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Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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1answer
14 views

Aspect angle of object

How can I find aspect angles with respect to ground of an object in at height $h$. Object is rigid body with pitch angle $\theta$ in body coordinate.
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2answers
51 views

Relativity from a basic assumption

Is it possible to derive Lorentz transformations just by assuming that if two spaceships are moving away from each other with a constant speed, it is impossible for them to tell who is moving, even if ...
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1answer
25 views

Cyclotron motion for an arbitrary velocity and constant magnetic field

I understand the classic examples of taking the magnetic field along the direction of a certain axis, and then analyzing the motion of a charged particle. This gives helical motion (for a general ...
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5answers
94 views

Varying constants in special relativity [on hold]

The first thing you learn in special relativity is: All "inertial" frames of reference are equivalent. There is a frame - invariant speed limit, usually called speed of "light." So, geometry is a ...
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0answers
35 views

Hamilton-Jacobi equation and method of solving it [duplicate]

So, this equation we get when we find canonical transformation that makes new hamiltonian=0. There are 4 main transformations: F1(q,Q,t), F2(q,P,t), F3(p,Q,t), F4(p,P,t). On practice and in every book ...
7
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1answer
470 views

Why do we use the Lagrangian and Hamiltonian instead of other related functions?

There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to ...
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1answer
47 views

How is four-velocity automatically normalized?

This is a page from Sean Carroll's Spacetime and Geometry.There is a line in this page which says that the four velocity is automatically normalized.This absolute normalization is a reflection of the ...
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1answer
31 views

Is the Jacobian different for different ${\cal L}^p$ norms?

(I posted this to the math stackexchange, but I've yet to receive an answer so I figured I should post here too, as this forum seems faster to respond and is full of knowledgable people.) Because the ...
2
votes
3answers
59 views

Lorentz transformation of vector field

Under a Lorentz transformation, a vector field transforms as: $A'_{\mu}(x')=\Lambda^{\nu}_{\mu}A_{\nu}(x)$ My question is, why is the Lorentz transformed vector field evaluated at $x'=\Lambda x$, ...
1
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2answers
83 views

Paradox are all 4D distances zero?

The 4 dimension distance from the origin of a point is $\sqrt{x^2+y^2+z^2-t^2}$. Which means the 4 dimensional distance on the light-cone is zero. Take a point A and a point B in the future at ...
0
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2answers
39 views

Is an inertial force needed to be added to second Newton's law when transforming from cartesian to cylindrical coordinates?

Let's consider I have a particle moving on the $x,y$ plane. On this particle acts the Lorentz force, however it does not necessary perform a rotation by the axes origin $(x,y)=(0,0)$. I would like to ...
2
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3answers
50 views

Changing Coordinate System Changes derivation (mechanics, dynamics)

I'm reviewing some mechanics, and having an issue where my choice of coordinate system results in an incorrect derivation, one that is clearly incorrect. I suspect there's a missing step in my ...
0
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3answers
59 views

Can a wave's Poynting vector be in the opposite direction compared to its direction of propagation?

Can a wave's Poynting vector be in the opposite direction compared to its direction of propagation, and if so, what physical implications does it have? As I understand, the poynting vector s can be ...
0
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1answer
31 views

Why should the position vector be noted as $R\hat{R}$ in spherical polar coordinates?

Why should the position vector be noted as $R\hat{R}$ in spherical polar coordinates? Now i did the calculation like this: $\vec R = R \sin\theta \cos\phi \hat{i} + R \sin\theta \sin\phi \hat{j} + R \...
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0answers
35 views

A Scalar Function Tranformation — Question on Notation in 't Hooft Document

I started reading a document by Gerard 't Hooft which can be found here. Right at the start I am puzzled by a simple expression. It is equation 2.2 showing how a scalar function transforms. I ...
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0answers
45 views

Coordinate transformations [on hold]

In Einstein's "The Meaning of Relativity" I don't understand the relation between $b_{\nu\alpha}$ in equation (3a) and $\lambda$ that pops up in equation (2b). I understand the fact that there's a ...
1
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2answers
85 views

What is the reasoning behind 1 step in Einstein's derivation of the Lorentz Transformation

In Einstein's book "Relativity" there is a wonderful derivation of the Lorentz transformation, requiring no more than high school algebra (pp. 117 - 121). It is quite clear but I do not understand ...
2
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4answers
105 views

Visualization of $ dtdx$ and $dxdy$ term in metric tensor

For the sake of simplicity, lets take a 2+1 dimensional spacetime. Lets take the metric $$ds^2 = g_{tt}dt^2 + g_{xx}dx^2 + g_{yy}dy^2 + g_{tx}dtdx + g_{xy}dxdy$$ What is the visualization or ...
0
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1answer
46 views

Another question about Four-velocity derivation

Consider,again (Question about derivation of four-velocity vector) the following: For a massive particle with position $x^{\mu}(t) = (x^{0},x^{1},x^{2},x^{3}) \equiv (x^{0},\vec{x})$ we define the ...
1
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2answers
65 views

Naive question about the need to construct the 4-velocity

Special relativity was not born as a 4-covariant theory. Instead, Einstein derivated the kinematical quantities without spacetime,therefore without 4-vectors. Consider then the following: First ...
0
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2answers
134 views

Doubts on covariant and contravariant vectors and on double tensors

I'm trying to study tensors. Given a coordinates transformation from cartesian to $u_i$ ones: $$ u_1 = u_1 (x,y,z) \qquad u_2 = u_2 (x,y,z) \qquad u_3 = u_3 (x,y,z) $$ I can write a vector $\mathbf{...
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2answers
56 views

Hamiltonian description of a system

I know that phase space is the Hamiltonian description of a system, where we deal with position and momentum in equal footing. My question is in this phase space are those position and momentum are ...
0
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1answer
88 views

Gamma factor in special relativity

I try to derive the Lorentz transformation of a Lorenzt transformation frame an inertial frame $O$ to the frame $O'$ of a moving particle at constant speed v. We have four vectors $\textbf{x}'=\Lambda ...
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0answers
59 views

Canonical coordinate

Sorry for my broken English. I'm a physics undergrad and quite poor at math. While reading a mechanics textbook, I've found something I cannot understand. There are coordinates, $(q,p,t)$ $\...
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1answer
38 views

Interdependence of time, the speed of light and distance in General Relativity

In the relation $$ \text{time}\cdot\text{speed of light} = \text{distance}\,,$$ by definition, the speed of light is constant. When bringing a clock, like in a GPS satellite, up into orbit, i.e. ...
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5answers
90 views

How to determine the direction of a vector?

I have been learning about vectors and acceleration recently and I still don't understand how to determine the direction of a vector. For instance, if we consider a freely falling particle and ...
0
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2answers
71 views

$ct' = γ(ct − βx)$

I was reading a paper on S.R. saw this eq: $ct' = \gamma(ct − \beta x)$. However I am confused by the Minkowski's diagram: We know $\tan \theta = \frac{x}{ct}= \beta$. However, wouldn't it also mean $...
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2answers
31 views

How to change a formula when changing the axis?

I have been lerning about acceleration and free fall recently, and we were told the formula for distance is $s=s_0+v_0t+\frac {1}{2}at^2$ (for movement along $x$-axis) but when we change the axis (to ...
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0answers
30 views

Isometries and coordinate transformations in the context of Noether's Theorem

If I have a theory defined on some manifold, my understanding is that the dynamical objects in the theory should carry a representation of the isometry group of that manifold. Moreover, the action $S$...
0
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2answers
64 views

Unit Vectors in physics

I'm reading the Massachusetts Institute of Technology: "Review of Vectors" , and I've found this: I can't see any relationship between the text that is highlighted in yellow and what's depicted in ...
2
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0answers
51 views

Pictures of Different Coordinate Systems in General Relativity

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
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0answers
69 views

How can you explain $cot(\alpha)= \dfrac{d}{d\theta}\cdot ln(r) = \dfrac{1}{r} \dfrac{dr}{dt}$ in a polar coordinate system? [migrated]

This alinea is about the $$cot(\alpha)= \dfrac{d}{d\theta}\cdot ln(r) = \dfrac{1}{r} \dfrac{dr}{dt}.$$ Where does the $ln (r)$ come from? How can you derivate it from that picture? I want to use that ...
6
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4answers
231 views

Why do we care only about canonical transformations?

In Hamiltonian mechanics we search change of coordinates that leaves the Hamilton equation invariant: these are the canonical transformations. My question is: why we want to leave the equations ...
0
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1answer
30 views

In case of torque on sliding block on horizontal surface, why $y$ component of $\vec{r}$ is taken negative?

I've attached the picture above. In case of the torque on a sliding block on horizontal surface why r component is taken negative.
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3answers
49 views

I have a question regarding the Painlevé-Gullstrand (PG) metric with factor 2

I have a question regarding the Painlevé-Gullstrand (PG) metric. If we have the line element in a radial fall we get: $$d\theta = d\phi = 0$$ $$ds^2 = -dT^2 + \left(dr+\sqrt{\frac{r_s}{r}}dT\right)^...
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4answers
108 views

How do we determine who is the “observer” when calculating time dilation? [duplicate]

I was reading about time dilation and I thought of a question. I really think that I am just misunderstanding something, so please bare with the question. But, I would appreciate if somebody could ...
1
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2answers
96 views

“It is the gist of general relativity that it admits, on an equal footing as it were, every possible coordinatization.”

The title is a quote from Hermann Weyl in a 1955 article: Weyl, Hermann. "Why is the world four-dimensional?" In Levels of infinity: Selected writings on mathematics and philosophy. Courier ...
6
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1answer
64 views

2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
6
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0answers
90 views

Twin paradox with one twin in orbit, one in radial free-fall

Many questions on Physics SE relate to the twin paradox, but I did not find any that ask this specific question. Suppose that object A is in a circular orbit around a spherically symmetric, non-...
1
vote
1answer
78 views

Rigorous definition of generalized coordinates

In Goldstein's classical mechanics and in many other books I haven't seen a rigorous definition of generalized coordinates. In a system of $N$ particles described by $\textbf{r}_1, \dots, \textbf{r}...
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2answers
69 views

Physical interpretation of FRW normal coordinates

The Friedmann-Robertson-Walker metric (I consider for notational simplicity the flat space case): $$\text d s^2 = \text d t^2 - a(t)^2\text d \boldsymbol{x}^2$$ can be brought to normal (Minkowski) ...
0
votes
1answer
71 views

Transformation of a Lagrangian

$$L(\lambda,\mu,\dot{\lambda},\dot{\mu})=\frac{m}{2}(\lambda^2+\mu^2)(\dot{\lambda}^2+\dot{\mu}^2)-\alpha \lambda^2\mu^2,$$ I'm supposed to express this Lagrangian through $x=\lambda^2-\mu^2$ $y=2\...
1
vote
1answer
108 views

How come $\frac{d}{dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}$ in Lagrangian mechanics? [duplicate]

It is written in the Goldstein's Classical Mechanics text that $$\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}=\sum_k \...
4
votes
1answer
56 views

General coordinate transformations?

Say I have a vector field expressed in Cartesian coordinates: $$\mathbf{A} = \sum_i A_i \mathbf{\hat{e}}_i$$ where the $\hat{\mathbf{e}}_i$ are the generalisation of the unit vectors $\mathbf{\hat i}, ...
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4answers
104 views

Why $g$ is not negative in potential energy $= mgh$ formula

For eg. A boy of mass 55 kg runs up a staircase of 50 steps in 10s. If the height of each step is 10 cm, find his power. My question is here why are we not taking g as negative as boy is moving up ...
0
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3answers
84 views

Step-by-step guide to finding the phase constant in simple harmonic motion

Working on some simple harmonic motion problems involving an oscillating spring/mass system ... the usual. I never really understood exactly how to find the phase constant for the $$x(t)=Acos(wt+phase ...
2
votes
2answers
127 views

Unlike rotation, why a $3\times 3$ translation matrix cannot be written in 3D? or can it be?

The effect of rotation in 3d on a vector, $\vec{r}=x\hat{x}=y\hat{y}+z\hat{z}$ is given in the form a matrix product:$$\vec{r}\to O\vec{r}$$ where $O$ is a $3\times3$ proper orthogonal matrix. Can we ...
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votes
1answer
88 views

Line element to polar coordinates [closed]

I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is: \begin{equation} \mathbf{v}=\frac{A}{r} \hat{r} + \frac{B}{r}\hat{\theta} \end{equation} So:...
1
vote
1answer
85 views

Difference betwee $(d\vec r/dt)_{fixed}$ and $(d\vec r/dt)_{not~fixed}$

$[![Picture ~from ~http://www.astro.uwo.ca/~houde/courses/PDF%20files/physics350/Noninertial_frames.pdf][1]][1]$ From the picture above, the set of coordinates $x_i$ are the ones that are not fixed ...
3
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2answers
46 views

Effect of Co-ordinate Change on Euler-Lagrange Equations for Scalar Fields

Consider a single scalar field $\phi$ on a manifold $\mathcal{M}$. Suppose in $\{x^\mu\}$ co-ordinates, the Lagrangian density is $\mathcal{L}(\phi, \frac{\partial \phi}{\partial x^\mu})$. This means ...