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Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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1answer
59 views

Having some trouble with acceleration in polar coordinates

So, I solved a question about acceleration in polar coordinates, but most people in my class (Classical Physics, first year at university studying Physics) disagree with my answer. So the question is ...
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1answer
241 views

What are the points in spherical coordinates?

Let's use the spherical coordinates so that $\vec P=(r, \theta, \phi)$. In this context i've read that it's possible to write $$\vec P'=\vec P + d\theta\ \vec e_\theta+d\phi\ \vec e_\phi+dr\ \vec e_r$$...
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27 views

Metric tensor $g$ for static gravitational field referred to static coordinate system

Assume there is static gravitational field. I want to deduce that there exists a coordinate system where $$g_{m0}=0, \quad m=1,2,3.$$ Is this a reasonable result? Why would it be contradictory if ...
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0answers
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Differentiation of metric tensor in new coordinate

I want to understand the explicit meaning of $g_{\mu'\nu',\lambda}=0$ where unprimed coordinates are coordinates of the the original coordinate systems and primed ones are for new coordinate system. ...
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1answer
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What's the meaning of the coordinates if we use a polar coordinate system?

In general, the coordinates of a vector are defined as the projections of it onto the coordinate axis. Moreover, in a polar coordinate system, the basis vectors $\hat e_\phi$, $\hat e_r$ depend on the ...
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2answers
70 views

Can we detect a cyclic coordinate by just inspecting the Lagrangian?

I'm reading through Susskind-Hrabovsky's Theoretical Minimum. On page 126, where they are talking about cyclic coordinates, an example is given: Suppose two particles moving on a line with a ...
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57 views

Trajectories in space

I want to say that a set $T$ of vectors in $R^{\,4}$ is a "trajectory" if there is an interval $I$, and continuous functions $x,y,z$ on $I$, such that $T$ is the set of $[t,x(t),y(t),z(t)]$ for $t$ in ...
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2answers
82 views

Direct derivation of point-like particle metric in GR

The usual way to derive metric of a point mass in general relativity is (to my knowledge) based on assuming specific form of the metric that reflects spherical symmetry and independence on "time" (...
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1answer
39 views

Lagrangian form of acceleration

Reading the Wikipedia article on Lagrangian mechanics I have problems to understand one basic proposition in the motivation of Euler-Lagrange equation. The article says: For me is unclear how the ...
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2answers
51 views

How to interpret negative time in Lorentz transformation?

I am somewhat confused about how to interpret negative time in Lorentz transformation. In the usual case of two reference systems S and S' where the distance X (the one that measures S) to an event, ...
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1answer
36 views

Interval invariance under galilean transformation

If we have the classical distance between two points in euclidean space, we define: $ S=(x^1)^2+(x^2)^2+(x^3)^2=\eta_{\mu \nu}x^\mu x^\nu $ If we want to make a coordinate transformaction of the ...
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1answer
50 views

Geodesic Equation from Coordinate Transformation

Let $\xi^a$ be the usual coordinates and $x^\mu$ the new coordinates, both flat. Now we know that since the metric is flat, $$ \frac{d^2\xi^a}{d\tau^2} = 0 $$ $$ \Rightarrow \ \frac{\partial}{\...
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1answer
60 views

Why does the choice of origin affect the wave function?

Consider a particle in a 1D-Box. The box ranges from x=0 to x=l in the first case, and from x=-l/2 to x=l/2 in the second case. The only difference I see is that the origin is shifted. On solving ...
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1answer
37 views

numerically integrating a trajectory in polar coordinates

So I've reduced my problem to not being sure how to integrate a trajectory in polar coordinates. Suppose I have a free particle and I express its Hamiltonian thus: $H =\eta_{ij}P^iP^j,$ where $\...
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0answers
54 views

How is the idea of an orthogonal matrix equivalent to $\lambda_{ij}$?

In my classical dynamics class, my professor showed how a vector under rotational coordinate transformation behaves. During the lecture, he used $\sum R_{ij} \mathbf{V_j} = \mathbf V_i'$ where $\...
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1answer
79 views

Different versions of the Robertson-Walker Metric

One form of the Robertson-Walker metric is $$ds^2 = c^2dt^2 - a(t)^2[d\chi^2+ S_k(\chi)^2(d\theta^2 + \sin^2\theta ~d\phi^2)]\tag{1}$$ $$\\$$ Considering curvature, where k = 0 , +1, -1 for flat, ...
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3answers
125 views

Notion of Present

Can't I sync all watches in spacetime and call this time slice the present? In Carlo Rovelli's book he tried to explain that the notion of the present is local only, which I could not follow.
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1answer
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Does changing the angle of a pendulum also shift the coordinate plane w.r.t which we give rectangular components to the $mg$ vector?

So given a simple pendulum, which makes an angle of 0 with the vertical axis in it's resting position.Now the pendulum is moved to a side by an angle $\theta$ with the vertical axis. The components of ...
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0answers
105 views

Hamilton-Jacobi Equation: Why does any $F(q,Q,0)=f(q,Q)$ lead to a solution?

I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0 $$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
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1answer
54 views

Question about computing Christoffel symbols

I am trying to calculate the Christoffel symbols in polar coordinates, and I am confused on one step. Given that I am here, for example: $$\Gamma_{r \theta}^{\theta}=\frac{1}{2} g^{\alpha \theta}\...
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2answers
118 views

Does the relative speed of time mean there is less energy where time is slower?

Time runs relatively slower near a planet than in outer space. Does this mean that there is less energy near the planet? Is there a relationship between energy and the speed of time? If so, this ...
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1answer
35 views

Non-holonomic constraints, degree of freedom and generalized coordinates

If a system has $N$ coordinates and $M$ number of holonomic constraints then number of degree of freedom $=N-M$ and generalized coordinates $=N-M$ too. But if there are $k$ non-holonomic constraints ...
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2answers
113 views

Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

What's the physical reason that all (inertial) observers agree on the value of the spacetime interval $$ds^2 = (c dt)^2 - dx^2 - dy^2 -dz^2 \, ?$$ What would be the physical implications if different ...
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1answer
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“Newtonian limit” property of special relativity

Books say that special relativity is indistinguishable from Newtonian mechanics when the speed of the primed frame ($v$) is small compared to the speed of light ($c$). This is what I mean by the "...
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3answers
418 views

Global Frame of Reference in General Relativity

People have been saying that a global frame of reference does not exist in General Relativity. However, from Wikipedia: In Schwarzschild coordinates ${\displaystyle (t,r,\theta ,\phi )}$ the ...
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1answer
51 views

Why is the force of gravity positive for an oscillating spring?

When analyzing the movement of a weight attached to a spring, many sources set up the force equation using newton’s second law as follows. $$mg-k(L+x)=ma$$ where $L$ is the length that the mass $m$ ...
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51 views

Math question about point transformations

I am trying to prove the classic problem to showcase Lagrangian's scalar invariant property. Namely, that if you have $x_i = \{ x_1, ...., x_n; t \}$ , you can then represent $L(x_1,....,\dot{x_1},.....
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2answers
55 views

Designating a coordinate system with multiple objects [closed]

So I am slightly new to physics, but am thoroughly enjoying the contextual thinking changes that physics brings about. My question is regarding coordinate system designation, on a group of individual ...
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5answers
764 views

Intuitive methods for representation of Cartesian Coordinates in terms of Spherical Coordinates as basis [closed]

I was going through Griffith's Electrodynamics and came upon an example, where he used that, $$\cos\theta \ \hat{r} - \sin\theta \ \hat{\theta} = \hat{z} $$ Now I admit I was confused for a while ...
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What is topocentric horizon coordinate system?

I was going through the book Orbital Mechanics by Howard Curtis and in the book he talks about topocentric horizon frame and its angular velocity about an inertial frame? What does it mean and how do ...
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3answers
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The role of Lorentz tranformations

My questions concern the role of Lorentz transformations in Special Relativity and General Relativity, as described in the following fragment of the series of GR lectures: https://www.youtube.com/...
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Frame-referenced time derivatives

I have reviewed and am familiar with the similar questions asked and answered previously on this forum, the various Wikipedia references and the derivation used by Kane and Levinson [2]. The ...
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1answer
76 views

Relativistic rockets paradox [closed]

The next problem is, as I understand, a spin on the ladder paradox which is extensively documented everywhere. There are two rockets ($A$ and $B$), of length $L_p=1$m which are traveling in opposite ...
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1answer
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Does the component of vector depend on the orientation of the axes?

the question was: A situation may be described by using different sets of co-ordinate axes having different orientations. Which of the following do not depend on the orientation of the axes? (...
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3answers
103 views

Time difference as a result of Lorentz boost

So I am given two clocks A and B moving in $S'$ frame with a velocity $V$ relative to $S$ frame. The two clocks are separated by a distance $L$ and are synchronized in $S'$ frame. The objective is to ...
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1answer
77 views

Hamiltonian operator in polar coordinates with momentum operators

The Hamiltonian operator for a free non-relativistic particle looks like $$ \hat{H} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m} \nabla^2. $$ In polar coordinates, the Laplacian expands to $$ \...
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30 views

Berry curvature from spherical polar coordinates to Cartesian coordinates

Let us consider the typical example of calculating the Berry phase: a spin in a magnetic field. We start with the Hamiltonian $H=-\textbf{B} \cdot \boldsymbol{\sigma} + B$ where $\textbf{B}$ is the ...
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1answer
330 views

Lorentz transformation boosts as matrices

I have seen Lorentz transformation boosts (say, along the x-direction) written as (in $c=1$ units): $$ \left[\begin{array}{cccc}{\gamma} & {-v \gamma} & {0} & {0} \\ {-v \gamma} & {\...
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3answers
65 views

Lorentz transformations for time co-ordinates (STR)

I am little bit confused with the implication of Lorentz transformations for time co-ordinates or atleast how to apply those! Consider 2 frames of reference $O$ and $O'$ in which $O'$ is moving with $...
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1answer
79 views

Time Dependence on Landau & Lifshitz's Proof of Poisson's Bracket Canonical Invariance

I'm reading Landau & Lifshitz's Mechanics and, at a certain point when discussing canonical transformations, they prove that Poisson brackets are canonical invariants. The proof starts with ...
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1answer
268 views

Is a Wick rotation a change of coordinates?

My understanding is that a Wick rotation is a change of coordinates from $(t,x) \rightarrow (\tau , x)$ where $\tau = i t$. In the $(t,x)$ coordinate system, the Minkowski metric has components $ \...
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1answer
64 views

What is the Rindler wedge?

Consider relativistically accelerating body along the coordinate $$x = \frac{c^2}{\alpha} \cosh\left(\frac{\alpha}{c} \tau\right) -\frac{c^2}{\alpha}$$ Why there is such "special" distance as $$d=\...
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25 views

Surface integral of vector field over cone, vertex not at origin

I have a vector field (originally given in Cartesian form). I need to find its integral over a cone with equation something like:$$1-z=\sqrt{x^2+y^2}, z>0$$ How do I proceed? It is not possible in ...
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1answer
35 views

Transformation of a vector's components in a time-dependent transformation

I know how the contravariant and covariant components of a vector transform when the coordinate system is changed (⇒ the known relation between the old coordinate system and the new one, I multiply by ...
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1answer
56 views

Kinetic energy in Lagragian mechanics [duplicate]

In my classical mechanics class they asked why the kinetic energy for an holonomic mechanical system has the homogeneous quadratic form. Of course for a autonomous standard system( system that is ...
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2answers
180 views

How is the global time coordinate $t$ (“observer at infinity” time) defined operationally e.g. in the Schwarzschild metric?

This is a question about coordinate time versus clock time / observed time, which I want to understand because I am teaching a GR course. Consider the Schwarzschild metric for specificity. I ...
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0answers
27 views

A problem about proof of Noether's theorem in Nakahara's Geometry, Topology and Physics [duplicate]

Nakahara's Theorem 1.1 (only the first half of the proof is taken) Let $H(q_k,p_k)$ be a Hamiltonian which is invariant under an infinitesimal coordinate transformation $q_k \rightarrow q'_k=q_k+\...
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1answer
50 views

Intrinsic curvature calculation

Gauss theorem egregium says that it is possible for the inhabitants of a 2d surface to calculate the surface curvature without knowing that it is embedded in a 3d euclidean space, simply calculating ...
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1answer
142 views

How to check if a generating function produces an identity transformation without substituting the CT equations in the Hamiltonian?

In chapter 9, Goldstein ($3^{rd}$ ed.) includes a discussion and a few "trivial special cases" of Canonical Transformation which keeps the form of the Hamiltonian unchanged and named it Identity ...
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7 views

Angular positions of points on a pole figure

I have to integrate a certain physical property of a crystal within a section of orientation space, Say between the space bound by directions <100>, <110> and <111> and so on. How can I ...