Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
2,669
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Is the massive Fierz-Pauli action invariant under changes of coordinates?
The action for linearlized gravity on a curved background has the form
$S_{LinGrav}=\frac{1}{16 \pi G} \int d^{4} x \;\sqrt{-g}\;\left[-\frac{1}{4} \nabla_{\rho} h_{\mu \nu} \nabla^{\rho} h^{\mu \nu} +...
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Decomposing velocities in the tetrad frame
Background
At each point in a manifold we can find an orthonormal set of basis vectors $\hat{\mathbf{e}}_\alpha$ such that
$$
g(\mathbf{e}_\hat{\alpha}, \mathbf{e}_\hat{\beta}) = \eta_{\alpha \beta}
$...
- 604
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What is the relevance of the Lorentz factor in general relativity?
The Lorentz factor is ubiquitous in Special Relativity and is used to express "how much the measurements of time, length, and other physical properties change for an object while that object is ...
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Double pendulum damping and spring forces
If I have the equations of motion for a double pendulum (from https://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf), can I include a time dependent damping and spring force ...
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Are expressions like "x axis," "x dimension," "x direction," "x plane," "x boundary," etc. all hyphenated? [closed]
I'm not a native speaker of English. I'm currently writing a journal article and want to make sure I follow all conventions properly. Are expressions like "x axis," "x dimension," &...
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Proof of the valence $\lambda$ of a canonical transformation equaling its Jacobian determinant [closed]
Let $Q(q,p),P(q,p)$ be a canonical transformation with valence $\lambda$. The following is intended to be a proof of the following relation:
$$\lambda = \frac{\partial(Q,P)}{\partial(q,p)}.$$
Let $F(q,...
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Metric tensor determinant under coordinate transformation
I've been studying GR through Wald's and Carroll's books, and I've been trying to derive one expression.
$$g(x^{\mu^\prime}) = \left|\dfrac{\partial x^{\mu^\prime}}{\partial x^{\mu}}\right|^{-2} g(x^\...
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Where is Schwarzschild solution valid?
The extrinsic Schwarzschild solution is a vacuum solution, meaning that it is valid for regions of spacetime where there is no matter or energy. This seems to imply that the Schwarzschild solution is ...
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Why is clock synchronisation such a big deal in physics?
I was reading Classical Mechanics : The theoretical minimum by Leonard Susskind, and he says
Assume that two clocks at different places can be synchronised.
I don't understand why one should do that....
- 369
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Net force acting on current loop in $B$ field linearly dependent $z$
Suppose there is a magnetic field $\overline B$, only in the $z$ direction and dependent linearly in $z$ coordinate e.g. $$\overline B=(0,0,B_0*z)$$ $B_0$ is a constant.
And a circular current loop ...
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What is the Schwarzschild metric in cylindrical coordinates?
I was researching online for different metrics of spacetime out of curiosity, and I found one that was said to be Schwarzschild metric in cylindrical coordinates:
$$ds^2 = -\left(1-\frac{r_s}{r}\right)...
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Tortoise coordinate transformation
The differential form $dr$ can be written $\left(1-\dfrac{2GM}{r}\right)dr^*$ where $r^*$ is the tortoise coordinate. Writing the Schwartzchild metric then gives
$ds^2$ = $\left(1-\dfrac{2GM}{r}\right)...
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Derivation of Hamilton-Jacobi (HJ) Equation
In the Derivation of Hamilton Jacobi Equation, I didn't understand the bold parts:
we can write (1) formally as,
$$
\frac{\partial F\left(q_i, Q_i, t\right)}{\partial t}=-H\left(p_i, q_i, t\right)=-H\...
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Sagnac effect and interval invariance
In this article there is one thing I don't get: the author considers (page 9) transformation of the coordinates $$(t,r,\theta)\mapsto (t,r,\theta+\omega t)$$ and then apparently uses invariance of the ...
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On state transformations and the requirement of space-time invariance in (non-relativistic) quantum mechanics
I am trying to follow the development in Ballentine's Quantum Mechanics: A Modern Development but am struggling a lot. Please excuse my attaching of a picture of the development, but my question quite ...
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Vector field coordinate transformation
On Carroll's spacetime and geometry book, page 67, the book gives the component form of vector field commutator
$$[X,Y]^\mu=X^\lambda\partial_\lambda Y^\mu- Y^\lambda\partial_\lambda x^\mu \tag{2.23}$$...
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Physical interpretation of linear non-Lorentz coordinate transformations
In school we are taught that Lorentz transformations relate coordinates of observers in uniform motion. Later we are taught that nonlinear coordinate transformations are associated with acceleration. ...
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Aren't the virtual work/virtual power principles in mechanics simply solving for the vector differential equation of motion in a preferred direction?
My conceptual understanding of the virtual work/virtual power principles is that, by hypothesizing "virtual displacements"/"virtual velocities", one can solve the equations of ...
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What are good books/chapters of books or articles to study canonical transformations in quantum mechanics at a graduate level?
I am looking for any kind of sources about canonical transformations in quantum mechanics in the operatorial formulation of the theory and its connection with the classical canonical transformation ...
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Is it meaningful to draw $ct$ and $x$ axes by two lines intersecting at right angles?
In the Cartesian coordinate system, the x-axis is really perpendicular to the y-axis, by construction. Also, under a rotation of the coordinate system, the transformed coordinate axes $x',y'$ remain ...
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How the equation of a projectile represents a parabola? [closed]
I am not able to prove that equation of motion of a projectile is parabola. The book simply says the given below is the equation of a parabola but does not clarify it
$$y= {\tan\theta}x - \frac{g}{2(...
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Why is cosmological time unique?
According to the definition I have encountered for the concept of cosmological time, it is defined in the following way:
The cosmological principle states that, at each location in the universe, it ...
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Understanding generalized coordinates [closed]
I understand that for each of N particles they have a position:
$$\vec{r}_i=\vec{r}_i (q_1, q_2, \dots, q_n, t)$$
where $q_1, q_2, \dots, q_n$ are n generalized coordinates. However I don't understand ...
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What should be the definition of a comoving frame in phase space?
In short, I think there are two types of comoving frame when talking about distribution function, since it is defined in phase space. Which one should be the real one? I suppose this question is ...
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General Lorentz transformation between two four-vectors
I am reading a paper that makes the claim that
$$
\Lambda^\mu {}_\nu
=
g^\mu{}_\nu
-
2 \frac{(V + W)^\mu (V + W)_\nu}{(V + W)^2}
+
2 \frac{W^\mu V_\nu}{V^2}
$$
is a proper Lorentz transformation,
...
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Why is Lorentz Transformation defined with one super and one sub index?
I came across the Lorentz transformation in tensor form, usually written as
$$\Lambda ^\mu _{\nu}$$
I understand that the first index usually corresponds to rows and the second to columns, and while I ...
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Contravariant Components (Susskind's book)
In his book about SR & classical field theory, Susskind generalizes from the differential of $X'$ (function differential) to any 4-vector. I got stuck there trying to figure out why it is ...
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How to deduce the general boost transformation matrix? [duplicate]
The general matrix for a boost in an arbitrary direction, is given by:
$$\Lambda (\vec{v})=
\begin{pmatrix}
\gamma & -\gamma\beta_x & -\gamma\beta_y & -\gamma\beta_z\\
-\gamma\beta_x & ...
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How are co-ordinate systems built physically in curved space-time?
How do we physically choose a co-ordinate system for making astronomical observations?
In a special relativistic system, the definition of relative velocity, clock synchronization is well understood ...
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What prevents the line element being Minkowskian in the vicinity of a point mass?
This is probably a naive question and I'm missing something really simple. The Schwarzschild solution has been constructed in consideration of the following requirements:
The field equations $ \frac{\...
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Why is the Centripetal Force Positive, if the acceleration is negative? [closed]
If $ma_r = F $, but F is positive and $a_r$ is negative, wouldn't this mean a negative mass? I understand why centripetal acceleration is or is defined to be negative, since it is in the opposite ...
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Why can we put these conditions on coordinates of worldsheet?
https://www.asc.ohio-state.edu/mathur.16/classicalstring.pdf
At first, I write some notations I need here.
$I=[0,1]$, $M$ means $(1,3)$ Minkowski space, smooth map $X:I\times I\to M$ is timelike ...
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Showing that the contravariant base vector transforms as a vector [closed]
I wanna show that $Z^a$ is indeed a contravariant vector in the same way I showed that $Z_i$ is indeed a covariant vector (see attached image).This is how I define $Z^a$ : $Z^a = \frac{\partial y^a}{\...
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Vectors and One-forms in Cylindrical Coordinates and the Angular Momentum
The angular component of the velocity of a particle in cylindrical coordinates has different units if we consider the vector component $v^{\phi}$ or the one-form component $v_{\phi}$:
$$ v^{\phi} = \...
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Frame of reference of a freely falling observer in General Relativity
Suppose I have an Einstein manifold (a manifold with a metric that solves the Einstein field equations. We can take $\Lambda = 0$ for simplicity in this example). The worldline of a freely falling ...
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Why do we consider the whole process, taking a single moment? Universality of the vector equation
I have considered a simple model for describing the elastic force vector of a spring. First, I chose a reference frame in an arbitrary way, then I drew the necessary vectors, we get a ratio that ...
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Deriving Lagrange equation with constraint
I'm having a hard time understanding the derivation of Lagrange equation from Newton's law when there is constraint (I'm ok with the basic case where there is only kinetic energy and potential ...
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Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$U= \frac{1}{2} \sum_{j,k} A_{jk} q_j q_k \quad \quad T= \frac{1}{2} \sum_{j,k} m_{jk} \dot{q}_j \dot{q}_k $$
and the lagrangian is
$$ \frac{\partial U}{\partial q_k} - \frac{d}{dt} \...
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Visualizing the conformal compactification diagram of $G$
I asked a question a year and 3 months ago on mathstackexchange but after 3 bounties and still no answer I've decided to try here. Here's the link: conformal compactification.
Construct a conformal ...
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To derivation the 2D flow ${\phi }_{2}-{\phi }_{1}$ and $ {\psi }_{2}-{\psi }_{1} $ in cylindrical coordinates
I try to derivation the 2D flow ${\phi }_{2}-{\phi }_{1}$ and $ {\psi }_{2}-{\psi }_{1} $ in cylindrical coordinates as follows, if there is a wrong concept of physics,please say some thing about it.
(...
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Is frame of reference a point of view? [duplicate]
The definition of frame of reference I found is :"It is just a coordinate". But in solving problem, my teacher always uses frame of reference by considering it at rest (although it is moving)...
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Rigid body motion in Arnold's book
During the study of the motion of a rigid body, in Arnold's book, two coordinates systems are introduced: one is fixed $k=\{O',\hat e_1',\hat e_2',\hat e_3'\}$ and one is inside the rigid body $K=\{O,\...
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Choosing coordinates in Lagrangian Mechanics
Consider the problem of a hoop rolling down an inclined plane, with the plane sliding (frictionless) in a horizontal motion.
I don't know how to choose the generalized coordinates for this system. In ...
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What comprises of a 'sensible' coordinate transformation?
I am doing the course on general relativity at my university and have been struggling with covariant and contravariant vectors. I understand that components of contravariant vectors transform in a way ...
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Canonical Transformation of Poisson Bracket [closed]
In Goldstein section 9.4(pg 381) it tells us that for a Hamiltonian that is not explicitly time dependent, transformations of $Q = Q(q,p), P = P(q,p)$ are canonical if $$\frac{\partial Q}{\partial q} =...
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Fraction with components of Lorentz transformation
I want to show how partial derivative transforms under a Lorentz transformation.
Since the partial derivative has a fixed definition with respect to the $x$-coordinate it stays unchanged: $\partial_\...
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How does one covert from Cartesian coordinates to Boyer-Lindquist coordinates?
I am new to physics stackexchange, but I have a question which I seem to have not been able to find an answer to.
I already know that the transformations from Boyer-Lindquist coordinates to Cartesian ...
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2
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How to understand the quadratic form of kinetic energy with $\dot{q}$ coefficients?
Kinetic energy can be written as:
$$ T=\frac{1}{2}\sum_{\alpha=1}^K\sum_{\beta=1}^K a_{\alpha \beta}(q)\dot{q}^\alpha \dot{q}^\beta$$
Where the object $a_{\alpha \beta}$ is a certain tensor. How to ...
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Coordinate transformation and gauge choice in gravitational waves [duplicate]
Assuming weak field : $g_{\mu\nu}=\eta_{\mu \nu}+h_{\mu\nu}$ and considering terms only linear in $h$
We get christoffel sybol as $\Gamma^\mu _{\nu \lambda}=\frac{1}{2}\eta^{\mu \alpha}(\partial_\nu ...
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2
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How is Coriolis acceleration in polar coordinate, different from Coriolis acceleration due to observation in non-inertial frame of reference?
In Kleppner and Kolenkow's book: An Introduction to Mechanics, on page 34 (pasted below) on the topic titled "Acceleration in Polar coordinates", it has been mentioned that:
"when $r$ ...
