Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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43 views

What does spacetime interval really mean? [duplicate]

Is there any simple way to intuitively understand spacetime interval, proper time and proper length?
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2answers
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Why do the tangent vectors of coordinate curves correspond to the coordinate basis vectors?

In Schutz's General Relativity Chapter 5, after he defines vectors in the modern view as tangents to a particular curve, he states the relationship between bases of different coordinates as: $$\vec{e}...
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How different observers measure time?

Suppose I have a coordinate system, call it $S$, in which an observer $O$ is not moving, and $O'$ is moving with constant velocity and another coordinate system $S'$ where $O'$ is not moving and $O$ ...
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1answer
62 views

Equation of motions for simple pendulum in cartesian coordinates instead of generalized coordinates [closed]

I am trying to write the equation of motions for a simple pendulum but instead of writing them in generalized coordinates ($\theta$), I want to write them in cartesian coordinates (x, y), as I will ...
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1answer
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How to define the components of the Poincare group?

I know that the Poincare group/inhomogeneous Lorentz group can be defined as: $$ x^\mu = (t,-x) \\ t \rightarrow t^\prime = \gamma x + \delta t + b^0 \\ x \rightarrow x^\prime = \alpha x + \beta t + b^...
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1answer
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Oblate spheroid conductor [closed]

Oblate spheroid coordinates $(\xi, \eta, \theta)$ are related to cylindrical coordinates $(r, \theta, z)$ as follows: $r=\sqrt{\frac{(\xi+a^2)(\eta+a^2)}{a^2-b^2}}, \theta=\theta, z=\sqrt{\frac{(\xi+b^...
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1answer
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Eigenfunctions and eigenvalues of particle in 2D box

A particle in a 2D potential box has two degrees of freedom. It is bound by the infinite potentials at the boundaries. Our professor asked us to resolve this into its respective $x$ and $y$ components,...
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The effects of Lorentz transformation on shape

Imagine a solid 3D cube. Now imagine that this cube is traveling close to the speed of light. To what degree will the spatial geometric properties of this object (or in general of any 3D object) ...
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What do bending moments look like after coordinate frame rotation?

Suppose a 3D body had certain bending moments about the x, y, and z axes. What would the bending moments about a different coordinate system look like? In my case, I have restricted myself to a planar ...
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3answers
174 views

Why do the Christoffel symbols not transform as a tensor?

Let $(P, \pi, M)$ be a principal $G$ -bundle. Given $A \in L(G),$ we define the vector field $X^{A} \in \Gamma(T P)$ by $$ \begin{aligned} X_{p}^{A}: \mathcal{C}^{\infty}(P) & \stackrel{\sim}{\...
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1answer
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Transformation of coordinate in Lagrangian

Lagrangian for a Central force problem is: $$\mathcal{L} = \frac{1}{2}\mu(\dot{r} + r^{2}(\dot{\theta}^{2} + sin^{2}\theta\cdot \dot{\varphi}^{2})) - U(r)$$ We know that angular momentum is defined as:...
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Rotation Reference Change Transformation CRTBP Python

I have created the following script that will propagate the dynamics of the CRTBP set for the Earth-Moon system. I have the initial conditions set for a vertical Lyaponov orbit. The goal I am trying ...
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1answer
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Time derivative of $\rm{atan2}$ when $x=0$

I want to take the time derivative of the $\rm{atan2}$ function to calculate an azimuth rate in spherical coordinates, given position and velocity in Cartesian $xyz$ coordinates. $$\rm{atan2}(y, x) = \...
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3answers
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Does Schwarzschild metric in Kruskal-Szekeres coordinates admit asymptotic ($r \to +\infty$) timelike observers?

I thank in advance whoever will answer my question. Schwarzschild metric in Schwarzschild coordinates in $\mathbb{R}^{1,3}$ is [1]: $$ds^2=-\bigg(1-\displaystyle\frac{2M}{r}\bigg)dt^2+\bigg(1-\frac{2M}...
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$\frac{0}{0}$ from Curvilinear Dirac Delta

The definition of the Dirac Delta in an arbitrary curvilinear coordinate: $$\delta(\vec{r})=\frac{\delta(x^1-x^1_0)\delta(x^2-x^2_0)\cdot \cdot\cdot \delta(x^N-x^N_0)}{h_1h_2\cdot\cdot\cdot h_N}$$ ...
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1answer
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Lorentz transformation in system of unit [closed]

In a system of units in which the velocity of light $c =1$, $$x' = 1.25x - 0.75t$$ $$y'= y$$ $$z' = z $$ $$t'=1.25t -0.75x$$ is a Lorentz transformation. Why is this so? That is, how can we ...
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Mathematical description of systems of reference - classical mechanics vs special relativity

Notation: In the following, $E^n$ denotes an euclidean space of dimension $n$ (an affine space with inner product $\langle\,\cdot\,,\,\cdot\,\rangle$ on the translation space). The answer to this ...
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1answer
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Evaluation of vertex function with Feynman Parameters

On page 191 of Peskin & Schroeder, they show that after using Feynman parameters $x, y, z$, the denominator of the integrand of the vertex function is: $$D = k^2 + 2k(yq - zp) + yq^2 + zp^2 - (x+y)...
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1answer
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Coordinate transformation of boundary condition

Let us suppose a heat transfer problem inside a cylinder of radius $r_a$. If we neglect changes along $z$ and $\theta$ directions, i.e. only a cross section of the cylinder, the problem can be ...
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2answers
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Where does this formula come from? [closed]

I am doing revision for my module stellar & galactic astrophysics and have come upon this formula which I cannot seem to derive. Could someone please explain where it comes from? "For an ...
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Rotation of crystal structure to match with match another structure of same compound/polymorph

Say, I have two crystal structures of a particular organic molecule, the crystal structures are basically identical, apart from a rotation and redefinition of the lattice vectors/angles, a simple ...
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1answer
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$F=ma$ if $ a < 0$

The equation $F= m\cdot a$ is known, but what if $a < 0$? Is the force exerted negative?
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0answers
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Difference between covariant derivative, gradient, Lie derivative, Lie bracket, Poisson bracket, Christoffel symbols. With examples and cases

I have some knowledge but when I started to examine these notions I was confused. I have a sess in my mind. Do someone clear up the difference between covariant derivative, gradient, Lie derivative, ...
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1answer
100 views

Dirac delta in spherical coordinates. What I'm doing wrong?

I must show that the integral $$\frac{1}{(2\pi)^{3}}\int_{\vec{k}}d^{3}k\frac{\cos(\vec{k}\cdot\vec{x})}{\left({\sqrt{|\vec{k}|^2+m^{2}}}\right)^{s}}=\delta^{3}(\vec{x})$$ when $s=0$ by using ...
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1answer
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How to derive the bank angle of an aircraft from its roll angle and pitch angle?

From Young (2017) (https://onlinelibrary.wiley.com/doi/book/10.1002/9781118534786) it is stated that we can define the bank angle ($\Phi$) of an aircraft as the angle between its Y body axis and the ...
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1answer
72 views

Second law in polar coordinates?

Consider a pendulum whose string is replaced with a spring. It is a system which can be described neatly using polar coordinates, where your radial component is $ l = l_0 + \Delta l $ and your ...
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1answer
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Coordinates systems and frames of reference in classical mechanics

I have some doubts about the way frames of references are introducted in Arnold's mathematical methods of classical mechanics. It is said that, given a set $M$, then $\phi_1:M \rightarrow \mathbb{R} \...
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2answers
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How to find position vector in Kepler orbit?

As the title says, I am trying to determine the position vector $r$, knowing true anomaly, semi-major axis, angular momentum and eccentricity vector. There is an equation describe the distance to the ...
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3answers
97 views

Christoffel symbol and covariant derivative

I came across the Christoffel symbols via the geodesic equation, and I understand the extrinsic form and the intrinsic form and can prove that they are identical: extrinsic form: $$\Gamma^{j}_{~ik}=\...
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1answer
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Kinematics in polar coordinates

Take a particle in polar coordinate system to follow the equations: $$\theta=\omega t$$ and $$r=r_oe^{-\omega t}$$ Now, the radial acceleration will be- $$a_r=\ddot{r}-r\dot \theta^2$$ which we get as ...
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2answers
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Understanding dependent/independent variables in physics

How does one determine the independent and dependent variables? What do the terms mean? Can they be derived from a formula? For example I saw in a textbook $F = k\Delta l$, Hooke's Law, that $F$ is ...
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2answers
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Number of variables in the Hamilton-Jacobi equation

In Goldstein's Classical Mechanics, while introducing the Hamilton-Jacobi equation, he argues that the equation $$H(q_1, ... , q_n; \frac{\partial S}{\partial q_1}, ..., \frac{\partial S}{\partial q_n}...
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2answers
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Translation of coordinates to generalised coordinates

The translation form $r_i$ to $q_j$ language start forms the transformation equation: $r_i=r_i (q_1,q_2,…,q_n,t)$ (assuming $n$ independent coordinates) Since it is carried out by means of the ...
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2answers
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Canonical transformation such that Hamiltonian of a freely falling body becomes $H'(P,Q)=P$

Can someone please help me with this problem I am unable to find a suitable generating function? The question says: To find a canonical transformation such that Hamiltonian of a freely falling body ...
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3answers
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Linear algebra as a gauge theory

Is linear algebra a gauge theory? Is the gauge transformation a change of basis? This was the explanation that I received: "Take the principal bundle to be the frame bundle $LM$ of your space $M$...
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1answer
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How do you find the proper separation between two spacetime points?

Suppose you had two points in space-time A and B, where A = (t1, χ1, θ1, φ1) and B = (t1, χ2, θ1, φ1). How would you use the FLRW metric to find the proper separation? In this case the points occur ...
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1answer
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Computing total power from $x$, $y$, $z$ components of Poynting vector

I have (real parts) of $x$, $y$, $z$ components of Poynting vectors on the surface of a sphere. Since the total outward power flow from the sphere involves integrating the normal component of a ...
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0answers
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What are the Lorentz Transformations between polar coordinates? Or can Lorentz Transformations be Non-Linear?

This question rises from the comments on @G Smith's answer's to this question https://physics.stackexchange.com/a/603032/113699 Precisely I was trying to understand the Lorentz Transformations between ...
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0answers
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Momentum Vectors in Bondi coordinates

In the Bondi-Sachs formalism, we can define the notion of 'retarded' time via a coordinate transformation of the usual Minkowski metric $$ d s^{2}=\eta_{\mu \nu} d x^{\mu} d x^{\nu}=-\left(d x^{0}\...
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2answers
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Lorentz Transformations for Polar coordinates or Inertial Frame in Polar Coordinates

Do polar coordinates define an inertial frame or not? Everywhere in GR, the authors of all the books talk about bring the metric to diag(-1, 1,1,1) which would show that a Local Inertial Frame exists ...
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What is meant by coordinate time? Isn't it time measured on a clock? If not then what does it measure?

In this question- Is proper time equal to the Invariant Interval or the time elapsed in the Rest Frame? @Dale in the comments says- no $dt$ is never physical time. It is always coordinate time. The ...
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1answer
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Tensor transformation Formula Proof

Ok so basically I am trying to prove that the following expression: Can be written using matrices like this: Any suggestions on how to approach this?
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2answers
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Why is the cross product of vectors in the given image like this? [duplicate]

I know the question seems kind of unclear but why (in the attached image) the cross product like this, I hope someone can clear my doubt.
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1answer
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Why is the solution of the radial Schrödinger equation valid at $r=0$?

The Schrödinger equation for a particle in a central potential is $$\left[\frac{p_r^2}{2m}+\frac{\ell(\ell+1)}{2mr^2}+V(r)\right]\psi(r,\theta,\varphi)=E\psi(r,\theta,\varphi).$$ This gives solutions ...
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1answer
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Is light affected by inertia?

A very popular way used by teachers to explain Einstein's theory that The speed of light is a Universal Constant, is to use an example as follows: Two observers moving relative to each other [let's ...
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1answer
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Lorentz transformation of four vector field

For a 4-vector field $V^\mu (x)$, the Lorentz transformed 4-vector field $V'^\mu(x')$ can be written as $$V'^\mu(x')={\Lambda^\mu} _\nu V^\nu(\Lambda^{-1}x')={\Lambda^\mu}_\nu V^\nu(x).$$ This can be ...
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2answers
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Understanding proper-time as “evenly spaced gradations” on the worldline?

In this video series on relativity, proper-time is explained as "evenly spaced gradations" along a particle's worldline. Here is a screenshot: Suppose I carry a clock with me that ticks at ...
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1answer
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Why is a metric necessary in SR?

Simple question since I don't fully understand the purpose a metric fulfills when looking at relativistic kinematics for example.
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2answers
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Can I use the spacetime interval even if origins are not equal at $t=t'=0$?

Can I use the relativity formula $(\Delta s)^2 = (c \Delta t)^2 - (\Delta r)^2$ even when observers did not start at the same origin at $t=t'=0$? I know this is constant, but is it constant even if ...
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1answer
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Euler angles - geometry

Hello, I don't quite see why should the angle between $\hat{\dot{\theta}}$ and the projection of $\hat{\dot{\phi}}$ onto the $x_0$, $y_0$ plane be a right angle. Does it have something to do with pure ...

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