Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
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Changing velocity even when acceleration is zero
Consider a particle performing circular motion with position given by
$$\vec r=r_0e^{\beta t}\hat r$$ such that $\dot\theta=\omega$, is a constant.
The velocity becomes
$$\vec v=\beta r_0e^{\beta t}\...
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Volume element in system phase space
Consider $N$ particles in $3D$, with coordinates $q_i$ and momenta $p_i$, so $\{q_1,p_1,q_2,p_2,...,q_{3N},p_{3N}\}$ are variables. Construct a phase space of the system, with axes $(q_1,p_1,q_2,p_2,.....
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Is the volume in general relativity independent or dependent on the coordinates?
The volume in curved space is calculated as:
$$V=4 \pi\int_{\Omega}r^2\sqrt{g_{rr}} d\Omega$$
Is this volume dependent or independent from the chosen coordinates? As I understand it should be ...
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Vectors in different coordinate systems [closed]
The answer for the first question is "The wind blows from $40.5^\circ$ NE". But I think it is impossible. The following is how I do the first question.
Set that $\mathbf{i}$ is the unit ...
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Are curvilinear coordinates inertial?
At 1:46:34 of this lecture by Frederic Schuller, Inertial coordinates are defined as ones which satisfy the following equation:
I am confused by the above equation because it would imply any ...
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Lagrangian Dynamics of an inverted Spherical Cart Pendulum
Introduction
I have to come up with a PD-controller for an inverted Spherical Cart Pendulum, therefore I tried to compute the Dynamics of such a Pendulum.
The Spherical Cart Pendulum is a hybrid ...
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How to calculate Proper Distance as an arc length in Schwarzschild metric?
I am trying to determine the method to calculate proper distance with constant time and radius in Schwarzschild Geometry. With only $\theta$ and $\phi$ being variable. I think it involves integrating ...
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Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?
Newton's second law is a coordinate agnostic statement, we can use it to calculate the forces in a coordinate system, and hence, the motion of the body in that coordinate system. However, depending on ...
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Confusion about the action variable definition
Suppose we have an integrable system consisting of a $2n$-dimensional phase space $M$ together with $n$ independent functions $f_{1\leq j \leq n }$ in involution. Suppose the level set
$$M_f = \{ (p,q)...
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Conservation and potential with non-cartesian forces
I understand how to determine if a force is conservative from
\begin{equation}
\nabla\times \mathbf{F}=0 \implies \mathbf{F}\text{ is conservative}
\end{equation}
When $F$ is in cartesian coordinates.
...
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Number of Degrees of Freedom of a Rigid Body System - Proof
Let us define the number of degrees of freedom of a material system as
the number of scalar parameters needed to know the position of each particle of the system with respect to any inertial frame of ...
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Contravariant Vector Component Transformation from Polar to Cartesian
I am new to tensors and I have just learned that the contravarient components of a vector transforms in the following way (using Einstein summation convention)
$$A^{'i}=\frac {\partial x^{'i}}{\...
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Change of Metric Under Coordinate Transformation
Under a local change of coordinates $x\to x'=x+\delta x$, the metric transforms as
$$g_{\mu \nu}^{\prime}\left(x^{\prime}\right)=g_{\lambda \rho}(x) \frac{\partial x^{\lambda}}{\partial x^{\prime \mu}}...
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How do we assume the direction of $u_{\theta}$ and $u_{r}$ in polar coordinate systems?
Is there a way to correctly predict the direction of the unit radial vector and the unit transverse vector in problems like the one below or is it just better to take a guess and solve the problem ...
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Cart Pole kinetic energy
As explained in [1], the kinetic energy of a Cart Pole is:
$$
\frac{1}{2} (M+m)\dot x^2 + \frac{1}{2} m L^2 \dot \theta^2 - m L cos(\theta) \dot \theta \dot x
$$
Where $m$ is the mass at the tip of ...
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Would a gyroscope have solved the longitude problem?
So I was thinking about the longitude problem, which was the problem of determining the longitude at sea. It caused great problems in sea navigation. See: https://en.wikipedia.org/wiki/Longitude#...
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Soldering Spinors in cylindrical coordinate
For my computation, I need to have Solderings in cylindrical coordinate(I think). I know what Solderings are in cartesian coordinate:
$\sigma^{\mu}_{A\dot{A}}=(\sigma^{0}_{A\dot{A}},\sigma^{1}_{A\dot{...
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Why can't we use integral of $x$, $y$ and $z$ in calculating moment of inertia
I've got no problems with calculating the moment of inertia/tensor of inertia of a cube using an integral over the lamina of a cube. However, I must be missing something obvious or making some sort of ...
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Dimensionless bulk coordinate in AdS?
I just had a look at the AdS-C metric that can be expressed as follows
$$\begin{equation}
ds^2 = l_4^2 d\sigma^2 + \frac{l_4^2}{l_3^2} \cosh^2(\sigma) \left( -f(r)dt^2 = f(r)^{-1} dr^2 + r^2d\theta^2 \...
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Is this hamiltonian of the form of some well-known physical system?
I'm doing a homework exercise and I'm asked whether some hamiltonian (that is the result of a canonical transformation of some other hamiltonian) is reminiscent of the hamiltonian of some well-known ...
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How would the following image look like, if we didn't use $ct$ for time?
I just wonder how spacetime would look like if we didn't use $ct$ for $t$ and we just used $t$ instead?
I guess the $t$-axis would just scale. Would that mean that, the hyperbolas would be very hard ...
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In canonical transformation, is there any rules or methods for finding the transformation $(q,p)\to(Q,P)$?
If we get two different Hamiltonian by using two methods of canonical formulation of theory and these two Hamiltonian are equivalent. How can I find the canonical transformation from which we can ...
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What is a signature of pp-wave metric?
pp-wave spacetime metric is defined in Brinkmann coordinates as $$ds^2 = H(u,x,y)du^2 + 2 du dv + dx^2 + dy^2.$$ Since it's lorentzian (https://en.wikipedia.org/wiki/Pp-wave_spacetime), I wonder what ...
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Question from Einstein's original paper on special relativity
Just for fun, I started reading the original paper by Einstein on the special theory of relativity. Well, what do you know? I got stuck in a place because he jumped a few steps. See if anyone can help ...
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Velocities - Equation 1.46 of Goldstein 3rd edition
In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein
uses the parametrization (equation 1.45')
$$\displaystyle{\vec{r_i}=\vec{r_i}(q_1,q_2, ..., q_n, t)}\tag{1.45'}$...
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Different AdS metrics in global coordinates
In David Tong's lecture notes I came across the folloing AdS metric in global coordinates
\begin{equation}
ds_3^2 = \left( \frac{dr^2}{\frac{r^2}{R^2} + 1 } - \left(\frac{r^2}{R^2} + 1 \right)dt^2 + ...
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Basis Vectors as Partial Derivatives Issues
I have been introduced a number of times to people defining vectors as derivatives of a curve, with basis vectors as partial derivatives, but I have several issues with this that make this formalism ...
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Can the metric tensor be treated as a linear transformation?
In general relativity, the metric tensor $g$ is a covariant, second rank, symmetric tensor that can be written down as a 4x4 matrix. The metric tensor generalizes the notion of distance between points ...
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Equations of motion only have a solution for very specific initial conditions
An exercise made me consider the following Lagrangian
$$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$
If I didn't make a mistake the equations of motion should be given by:
...
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Is the proper time invariant while going from stationery frame to freely falling frame in Genral Relativity?
While reading GTR, I found the following calculation:
Consider an arbitrary gravitational field and let us take $x^{\mu}$ as the stationery/lab frame and $\xi^{\mu}$ as the freely falling frame, where ...
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How to find canonical transformation to achieve desired Hamiltonian?
I am trying to find a way to transformation that will turn a Hamiltonian from one form into another form:
$$(1)\;\;\;H=p^2+e^x\rightarrow\bar{H}=p'^2.$$
I don't know of any systematic ways to do this. ...
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Choosing diffeomorphisms for the pullback metric in the Weak Field approximation
In the weak field approximation of the EFEs $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we take $g_{\mu\nu}\approx \eta_{\mu\nu}+h_{\mu\nu}$. The $\eta_{\mu\nu}$ term is just the flat space Minkowski metric and $...
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How is a locally inertial frame possible in Principle of Equivalence of General Relativity?
The principle of equivalence states that it is possible to choose a locally "inertial" coordinate at every space-time point, in presence of an arbitrary gravitational field, where the ...
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Coordinate singularity in general relativity and smooth structure of a manifold
I'm a bit confused by the notion of coordinate singularity, or perhaps relatedly, the differential geometry behind GR.
In my elementary understanding of differential geometry, one starts with a ...
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How does one generate a formula(s) for an artificial satellite's location at time $t$ given a set of known locations at discrete times?
Given a set of spatio-temporal points in space {(Lat_1, Lon_1, Alt_1, Time_1), ... (Lat_n, Lon_n, Alt_n, Time_n)}, I need a function (i.e. a formula) that takes these points (or subset of points) as ...
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Relativity Proof with Defining interval [duplicate]
I have started reading Landau & Lifshitz Vol. 2 (fields theory) and I've got confused about something I read.
to prove the Lorentz transform, it defines interval:
$$ds^{2} = c^{2} dt^{2} - dr^{2}\...
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String theory: Conformal invariance and Conformal Killing Vectors
I am confused by the relation between the invariance of the Polyakov action under conformal transformations and the Conformal Killing Vectors (CKVs) appearing during the process of quantization.
Let ...
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Does the spacetime curvature in the vicinity of a massive body increase, decrease or remain unchanged with the increasing velocity of an observer?
Does the spacetime curvature in the vicinity of a massive body such as the sun increase, decrease or remain unchanged with respect to an observer's increasing velocity relative to that massive body?
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When defining a coordinate system, does it matter if it is right- or left-handed?
When you are defining a coordinate system when solving a problem, do the coordinates need to be right-handed to obtain a correct solution? I feel like the answer is no because the directions of ...
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(Why) is the real life observation space spherical?
It is a quite philosophical question. To explain it, let's consider a dish antenna radiating a EM field like in the following picture:
Let's image you are an observer who can look the antenna ...
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Step in derivation of Lagrangian mechanics
There is a step in expressing the momentum in terms of general coordinates that confuses me (Link)
\begin{equation}
\left(\sum_{i}^{n} m_{i} \ddot{\mathbf{r}}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\...
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In non-relativistic QM are coordinate systems $(\vec{r_1},t)$, and $(\vec{r_2},t)$ indistinguishable if $\vec{r_2}=\vec{r_1}+\vec{u}t$?
As I understand it non QM reduces to classical physics when planks constant is negligible compared to the relevant action, and in non relativistic classical physics coordinate systems $(\vec{r_1},t)$, ...
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Equations of Motion and Minimization of Spacetime Interval
I'm trying to show that the extrema of a path in spacetime, as given by the spacetime interval (or length if just considering space) is the one that solves the equations of motion.
Let a path be given ...
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How coordinate axes of a reference frame moving with a vel. $v$ or acceleration w.r.t another F.O.R are different from the other's coordinate axes?
According to definition of inertial frame of reference (F.O.R)--Any other frame of reference at rest or in uniform translational motion (that is, motion in which the respective coordinates axes in the ...
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What's the physical content in the invariance of spacetime interval in GR?
Spacetime interval in one co-ordinate system is given by : $$g_{\mu \nu} dx^{\mu} dx^{\nu} \tag{1}$$
$dx$ is some infinitesimal displacement vector between two events.
Spacetime interval after a ...
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Energy change under point transformation
How do the energy and generalized momenta change under the following
coordinate
transformation $$q= f(Q,t).$$
The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times ...
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Lorentz boost as a conformal transformation
The conformal group is the set of transformation that preserve angles. With this idea, then a conformal transformation is such that $x\rightarrow x^\prime$ and
$$
g^\prime_{\mu\nu}(x^\prime) = \Omega(...
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How is it that adding a random field to the partial derivative results in a tensorial operation?
We know that the partial derivative of a tensor is not a tensor. But how is this problem fixed by adding to the partial derivatives, a field of Christoffel symbols? Christoffel symbols are a ...
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Bra ket notation in spherical coordinates
A position eigenket can be written using a tensor product of individual Cartesian eigenkets as $\mathbf x=|x\rangle \otimes|y\rangle
\otimes |z\rangle$
Can I also using spherical coordinates write the ...
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Anti-holomorphic contribution to 2d conformal algebra
I am reading Ginsparg's notes on 2D-CFT, and I am deeply confused about why Ginsparg states after (1.8) that the conformal algebra for 2d Euclidean space consists of two copies of the Witt algebra.
My ...
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