All Questions
Tagged with homework-and-exercisescoordinate-systems
413 questions
0votes
2answers
155views
Is relative motion of Sun as seen from Earth a circle? How to derive it mathematically?
Assume a perfect solar system i.e. Earth revolves around the Sun in a circle of radius R, Sun being at the center. The locus of Earth is $x^2 + y^2 = R^2$ I have a moving coordinate system whose ...
user495133
5votes
5answers
472views
Derivatives of conjugate momenta wrt generalised coordinates in Lagrangian mechanics
I am struggling understanding the validity of the solution that my professor gave to the following exercise. A heavy symmetric top rotating about a fixed point has Lagrangian: $$ \mathcal L=\frac{I_1}...
3votes
1answer
151views
How to describe the motion of inertial particles in non-inertial frames?
Imagine a rotating observer represented by the basis $B = \{\mathbf{\hat r}, \boldsymbol{\hat \varphi}\}$. This observer wants to describe the movement of a particle moving on a straight line ($\...
- 1,821
3votes
2answers
285views
The inverse of a specific metric tensor [closed]
I am studying general relativity and here is a problem I encountered: Suppose $$ \mathrm{d}s^2=-M^2(\mathrm{d}t-M_i\mathrm{d}x^i)(\mathrm{d}t-M_j\mathrm{d}x^j)+g_{ij}\mathrm{d}x^i\mathrm{d}x^j $$ or ...
- 129
0votes
1answer
67views
Euler Angles Representation
I have an exam in two days in classical mechanics and I've encountered a question about Euler Angles that I did not completely understand: The question goes like this: Given two similar bodies as ...
- 33
0votes
1answer
145views
Proper time in moving frame of reference
I try to figure out how to calculate the proper time between two events for a moving frame of reference. Knowing that the proper time is $$(d \tau)^2 = \frac{- (ds)^2}{c^2},$$ and thus it is the same ...
2votes
0answers
71views
3-point function CFT
How can I compute the three-point function $<\phi^{(h_i)}(z_i) \phi^{(h_j)}(z_j) \phi^{(h_k)}(z_k)> $ in CFT using this conformal transformation $f(\xi)=\frac{(z_j-z_i)}{(z_j-z_k)} \frac{(\xi - ...
1vote
0answers
84views
Confusion on coordinate transformation matrix derivatives and Christoffel symbol transformation
Consider the coordinate transformation from $x^{\mu} \to x^{\bar{\mu}}$ given by the transformation matrix $\Lambda^{\bar{\mu}}_{\mu}$, and $\Lambda_{\bar{\mu}}^{\mu}$ for the inverse transformation. ...
- 41
0votes
3answers
83views
How and why should we properly assign signs to objects in motion in opposite directions? [closed]
On a foggy day, two car drivers spot each other, when they are just 80 m apart. They are traveling at 72 km/h and 60 km/h respectively towards each other. Both of them simultaneously apply brakes, ...
1vote
0answers
71views
GR: angular velocity in circular orbits
Starting with the Schwarzschild metric: $$ A = 1-\frac{2m}{r} $$ $$ \mathrm{d}\tau^2 = A\mathrm{d}t^2 - \mathrm{d}r^2/A -r^2\mathrm{d}\theta^2 - r^2\sin^2{\theta}\mathrm{d}\phi^2 $$ I want to ...
- 871
4votes
4answers
451views
Choice of Generalized Coordinates
I am working on solving problem 5.24 from the $3^{\mathrm{rd}}$ edition of Goldstein's Classical Mechanics: A wheel rolls down a flat inclined surface that makes an angle $\alpha$ with the horizontal....
4votes
1answer
95views
Circumference of ellipse in post-Newtonian metric
The post-Newtonian metric, in harmonic coordinates, is: $$\tag{1} \mathrm{d}s^2=-\left(1+\dfrac{2\phi}{c^2}\right)c^2\mathrm{d}t^2 + \left(1-\dfrac{2\phi}{c^2}\right)\mathrm{d}\mathbf{x}^2$$ where $\...
- 369
0votes
1answer
144views
Rotating a system
brekely physics book chapter 2 page 30 , a question about rotating a system by $ \frac{\pi}{2} $ around the z axis clockwise direction and writing vectors according to the new axis after rotation ...
- 305
1vote
0answers
97views
Impact Parameter as function of orbital parameters
In the case of a binary system (pulsar+companion), the impact parameter $\textbf{b}$ is the projection of the binary separation $\textbf{r}$ on the sky plane: \begin{equation}\tag{1} \textbf{b} = \...
- 369
0votes
1answer
93views
What is the equation if that projection starts SHM on the $x$-axis from extreme position?
Consider A particle performing Uniform Circular Motion. We know that its projection on diameter performs SHM. Then, if that projection starts SHM on the y axis from mean position, then $y=A\text{sin}(...
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