k}\cdot\vec{x})}{\left({\sqrt{|\vec{k}|^2+m^{2}}}\right)^{s}}=\delta^{3}(\vec{x})$$
when $s=0$ by using ...
0
votes
1answer
36 views
How to derive the bank angle of an aircraft from its roll angle and pitch angle?
From Young (2017) (https://onlinelibrary.wiley.com/doi/book/10.1002/9781118534786) it is stated that we can define the bank angle ($\Phi$) of an aircraft as the angle between its Y body axis and the ...
1
vote
1answer
72 views
Second law in polar coordinates?
Consider a pendulum whose string is replaced with a spring.
It is a system which can be described neatly using polar coordinates, where your radial component is $ l = l_0 + \Delta l $ and your ...
1
vote
1answer
125 views
Coordinates systems and frames of reference in classical mechanics
I have some doubts about the way frames of references are introducted in Arnold's mathematical methods of classical mechanics.
It is said that, given a set $M$, then $\phi_1:M \rightarrow \mathbb{R} \...
0
votes
2answers
42 views
How to find position vector in Kepler orbit?
As the title says, I am trying to determine the position vector $r$, knowing true anomaly, semi-major axis, angular momentum and eccentricity vector.
There is an equation describe the distance to the ...
0
votes
3answers
97 views
Christoffel symbol and covariant derivative
I came across the Christoffel symbols via the geodesic equation, and I understand the extrinsic form and the intrinsic form and can prove that they are identical:
extrinsic form:
$$\Gamma^{j}_{~ik}=\...
2
votes
1answer
70 views
Kinematics in polar coordinates
Take a particle in polar coordinate system to follow the equations:
$$\theta=\omega t$$
and $$r=r_oe^{-\omega t}$$
Now, the radial acceleration will be-
$$a_r=\ddot{r}-r\dot \theta^2$$ which we get as ...
2
votes
2answers
175 views
Understanding dependent/independent variables in physics
How does one determine the independent and dependent variables?
What do the terms mean?
Can they be derived from a formula?
For example I saw in a textbook $F = k\Delta l$, Hooke's Law, that $F$ is ...
1
vote
2answers
46 views
Number of variables in the Hamilton-Jacobi equation
In Goldstein's Classical Mechanics, while introducing the Hamilton-Jacobi equation, he argues that the equation $$H(q_1, ... , q_n; \frac{\partial S}{\partial q_1}, ..., \frac{\partial S}{\partial q_n}...
1
vote
2answers
44 views
Translation of coordinates to generalised coordinates
The translation form $r_i$ to $q_j$ language start forms the transformation equation:
$r_i=r_i (q_1,q_2,ā¦,q_n,t)$ (assuming $n$ independent coordinates)
Since it is carried out by means of the ...
2
votes
2answers
85 views
Canonical transformation such that Hamiltonian of a freely falling body becomes $H'(P,Q)=P$
Can someone please help me with this problem I am unable to find a suitable generating function?
The question says:
To find a canonical transformation such that Hamiltonian of a freely falling body ...
-1
votes
3answers
116 views
Linear algebra as a gauge theory
Is linear algebra a gauge theory? Is the gauge transformation a change of basis?
This was the explanation that I received:
"Take the principal bundle to be the frame bundle $LM$ of your space $M$...
0
votes
1answer
73 views
How do you find the proper separation between two spacetime points?
Suppose you had two points in space-time A and B, where A = (t1, Ļ1, Īø1, Ļ1) and B = (t1, Ļ2, Īø1, Ļ1). How would you use the FLRW metric to find the proper separation? In this case the points occur ...
0
votes
1answer
28 views
Computing total power from $x$, $y$, $z$ components of Poynting vector
I have (real parts) of $x$, $y$, $z$ components of Poynting vectors on the surface of a sphere. Since the total outward power flow from the sphere involves integrating the normal component of a ...
0
votes
0answers
37 views
What are the Lorentz Transformations between polar coordinates? Or can Lorentz Transformations be Non-Linear?
This question rises from the comments on @G Smith's answer's to this question https://physics.stackexchange.com/a/603032/113699
Precisely I was trying to understand the Lorentz Transformations between ...
3
votes
0answers
63 views
Momentum Vectors in Bondi coordinates
In the Bondi-Sachs formalism, we can define the notion of 'retarded' time via a coordinate transformation of the usual Minkowski metric
$$
d s^{2}=\eta_{\mu \nu} d x^{\mu} d x^{\nu}=-\left(d x^{0}\...
1
vote
2answers
94 views
Lorentz Transformations for Polar coordinates or Inertial Frame in Polar Coordinates
Do polar coordinates define an inertial frame or not?
Everywhere in GR, the authors of all the books talk about bring the metric to diag(-1, 1,1,1) which would show that a Local Inertial Frame exists ...
8
votes
4answers
768 views
What is meant by coordinate time? Isn't it time measured on a clock? If not then what does it measure?
In this question-
Is proper time equal to the Invariant Interval or the time elapsed in the Rest Frame?
@Dale in the comments says-
no $dt$ is never physical time. It is always coordinate time. The ...
0
votes
1answer
53 views
Tensor transformation Formula Proof
Ok so basically I am trying to prove that the following expression:
Can be written using matrices like this:
Any suggestions on how to approach this?
0
votes
2answers
43 views
Why is the cross product of vectors in the given image like this? [duplicate]
I know the question seems kind of unclear but why (in the attached image) the cross product like this, I hope someone can clear my doubt.
0
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1answer
72 views
Why is the solution of the radial Schrödinger equation valid at $r=0$?
The Schrƶdinger equation for a particle in a central potential is $$\left[\frac{p_r^2}{2m}+\frac{\ell(\ell+1)}{2mr^2}+V(r)\right]\psi(r,\theta,\varphi)=E\psi(r,\theta,\varphi).$$
This gives solutions ...
2
votes
1answer
63 views
Is light affected by inertia?
A very popular way used by teachers to explain Einstein'sĀ theory thatĀ The speed of light is aĀ Universal Constant, is to use anĀ example as follows:
Two observers moving relative to each other [let's ...
1
vote
1answer
47 views
Lorentz transformation of four vector field
For a 4-vector field $V^\mu (x)$, the Lorentz transformed 4-vector field $V'^\mu(x')$ can be written as
$$V'^\mu(x')={\Lambda^\mu} _\nu V^\nu(\Lambda^{-1}x')={\Lambda^\mu}_\nu V^\nu(x).$$
This can be ...
0
votes
2answers
92 views
Understanding proper-time as “evenly spaced gradations” on the worldline?
In this video series on relativity, proper-time is explained as "evenly spaced gradations" along a particle's worldline. Here is a screenshot:
Suppose I carry a clock with me that ticks at ...
1
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1answer
49 views
Why is a metric necessary in SR?
Simple question since I don't fully understand the purpose a metric fulfills when looking at relativistic kinematics for example.
2
votes
2answers
46 views
Can I use the spacetime interval even if origins are not equal at $t=t'=0$?
Can I use the relativity formula $(\Delta s)^2 = (c \Delta t)^2 - (\Delta r)^2$ even when observers did not start at the same origin at $t=t'=0$? I know this is constant, but is it constant even if ...
1
vote
1answer
40 views
Euler angles - geometry
Hello, I don't quite see why should the angle between $\hat{\dot{\theta}}$ and the projection of $\hat{\dot{\phi}}$ onto the $x_0$, $y_0$ plane be a right angle. Does it have something to do with pure ...
