Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
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Fixed coordinate frame as limit of rotating coordinate frame
I have a question about a fixed coordinate system as limit of rotating system. Consider for example a pendulum. The Lagrangian in the rotating frame is given by
\begin{equation}
L(\mathbf{r}, \mathbf{\...
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Asymmetry in Hamilton Equations
I noticed that in deriving Hamilton equations from the total deriveative of the Hamiltonian with respect to time, for the first equation $$\frac{dx_k}{dt}=\partial_{p_k}H$$ we do not need Lagrange's ...
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Why do we need invariants to represent real life quantities?
Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical ...
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Gauge Transformation of the Inverse Metric
The gauge transformation of the metric tensor is supposed to be
$$\delta_\xi g_{\mu\nu}=\xi^\rho\partial_\rho g_{\mu\nu}+\partial_\mu\xi^\rho g_{\rho\nu}+\partial_\nu\xi^\rho g_{\mu\rho}$$ with $\...
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Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta?
Given the Hamiltonian $H(x,p)$ of a system. Is there always a coordinate transformation such that the new Hamiltonian is $K(x',p')=K(p')$?
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Use of direction of downward velocity in equation of motion
[NOTE: I am not asking anyone to solve the question below but to point out where I might be wrong]
Q- A skier jumps from a horizontal track and lands on a steeper track with a launch angle of ∅=11.3°(...
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How to find angular velocity of a spherical object in spherical coordinates
How angular velocity vector is calculated here?
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Can I contract index in this expression?
I'm reading Carrol text on general relativity, on page 96 they arrive to the term
\begin{equation}
\frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\...
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Why does the Schrödinger equation work so well for the hydrogen atom despite the relativistic boundary at the nucleus?
I have been taught that the boundary conditions are just as important as the differential equation itself when solving real, physical problems.
When the Schrödinger equation is applied to the ...
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Torsion and Curvature in general relativity
in Landau's book, field theory, the famous physicist stated, "Because of equivalence principle， there should be a 'Galileo' frame, in which the Christoffel symbols should be zero, thus the torsion ...
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Variations of tensors are tensors?
Recently I posted a question about variation of metric.
I thought I understood it and talked with my friend about it.
After that he said he's not convinced because he can't prove variation of metric ...
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Why are only the Lagrangian and Hamiltonian used in mechanics? [duplicate]
Why is it that we have a closed set of four functions, connected by Legendre transforms, in thermodynamics but nobody ever mentions but two of the corresponding functions in mechanics?
I've read that ...
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Misunderstanding of lowering indexes using Euclidian metric
One may define a vector field in $R$ and see how its components transform under a basis transformation.
$ v= v^{u}\partial _{u} $
In principle, the components transform as contravariant such that ...
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Area of Kerr-Newman event horizon
I want to calculate the area of event horizon for a Kerr-Newman black hole by using boyer's coordinates.
I searched a lot from web, but I could not find any information about calculating event ...
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Choosing the sign for speed in Lorentz transformations
I am studying the Lorentz transformations and while I [believe] I understand the principle of it (you’re sorta “translating” from one frame of reference to the other), I am having a lot of trouble ...
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Polar form of Newton's Second Law
Can someone please show the steps for deriving the total energy of a particle from the Polar form of Newton's Second Law?
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Mcintyre Quantum Mechanics - Angular Momentum Conservation
I have two questions regarding this topic.
1.
I captured the part of the section I'm referring to. If I didn't my question would probably not make sense. My first question is to the second to last ...
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What does a negative acceleration mean? Is the object slowing down, changing direction, or both?
I am confused about such things as negative velocity, acceleration, and displacement and what the negative indicates.
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How to determine if a tensor is covariant or contravariant?
In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a ...
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In the Schwarzschild Metric does $\sin^2\theta{\Delta}\phi^2={\Delta}\phi^2\sin^2\theta$?
In the Schwarzschild Metric as the spacetime interval between two points in spacetime approaches $0$ for any ratio between the length of time and space the spacetime interval between the points in ...
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Difference between coordinate systame and frame of reference in relativity
I have taken a relativity course, and am not quite clear as to whether the notion of a frame of referene is independent of the coordinate system. At the moment, I think that any frame of reference (of ...
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Do canonical transformations form a group?
In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
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Radial component of velocity at extreme distances
Suppose I am given that a planet's position with respect to some star is of the form $\textbf{r} = r\textbf{e}_{r}$. Then of course $\textbf{v} = \dot{r}\textbf{e}_{r} + r\dot{\theta}\textbf{e}_{\...
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How to calculate spherical coordinate components of dipole field?
I understand well enough how to calculate the radial and tangential components in spherical coordinates at a point due to a magnetic dipole field using the magnetic potential gradient ($\...
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Line Element Transformation
This is just something that I've made up to see if I understand the method.
If I have the line element:
$$ds^2 = dr^2 + r^2\,d\phi^2$$
and I want to carry out a transformation with $r = \dfrac{...
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Minkowski diagram
I have not well understood the picture of geogebra regarding the angle of time (t') that is inclined compared to (t) of 26.57°angle .
In the picture we see that the velocity is setted at 0.5c, for ...
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Lorentz-transformation
I don't understand how to derive the matrix representing the Lorentz-transformation given two systems S and S':
$$x' = \Lambda x$$
these transformations do not leave the differences $\Delta x^\mu$ ...
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Is a change of coordinates the same thing as a change of chart on a manifold?
I am familiar with coordinate transformations in the common case. (Say, polar to cartesian and back)
I have recently been introduced to the definition of a differentiable manifold. Is it correct to ...
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Differential geometry: If $\vec v = v^i \vec e_i$, then why is $\vec r = r \vec e_r$ in spherical coordinates?
In differential geometry (and later carried over to GR) any abstract vector $\vec v$, exists on its own vector space.
We can then choose to represent this vector in a coordinate basis $\vec v = v^i ...
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Metric tensor: Why relate it to Cartesian/Minkowski coordinates?
Why does the metric tensor always relate to cartesian coordinates?
Let's take the simple case for the metric tensor in 3D-space without a time dimension,
$g_{ij}=
\begin{bmatrix}
1 & 0 &...
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Do objects move in 2 directions at once?
If a velocity vector of an object can be divided into an x and y component relative to a second object's position, and both objects have gravity that attracts both objects to each other. We then know ...
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Is it possible to derive $2\times 2$ Lorentz transformation matrix from only eigenvectors?
As a preface, I am somewhat familiar with year 1 linear algebra but not too familiar with how one makes the connection to Lorentz transformation matrices so I apologize if the answer is obvious.
One ...
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Coordinate transformation of basis vectors
The question
Let $e_a$ be the coordinate basis vectors in a manifold described by coordinate system $x^a$. The vector displacement between two nearby points is given by
\begin{equation}
ds=dx^ae_a=dx'...
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Proper time of an object crossing event horizon in Kruskal coordinates
So I am reading a paper on a certain black hole paradox. The specifics actually don't matter, but if you want context (p16): black hole thought experiment. An object falls into a black hole. The ...
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A paradox about canonical transform preserving Poisson bracket?
Let $q,p$ denote the position and momentum. Consider a transform generated by $g$:
$q' = q + \epsilon \{q,g\}---(1a)$
$p' = p + \epsilon \{p,g\}---(1b)$
Then:
$\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
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Formal Definition of Dot Product
In most textbooks, dot product between two vectors is defined as:
$$\langle x_1,x_2,x_3\rangle \cdot \langle y_1,y_2,y_3\rangle = x_1 y_1 + x_2 y_2 + x_3 y _3$$
I understand how this definition ...
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Do Bianchi identities hold in all coordinates?
I understand by expanding out the Riemann tensor, that the Bianchi identities can be derived within a local inertial frame (LIF) by taking the partial derivatives of the Riemann tensor relations in a ...
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Confusion Einstein notation polar coordinates
I'm having issues using Einstein notation in polar coordinates in flat space, I must be missing something basic.
Consider the following example. Take the following metric on a 2+1 spacetime; $ds^2 = ...
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Curved space-time and metric tensor
I'm studying about curved spaces and I read that a manifold is flat if there a coordinate system such that the metric tensor is constant everywhere.
Then I also read that when the space-time tensor ...
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How to show the metric is unchanged under an orthogonal transform?
Say we have the transform such that $x^i \rightarrow (x')^{i'}=M^{i'}_i(x)$
where $M$ is an orthogonal rotation matrix
I've been asked to show that a general metric $g_{ij}(x)$ invariant under the ...
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Journey away and back at close to the speed of light $c$
If a person drives away from a clock tower at a speed close to
$c$ for one whole day, and then drives back to it at the same speed (for another day), what would he see during each of the journeys?
I ...
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Gram-Schmidt procedure for generating orthogonal generalised coordinates
For a general natural system, the kinetic energy part of the lagrangian may be written as $$T = \frac{1}{2}\sum_{ij} a_{ij}(q_{1}, q_{2}, ..., q_{n})\dot{q}_{i}\dot{q}_{j}.$$ For $n = 2$, $$T = \frac{...
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How to properly align my relaxed molecule onto my gold electrode?
the question is described below since it can be considered a chemistry or physics question: How to translate from one plane to another? originally posted on mathematics SE
I am trying to get ...
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Why the zero-order term in a variational transformation of coordinates should be identically the same as the old coordinates?
In the Ref.[1, page 61] the author proposes that transformations between two coordinate systems can be described by a continuous parameter $\varepsilon$ such that when $\varepsilon=0$ the original ...
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Is the concept "space" actually needed?
I started making my mind around space and time and recently came to a point where I wondered if the concept of "space" is actually needed to describe physical processes at all and not just some ...
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Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
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Thermodynamical conjugate variables
In thermodynamics the potentials are typically only a function of 2 variables, say
$$U=U(S,V)$$
with entropy $S$ and volume $V$. I see that conjugate pairs $S,T$ or $p,V$ always have the unit of ...
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Angular velocity of a pendulum in Cartesian coordinates
Hello I have a problem how to write down equations for the pendulum correctly. Say I would use Cartesian coordinates $x, y$ representing the position of the mass. Then the velocities would be usually ...
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Length contraction of an accelerating rod
Let us assume a rod of length, $4l$, initially at rest. The coordinates are defined s.t - (assuming uniform density)
$$X_{back-end}(0) = -2l$$
$$X_{front-end}(0) = +2l$$
$$X_{mid-point}(0) = 0$$
...
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