Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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Transforming the diamond structure Raman tensor to [111] crystal orientation

I'm writing my physics bachelor on the Raman scattering effect in solids. I'm trying to evaluate the scattering intensity response to varying polarization angle. This is the well known linear ...
Eslam Aboelfadl's user avatar
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Equation of Motion Invariance in Galilean Mechanics

Consider a particle moving freely, where $\vec{r}(t)$ is the position of the particle. Suppose I move into a frame with $$\vec{r}' =\vec{r} + \epsilon \vec{F}(\vec{r}, t)\tag{1},$$ where $\epsilon$ ...
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Derivation of lagrange equation in classical mechanics

I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
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Robertson-Walker metric exercise [closed]

I'm trying to solve an exercise from my astrophysics and cosmology class, the request is the following, starting from the RW metric expression: $$ \begin{equation*} ds^2=c^2 dt^2 - a^2 \left ( \frac{...
Lip's user avatar
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How to choses coordinate systems and change between them? [closed]

I understand that when choosing a system for the problem that interests me I need to consider all the things that effect what I want to calculate and try to pick the thing that fits my interests the ...
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Deaccelerating to the Speed of Light

The trajectory of an observer with a uniform proper acceleration $a$ (Rindler) in an inertial frame $(t,z)$ can be described by the hyperbola \begin{equation} \left(z+\frac{\gamma_{0}}{a}\right)^{...
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Physical interpretation of the two possible roots for the isotropic Schwarzschild coordinate $r'$

I am trying to deep dive and study the isotropic Schwarzschild coordinates, whose line element is written for particles lying onto the equatorial plane $\theta=\pi/2$ as: $$ds^2 = -\left(\dfrac{1-\...
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How is the trajectory of a star found relative to the Sun?

So i know we can get radial velocity by measuring blue shift and then we can use the distance to the star and its proper motion to get its tangential velocity. In the case of Bernards star, its ...
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Hamiltonian transformation correction [closed]

In summary, I got an unexpected result when transforming the Hamiltonian of a simple harmonic oscillator, and am hoping someone will be able to correct my work below. I am trying to practice doing ...
Elijah's user avatar
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Do the Lorentz transformations imply a free particle is also accelerating? [closed]

Consider the following two points, or events as they are more commonly called, in SpaceTime: Event 1: $(x,t) = (0,0)$ Event 2: $(x,t) = (a,0)$ As you can see they are merely two separate locations, ...
lee pappas's user avatar
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Value of $p^{2}$ for little groups

I am looking at Weinberg, The Quantum Theory of Fields, Volume 1 page 66. In this table, the author mentions various little groups of the Lorentz group. Orthochronous Lorentz transformations must ...
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Finding radial acceleration from $xy$ vector cordinate [closed]

I know that is a silly question but i cant figure it out. Suppose we have $$ \textbf{R} = A i + B j $$ and want to find the radial acceleration. We know that the radial acceleration is $$ \ddot{r} -...
Zahra.sh's user avatar
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Non-orthogonal frames

Fix $4$-dimensional Minkowskian spacetime $(\mathbb{R}^4, \eta)$. As in my previous posts, a reference frame is then simply a choice of basis vectors $\{e_{\mu}\}\subset \mathbb{R}^4$ such that $\{e_0\...
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Do the Lorentz transformations lead to negative amounts of time? [closed]

Consider the following two points, or events as they are more commonly called, in SpaceTime: Event 1: $(x,t) = (0,0)$ Event 2: $(x,t) = (a,0)$ As you can see they are merely two separate locations, ...
lee pappas's user avatar
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Electric Potential Energy integral: where's the Jacobian matrix? [migrated]

Say I have a point charge somewhere of charge $Q$ and I want to calculate the work it'd take to bring another charge, of charge $q$ this time, into a distance of $R$ from it. I know the final results ...
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When we apply these concepts to physics, where do we put the UNITS in vector spaces and manifolds? Do units have a clear mathematical meaning?

We know that the space of all displacements is a vector space. The vector space is defined as a mathematical object $(V,k,+,\cdot)$ such that it satisfies the 8 properties, where $k$ is a field. We ...
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Avg. velocity in plane polar coordinates

It's fairly easy to describe velocity, acceleration and displacement in plane polar coordinate system is the time interval is approaching to 0, but how can we calculate velocity, acceleration and ...
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Signs terms including acceleration

Am I right in saying that accelerations can only be given a sign if a coordinate system is defined in relation to which they are described? Is this idea applied to any vector quantity? Is there a ...
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Coriolis Acceleration in Local Cartesian Coordinates

Generally, the coriolis acceleration is given as $-2\vec{\Omega}\times\vec{v}$ Just as $\vec{\Omega}$ or $\vec{v}$, the coriolis acceleration can be rewritten in local cartesian coordinates (edited): $...
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Will a distant observer really see an object that has fallen close to a black hole freeze in time?

I'm currently taking my first course in general relativity, and I was wondering: We know from the schwarzschild metric that for a (far away) observer looking at an object falling towards a black hole, ...
Tristan Diotte's user avatar
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On the distinction between frame of reference and observer

A Stack Exchange answer illustrates reference frame and observer as follows: A frame of reference means a co-ordinate system and an observer is someone using that co-ordinate system. For example I ...
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Variation of action under coordinate transformations

I am currently studying General Relativity from M.P. Hobson's "General Relativity: An Introduction for Physicists" and I had difficulty in understanding some concepts in variational field ...
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Is it possible to label particles of a continuum body?

In basic continuum mechanics (e.g. fluid dynamics), we label particles of the continuum, i.e., each particle can be identified by a label, e.g., $p$. Then other quantities are defined accordingly, e.g....
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Intuitive explanation of COSMIC TIME?

I came across the following statement, while studying a Newtonian model for cosmic expansion: "If $R(t)$ is the scaling factor, we can define the Hubble parameter as $H(t)=\frac{\dot{R(t)}}{R(t)}...
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Problem in calculation of spherically symmetric Laplacian in electrodynamics

I have come across the following operation in two electrodynamics textbooks, which I find problematic: When evaluating an integral over a Laplacian in a spherically symmetric function, the radial term ...
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How do physicists determine where to place the world or inertial frame when describing the equation of motion of an object?

For example, I have a pendulum as shown in the diagram above. I would like to write down its equation of motion. To do this, I must define a world frame (or inertial frame, or origin). But this is ...
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How does length contraction apply while distances increase in the Lorentz transformation?

I am having trouble settling the difference between the math in the length contraction equation and the Lorentz transformation. Say there is a piece of wood traveling near the speed of light and you ...
szammyboi's user avatar
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Determining a direction/projection/heading for an object in 3D space after it has been rotated

I hope this is a good place to find an answer. Regardless, Worldbuilding looks like a great forum to follow. I'm not a math wiz and I've tried to tackle this problem many times over the past decade or ...
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What equation governs the time boost of a distant object caused by local acceleration?

This isn't the time dilation aka rate change, but rather due to perspective change, where is Sam looking on Sally's worldline? For example, imagine Sam is 1000 lightyears from Sally and Sally is ...
Daniel Needles's user avatar
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Can we define time as a field? [closed]

The main objective is, can we relate time in terms of a field, I know time differs in many properties from an usual field. But I always imagine time as an forward moving field and we all know it is ...
Ash's user avatar
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Solutions of Laplace's equation for stream functions in cylindrical coordinates [closed]

I was reading Fluid Mechanics by Richard Fitzpatrick. Somewhere in the book, he tried to solve inviscid flow past a semi-infinite wedge https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node76....
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Infinitesimal translation of the scalar field (QFT) [duplicate]

I'm following my professor's notes on QFT, and I cannot understand this passage. It's about an infinitesimal transformation for the coordinates of a scalar field $\phi$. The passage reads: Let us ...
Martin and Friends's user avatar
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Geodesics on a 2D flat round metric/2 contradicting results

I was solving the geodesics of $$ds^2=dr^2+r^2d\phi^2$$ for an exercise. Using as an affine parameter the coordinate $\phi$ I arrived at the equations: $\frac{d^2r}{d\phi^2}-r=0$ $\frac{1}{r}\frac{dr}...
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Is small perturbation in axial direction directly analogous to radial direction for cylindrical coordinate?

In cylindrical coordinate, the stability for a cylindrical liquid column/ligament can be analysed using perturbation theory by applying small perturbation in radial direction as follow; $$\rho(z,t)=\...
jamill1283's user avatar
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Question on the transformation from Boyer-Lindquist to Kerr-Schild coordinates, for a modified Kerr metric

From Kerr metric, we do know that there exist a function with the form of: $$\Delta = r^2 - 2 M r + a^2 \tag{1}.$$ Following $[1]$, I did understand the coordinate transformation from Boyer-Lindquist (...
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AdS compactification of Minkowski space

I am trying to understand the paper "Anti De Sitter Space And Holography" by E. Witten (cf. https://arxiv.org/abs/hep-th/9802150). One of the first point it makes is that "Minkowski ...
Ignacio Garrido González's user avatar
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CFT In Embedding Space

I am trying to figure out how a translation or a conformal transformation explicitly look like in embedded space. Given a CFT in Euclidian (or Minkowski) coordinates $x^\mu$ we can embedded them in $d+...
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Can (extended) canonical transformation involve change of time?

A map from $(q,p)$ to $(Q,P)$ is called an extended canonical transformation if it satisfies $$ \lambda(pdq-H(q,p,t)dt)-(PdQ-K(Q,P,t)dt)=dF $$ Here, to include the change of $t$, let us use $$ \lambda(...
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An example of symplectomorphism that is not a canonical transformation

I want to check my understanding on the difference between symplectomorphism and canonical transformation. This is a follow-up of my previous post. (A) A map $(q,p)$ to $(Q,P)$ is called a ...
watahoo's user avatar
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Rescaling the null coordinates

Given a $4$-dimensional spacetime described by four coordinates $(t,r,\theta,\phi)$, we usually define the null coordinates by, \begin{equation} u = \frac{t-r}{2}, \quad v = \frac{t+r}{2} \end{...
mathemania's user avatar
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Is the metric distance invariant under any coordinate transformation?

Consider an infinitesimal coordinate transformation, $$ x^{\mu} \rightarrow x'^{\mu} = x^{\mu} + \epsilon^{\mu}(x). $$ We can show that the metric tensor under such a transformation, up to first order ...
ratchet411's user avatar
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Any concrete example of symplectomorphism that is not a canonical transformation? [duplicate]

I want to understand the relation between several different definitions of canonical transformation. I am studying the answer by Qmechanic in this post Let us define a canonical transformation as a ...
watahoo's user avatar
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1 answer
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Clarification on order of transformations

Suppose I have two rigid bodies, B1 and B2, I have the position of B2 relative to B1 as a standard homogenous transformation T_12 Similarly I the transformation of B1 relative to the origin as T_01. ...
FourierFlux's user avatar
6 votes
2 answers
282 views

Generalized vs curvilinear coordinates

I am taking the course "Analytical Mechanics" (from on will be called "AM") this semester. In our first lecture, my professor introduced the notion of generalized coordinates. As ...
R24698's user avatar
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2 answers
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How to change generalised coordinates in a Lagrangian without inverting the coordinate transformation?

Given a Lagrangian using the standard cartesian coordinates. $$ \mathcal{L} = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - \frac{1}{2}k(x^2 + y^2) $$ How to move to the hyperbolic coordinates given as $$2 x ...
Lost_Soul's user avatar
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1 answer
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Transformation of derivatives of coordinates

I am quite new to this topic. Please bear with me. Suppose we are given a transformation of both time and space coordinate's derivatives as $$ \partial_t\to D_t=\partial_t-f(t,x)\partial_t\\ \nabla\to ...
Luqman Saleem's user avatar
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Invariant metric

Suppose I have a metric that looks like the Lobachevsky upper-half space $$ds^2 = \frac{L^2}{z_0^2}((dz_0)^2 + \sum^d_{i=1}(dz_i)^2).$$ I now want to show that this metric is invariant under the ...
Geigercounter's user avatar
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1 answer
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Which version of the equivalence principle affects the coordinate dependency of the Landau–Lifshitz pseudotensor?

We know that the energy-momentum of gravity can be defined by a pseudotensor called the Landau-Lifshitz pseudotensor, which is coordinate dependent. In fact, the gravitational stress–energy will ...
Manuel's user avatar
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1 answer
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General formulation of "X transforms like an X"

It has been discussed several times on this site the phrase "a tensor is something that transforms like a tensor". I'm comfortable with both the mathematical formalism and the physical ...
Trebor's user avatar
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1 vote
1 answer
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Spherical coordinate of a vector when divergence of the vector is zero

$\nabla \cdot \mathbf{\delta u_{perp}} = 0$ where $\mathbf{\delta u_{perp}}$ is a function of both x and y coordinates and perpendicular to z axis. Moreover, $\delta u_{perp}$ along z axis is $0$. I ...
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