Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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What exactly is the Legendre transformation? [duplicate]

Goldstein et al's Classical Mechanics states that: The Hamiltonian $H(q,p,t)$ is generated by the Legendre transformation $$ H(q,p,t) = \dot{q}_i p_i - L(q,\dot{q},t). \tag{8.15} $$ But I don't ...
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What are the Shallow water equations in differential form and cylindrical coordinates?

I am ignoring bed height or any other extra terms. The version of SWE when written in cartesian is: $$ \begin{pmatrix} h \\ hu \\ hv \\ \end{pmatrix}_t + \begin{pmatrix} hu \\ hu^2 + \frac{g}{...
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Coordinate independence in spacetime

As far as I know, the $n$ coordinates $(x_1, x_2, ..., x_n)$ chosen to describe an $n-$manifold have to be mutually independent $\to$ the mutual derivatives must equal $0$ (for example, $\frac{dx_1}{...
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Christoffel symbols in general coordinates

In order to understand the meaning of covariant derivative, I have seen the following argument. Let us consider a covariant vector $V_\mu$. We would like to understand whether $$T_{\mu\nu} = \frac{\...
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Sagnac Effect and clocks synchronization

I read this from "The Sagnac effect and its interpretation by Paul Langevin", in cylindrical coordinates $$ ds^2=c^2t^2-dr^2-r^2d\theta^2 $$ ...This transformation means that the observer O (...
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Canonical Transformations that are Complex

I'm self studying through a book that has the following question. The book gives the answer, but I'm trying to understand why: Under what condition is the following transformation NOT canonical? $$Q =...
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78 views

A proof of Liouville’s theorem

I have found a proof of Liouville's theorem on the internet, which fits me very well except one step I don't understand, the derivation is as follows: In the derivative, it must have used the ...
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35 views

How to find a coordinates transformation on $A\,\mathrm dS_2$?

I have the following 2D metrics (describing the $AdS_2$ spacetime), which are supposed to be the same in different coordinates: \begin{align} \mathrm ds^2 &= \mathrm dt^2 - \sin^2{\!\omega t} \, \...
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1answer
50 views

Spacetime interval calculation - What am I doing wrong?

I'm calculating spacetime intervals in standard setup with frames A and B where t = t' = 0 and x = x' = 0, and all the motion is going only in x direction. Objects A and B move in relative velocity ...
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88 views

The complex form of Hamilton canonical equations

I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that If, however, the conjugate variables $q_k$, $p_k$ are replaced by the complex ...
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What is the magnitude of a tensor property in a fixed direction?

If I have a physical property represented by a $3 \times 3$ tensor, how can I find its magnitude in a particular direction, say $(\phi, \theta)$ in spherical coordinate system?
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Lorentz boost in light-cone coordinates

Consider a particle with momentum $p^{\mu}=(p^+,p^-,p_{\perp})$, where the momentum is written in light cone coordinates defined as, \begin{align} n^{\mu}&=(1,0,0,1)& \bar{n}^{\mu}=(1,0,0,-1) ...
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Null wave vector in different coordinate systems

The wave vector of a photon in a covariant form is, $$ k^{\mu} = (k^t, k^i)$$ where the index $i$ labels the spatial indexes. This can be written as, $$k^{\mu} = \frac{\omega}{c} \left(1, \hat{n} \...
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Spherical polar coordinates in a tetrad frame

I am looking at a paper which writes the spatial components of a vector $S_i$ in terms of spherical polar coordinates w.r.t the local tetrad frame as (Eq 33 in the linked paper), $$ S_1 = s \sin \...
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Coordinate Transformation of Torque

According to angular-velocity-expressed-via-euler-angles you can express angular velocity in euler angles. Would the coordinate transformation be the same if I were to convert torque vector $\vec{\tau}...
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1answer
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For a physical pendulum, why do you use an angular coordinate system when the centre of mass translates too?

I am trying to understand why you can use $F=ma$ for a simple pendulum, yet need the rotational equivalent for a physical pendulum. I understand it is because the rigid body can rotate too, whereas ...
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2answers
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Magnitude of basis vectors in Kerr metric

I am following some work in David McMahon's 'Relativity Demystified'. In it, he gives an example of some line element in flat spacetime in polar coordinates $$ \mathrm ds^2 = \mathrm dr^2 + r^2 \...
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4answers
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Motion between two particles in a relative manner

Suppose a particle A is travelling in east direction with velocity of x m/s and another particle B is travelling with velocity y m/s in the west direction. Why does the the particle B appears to move ...
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51 views

Derivation of equation of motion for mechanical seismograph

Consider a simple seismograph consisting of a mass $M$ hung from a spring on a rigid framework attached to the earth, as shown in the picture. The motion of the mass is apparently described by the ...
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Transformation to tetrad frame

I have some vector components as measured in the comoving tetrad frame $V^{(\mu)}$. This vector exists at coordinates $x^{(\mu)}$, which is different from the origin of the tetrad coordinate system. ...
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1answer
33 views

Change of basis in a Euclidean space

I am trying to compute the change in the contravariant components of a vector when the basis is changed from Cartesian (standard basis) to spherical polars. I understand that a general vector $\...
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Do gauge fields not transform like functions of the coordinates under translations?

By "transform like a function of the coordinates," I mean that under an infinitesimal translation $x^\mu \to x^\mu + \epsilon^\mu$, to first order in $\epsilon^\mu$ the function $f(t,\mathbf x)$ ...
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Frame-dependence of the gravitational field pseudotensor

What seems to be the common consensus in physics is that a gravitational field does not have a stress energy tensor due to the equivalence principle, but rather a pseudotensor. Is this pseudotensor ...
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What does a zero determinant of the metric tensor in a space means?

Does it show that a coordinate transformation occurrs, and how?
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How do you work out the coefficients of the metric tensor?

The definitions of covariant and contravariant tensor quantities are that they transform as $A' ^i=\frac{\partial x_j}{\partial x'_i} A^j$ and $A'_i=\frac{\partial x'_i}{\partial x_j}A_j$ respectively,...
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1answer
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What is the advantage of using a polar coordinate system with rotating unit vectors?

What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors $\mathbf{ \hat{r}}$ and $\mathbf{\...
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Is the Minkowski metric coordinate independent?

Suppose I have some vector $\mathbf{P} = p^{\mu} e_{\mu}$. Now, for a flat spacetime, the contravariant components can be lowered via the Minkowski metric, $$ p_{\mu} = \eta_{\mu \nu} p^{\nu}.$$ My ...
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1answer
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Variational principle if coordinate transformation depends on fields

Assume we have a Lagrangian that is given in terms of Lagrangian density. $$ L = \int \mathcal{L} (\Phi, \partial_{\mu}\Phi, x) d^N x $$ Also assume that $\Phi : \mathbb{R}^N \to \mathbb{R}^N$ and ...
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What “luminosity distance” means in a general spacetime?

In the paper "Asymptotic Symmetries in Gravitational Theory" by R. Sachs from 1962, the author says the following: In analyzing gravitational fields it is sometimes useful to introduce coordinates ...
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Stress-strain relation in spherical coordinates

Hey so I have a timeless question but I didn't find a source to check it. Would you have a precise reference (or if you can write it that's also perfect for me) for the stress strain relation in ...
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Legendre transform and coordinate system independence

I'm self-learning analytical mechanics. Consider a classical mechanical system. Even if it's clear to me that via (the usual) Legendre transform we can get a unique Hamiltonian function from a ...
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1answer
48 views

How is it possible to have four types of generating functions?

Since the Hamilton's equations of motion remain unchanged in form under a canonical transformation $(q,p)\to (Q,P)$, the Lagrangians must differ by a total time derivative of a function of $q,t$. In ...
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Free-float frames and inertial frames [duplicate]

In the book Spacetime Physics by Taylor and Wheeler, a free-float reference frame is defined as a reference frame where every free particle initially at rest with respect to that frame remains at ...
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2answers
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Clock ticks faster with altitude

Suppose there are two identical drones in a weak and uniform gravitational field that are initially at the same place and with synchronized clocks. Assume drone A first rises vertically for 10 ...
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1answer
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Observers at rest and simultaneity

Suppose we have two observers A and B and they are at rest. Observer A observes two objects falling from height H (A has same distance between the two objects). Does observer B will measure different ...
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Spin connection Kruskal space-time

Would anyone give me some references where I can find the components of the spin connection in Kruskal space-time?
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3answers
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Still confuse about tensor

In special relativity, a four-vector $\mathbf{x}$ in an inertial frame is related to $\mathbf{\overline{x}}$ through a Lorentz transformation $\mathbf{\Lambda}$: \begin{align} \overline{\mathbf{x}}...
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A concise definition of a frame of reference in Newtonian mechanics?

I've read Wikipedia's entry on frame of reference and also followed all of the references cited in the text (Salençon, Brillouin, Norton, etc) but I'm struggling to find any concise definition in all ...
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How to transform 6 DOF motion

Question Suppose there are two coordinate system A and B with gyro or vision sensor. Coordinate B is rigidly mounted on coordinate A with rotation $R$ and translation $t$ expressed in global ...
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1answer
61 views

Diffeomorphism in static spherically symmetric space-time

In a static, spherically symmetric space time we can choose the coordinates so that the metric takes the form: $$ds^2=-A(r)dt^2+\frac{dr^2}{B(r)}+C(r)\,[d\theta^2+\sin^2\theta\,d\varphi^2]$$ Sometimes ...
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How to transform a lagrangian after a change of coordinates?

Let's consider a generic lagrangian density in classical field theory: $$L(\phi(x),\partial_{\mu} \phi(x))$$ Now suppose I want to find the lagrangian for the same system with respect to another field ...
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In Bondi gauge/coordinate, how to obtain covariant derivative of the 2-dimemsional sphere using null projection operators?

In Bondi coordinate , the Bondi-Sachs metric is written as $$ ds^2=-\frac{V}{r}e^{2\beta}du^2-2e^{2\beta}dudr+r^2h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu) $$ Before any decomposition, the covariant derivative $...
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Motivation for the abandonment of inertial frames in general relativity [closed]

Inertial frames are at the core of special relativity. The laws of physics are supposed to be the same among them and free particles follow rectilinear paths in spacetime or simply stay at rest. Just ...
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Derivation of 4-velocity - where does this equation come from?

I'm a student working through a special relativity book. In a chapter of the book, they are deriving each of the components of a 4-velocity vector in terms of ordinary 3-velocity. (Here $\vec{x}$ ...
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What's the position vector for an ant on a sphere?

Imagine an ant on a sphere that perceives only two dimensions. Is there a coordinate system that allows the ant to describe the position with the position vector?
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General Relativity - Confusion between choosing basis (orthonormal & coordinate) and coordinate transformations

I am reading the book 'Gravity' by Hartle and presently I am at the section discussing orthonormal and coordinate bases. I am confused about a few points I had read previously and can't exactly ...
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Why the velocity $v$ is taken as value and not as definition in special relativity equations?

Why the velocity $v$ is taken as a value and the definition of velocity not applied on a relativistic equations? The equations of time dilation and length contractions as we know are $$L = {L_0}{\...
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Is four velocity always given by $U^{\mu} = d x^{\mu}/d\tau$?

I was taught that four-velocity is defined as $${\bf U} = \frac{d \bf x}{d\tau}$$ and that it has the components $$U^{\mu} = \frac{d x^{\mu}}{d\tau}$$ where $d\bf x$ is the four displacement and $\...
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1answer
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Some kind of slower time principle [duplicate]

I'm always trying to find underlying principles, like that the force is always directed toward a (locally) lower potential energy and alot of stuff like that. Recently I've begun to gain some layman ...
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1answer
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Simple Pendulum in Cartesian Coordinates

Riffing on the question in Simple Pendulum Why Generalized Coordinate Always Angle? , I'm trying to write down Newton's law for a simple pendulum in Cartesian coordinates. (I'm doing this as an ...