Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
-3
votes
0answers
32 views
A thought experiment to test Lorentz symmetry
Following is a simple thought experiment with matching Lorentz transform that clearly breaks Lorentz symmetry…. Or maybe I’ve just got my numbers wrong.
Questions are at the end.
Here is the ...
2
votes
0answers
32 views
Using action-angle variables in non-periodic system
I'm a little confused by the discussion in the last section $\S 52$ of Landau and Lifshitz's (Classical) Mechanics. Here, they consider finite motion of a system whose Hamilatonian is separable (for a ...
1
vote
2answers
36 views
Use of generating function in canonical transformation
In the theory of Canonical transformations, initially we use the fact that the new and the old system of $(q_i, p_i)$ with the Hamiltonian $H$ satisfy the modified Hamilton's principle. Now here, the ...
9
votes
4answers
766 views
Twin paradox in curved space time [duplicate]
In a flat space, where special relativity works, a travelling body can only return to the same point if we apply some kind of acceleration to the body.
So twin paradox is not a paradox because a ...
-2
votes
0answers
37 views
tell me what is the relationship between lorentz and Galilean transformation [on hold]
where came the Gamma
tell me the relationship between lorentz and
Galilean transformation
1
vote
1answer
55 views
Vectors transforming under change of coordinates
I was watching a lecture on tensors and the professor said that a defining feature of a vector $v$ is that it transforms under a coordinate transformation $x^{\mu} \rightarrow x^{\mu'}$ as
$$v^{\mu'}...
0
votes
1answer
94 views
Using $v_f = v_0 +at$ for objects in free fall [on hold]
I have a question about the difference of using $v_f = v_0 +at$ and $s = v_0t + \frac{1}{2}at^2$ for objects in free fall. I'm trying to solve a problem where there's a ball rolling along an inclined ...
-1
votes
0answers
62 views
Integrate isotropic coordinates [on hold]
When I take isotropic coordinates at the general metric for a static-spherically symmetric spacetime, finally you get the following integral:
$$\dfrac{dr'^2}{r'^2}= \dfrac{dr^2}{r^2 (1-2M/r)}$$
i.e:
...
0
votes
0answers
16 views
Generalized coordinates for a floating planar linkage
I am trying to derive the Euler-Lagrange equation of motion for a planar bipedal robot that consists of the body (a link) and 2 two-link legs for a total of five links.
$$D(q)\ddot{q}+C(q,\dot{q})\...
7
votes
3answers
3k views
Which observer measures proper time in the twins paradox?
I know that proper time is defined as the time which the clock moving relative to that observe shows. That is, a clock attached to observer A will always be As proper time. I also understand that this ...
0
votes
0answers
15 views
Movement expressed in polar coordinates
The movement of a certain particle in the $OXY$ plane is given in polar coordinates by means of the equations:
$$p(t)=p_{0}e^{\Omega t}$$ $$\theta(t)=\Omega t$$
And I want to calculate the speed ...
1
vote
2answers
89 views
Positive work along path
Consider I have a simple formula for the work along some path (in 1 dimension):
$$W~=~\int_{x_0}^{x_1}\vec{F}\cdot d\vec{x}.$$
If I now move from left to right ($x_1 > x_0$) along the axis (...
2
votes
1answer
42 views
Derivation of Hamilton-Jacobi theory using canonical transformations
The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function.
Is it possible to use a another type of generating function, ...
1
vote
2answers
77 views
Determinant of the metric tensor
After a change of coordinate system on flat space from $x\rightarrow y$, we have the metric tensor:
$$g_{\mu \nu} = \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\...
1
vote
1answer
73 views
Action angle variables and Action
Action given by principle of least action ($S$) and action variable given by action angle variable theory ($J$) are same?
1
vote
0answers
54 views
Way of splitting spacetimes into space and time
On several occasions, we would like to separate the components of the metric into three sets ($g_{00}$,$g_{0\alpha}$,$g_{\alpha \beta}$). Please refer to exercise $4.2$ of "Gravitation, Foundations ...
1
vote
1answer
23 views
Generating function in action-angle method and Hamilton-Jacobi theory
I think that in action angle method, generating function which generates such a canonical transformation does not explicitly depend on time, so new and old hamiltonians are equal. But in H-J method, ...
1
vote
1answer
85 views
Electric field at any point due to a continuous charge distribution
I am reading Purcell and Morin's Electricity and Magnetism 3rd Edition.
Equation ($1.22$):
$$\vec{E}(x,y,z)=\dfrac{1}{4 \pi \epsilon_0}
\int \dfrac{ρ\ (x^\prime, y^\prime, z^\prime)\ \hat{r}\ dx^\...
-2
votes
1answer
53 views
Integration Using Spherical Coordinates [closed]
So I had to find the moment of inertia of a hollow sphere of mass $M$, radius $R$, and negligible thickness.
$dI=R^2 \cdot dm$
where $dm = \dfrac{M}{4\pi R^2}\cdot R^2\sin(\theta)\cdot d\theta\cdot ...
3
votes
0answers
65 views
Difficult coordinate transformation
I am trying to introduce a tortoise coordinate for a modified Schwarzschild metric
$$\mathrm{d}s^2=\left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right) \mathrm{d}t^2 + \left(1-\frac{2M\mathop{}\!\...
0
votes
0answers
28 views
From cartesian to internal coordinates for the classical molecular vibrational problem [closed]
If we suppose to have a generic molecule made by $N$ nuclei, describing them with cartesian coordinates, the Lagrangian of this system, using an harmonic approximation for the potential energy is:
$$L=...
1
vote
2answers
76 views
How can the $v$ coordinate be null if $g_{vv}\neq 0$?
I'm probably missing something very basic here. As far as I know, a coordinate is called null when its coordinate lines are null. This that if $(M,g)$ is spacetime and $x^\mu$ a coordinate chart, the ...
5
votes
3answers
203 views
Extracting the 3D coordinates of a moving object from a video
Take a look at these two pictures, which are stills from a video which demonstrates magnus effect in football:
I want to extract the coordinates of this ball in 3D space from this video. These are ...
1
vote
2answers
82 views
Proof that $\vec {r(t)}=\vec r_0 + \vec v_0 t + \dfrac{1}{2} \vec a t^2$ for uniformly accelerated motion
Displacement of a particle moving through $ x $ axis is given by $$ x(t)= x_0 + v_0 t + \dfrac{1}{2} at^2 $$ Can we deduce from it that $$ \vec r(t)=\vec r_0 + \vec v_0 t + \dfrac{1}{2} \vec a t^2$$ ...
1
vote
1answer
56 views
Are Maxwell's equations valid in a rotating frame?
Maxwell's equations are covariant under Lorentz transformations. Are they covariant under going to a rotating frame and if not how do they look?
4
votes
5answers
156 views
Why we use vectors?
When we say that the position of an object is +5m on x axis why we need to use vectors? I mean could we don't use vectors and just say +5m on x or y or z axis instead of writing 5*unit vector either $...
2
votes
1answer
41 views
Transitioning between Lagrangian and Eulerian fluid variables
I'm having trouble understanding the motivation between wanting to transition between the two descriptions.
I figure that switching from Eulerian to Lagrangian fluid variables is useful if we want to ...
0
votes
0answers
33 views
Basis Vectors and Degrees of Freedom
I was wondering if there is a connection between the concept of basis vector in $\mathbb R^n$ and the degree of freedom available to a particle. If It is, please tell me how can I know one by knowing ...
1
vote
1answer
36 views
Where is the error in this calculation of net curl for simple magnetic field?
I wasn't sure whether to post this on MSE, but PSE seems more appropriate.
Let B be a static magnetic field in spherical coordinates, defined as $B=r\hat{\theta}$. Then, it's curl is $$\nabla \times ...
1
vote
1answer
38 views
Restriction Forces in Lagrangian Mechanics
I was recently preparing for a test on Classical Mechanics and a friend of mine started wondering if there was any method through which we could obtain the restriction forces acting on a certain ...
1
vote
2answers
68 views
Rindler Coordinates and homogeneous Gravity Field
I understood from the equivalence principle that an accelerated observer in free space is equivalent to a stationary observer in a gravitational field.
As far as I understood further, this means to ...
1
vote
1answer
63 views
Integrating over Euler Angles
I have a $6\times6$ matrix having its elements being functions of Euler's angles (ZXZ rotation scheme) representing a tensor physical property. To find the average of the tensor property, I need to ...
0
votes
2answers
54 views
Why does transforming from equatorial coordinates to galactic coordinates require 3 equations?
My question seems self explanatory but still:
In transforming equatorial coordinates to galactic coordinates or the reverse you have to solve a well known system that has three equations.
Since ...
0
votes
3answers
48 views
Electric field in spherical coordinates
Let's say I want to find the line integral of the electric field along some path $ab$ as shown here
I imagine taking small segments $dl$ of that path from $a$ to $b$, but as I imagine that, I ...
1
vote
1answer
24 views
Coordinates in 1970 Labeyrie article
I'm reading the 1970 article by Labeyrie on speckles in astronomy and I encountered a small problem. The author uses two sets of coordinates, $(\alpha, \beta)$ and $(x, y)$, connected by Fourier ...
1
vote
1answer
40 views
Translation operator in polar coordinates
Accordingly to Stone's theorem the generator of a strongly continuous one-parameter unitary group is self-adjoint.
The translation operator in the radial direction in Cartesian coordinates is ...
1
vote
0answers
35 views
Interpreting Kruskal for cosmological spacetime
I'm dealing with a metric of the form:
$$ds^2 = -f(r)dt^2 + a(t)(1/f(r)dr^2 + a(t)r^2d\Omega^2)$$
Where $f$ is just $f = 1 - 2m/r$. The first thing I did was to introduce a conformal time coordinate $...
0
votes
1answer
49 views
Convert sexagesimal to decimal
I've been studying astronomy and I've encountered 3 different (sexagesimal) ways to write angles.
hh mm ss - hours minutes and seconds
dd '' '''' - degrees, arcminutes and arcseconds.
+/- dd mm ss -...
0
votes
0answers
19 views
Do Newton's laws imply independence of generalised coordinates and generalised velocity? [duplicate]
When deriving lagrangian formulation from Newton's laws, We use the fact that generalized velocity and generalized coordinates are independent of each other. Is there a mathematically rigorous proof ...
0
votes
1answer
63 views
How to compare the observation with the theoretically predicted result?
On the Wikipedia Article on “Geodesics in general relativity”, it says the following: “Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved 4-D ...
1
vote
2answers
75 views
How do I look for (possibly) all coordinate transformations with a given metric?
From what I learned in tensor calculus so far, coordinate transformations are supposed to preserve the metric of the space. (Here I used GR notation, but the metric doesn't have to be the spacetime ...
1
vote
1answer
55 views
Why radial acceleration is expressed as the negative of centripetal acceleration?
My book says that:$$a_r=-a_c=-\frac{v^2}{r}$$Why radial acceleration is expressed as the negative of centripetal acceleration? Shouldn't they have the same direction and magnitude?
-1
votes
2answers
37 views
Problem on deriving canonical transformation condition
I'm trying to compute how a canonical transformation should be, given that preserve the symplectic form and trying to recover the condition on the Poisson Bracket. I then start with
$$\omega=\stackrel{...
0
votes
0answers
20 views
Orthogonal vector to $t=$const. hypersurface in Kerr-Schild coordinates
How do I find the orthogonal vector for the t= const. hypersurface in the
Kerr - Schild metric? I know that I start by dt=0 but where do I go from there?
0
votes
1answer
62 views
Expected momentum of ground state hydrogen $<p>$
I am trying to calculate the expected momentum of an electron in the ground state of hydrogen atom. This is the wave function.
So far I have done this:$$\iiint_V \Psi^* (-i\hbar) \frac {d\Psi} {dr} ...
2
votes
2answers
58 views
Definition of velocity in classical mechanics
Let $(r_1,r_2,r_3)$ be the coordinates of a particle $r$ in the coordinate system $\phi$. Let $\{\hat{e_1},\hat{e_2},\hat{e_3}\}$ be the coordinate basis of $\phi$. Why do we define the velocity $v$ ...
3
votes
2answers
430 views
The location of an object is gauge dependent. Therefore, it's not measurable?
The location of an object $x$ depends on how we choose our coordinate system. If we move the zero point, $x$ also changes. However, since we have translational invariance, we can always do such shifts ...
6
votes
3answers
195 views
In electromagnetism, why does nature prefer the right-hand rule over the left-hand rule? [duplicate]
At school I learnt the Right-hand rule to remember the resulting direction of different phenomena, such as geometrical cross products, mechanical torque, or the direction a screw will move when ...
0
votes
3answers
58 views
How to determine the direction of instantaneous acceleration in a 2D motion? [duplicate]
How do we determine the direction of instantaneous acceleration when the body is moving in a plane (or a 3D space)? This question has been truly bothering me for nearly two weeks. I looked it up, ...
0
votes
0answers
31 views
Partial derivative in generalized coordinates [duplicate]
Sorry for my broken English.
I'm a physics undergrad and quite poor at math.
I just started to learn analytical mechanics and it really confuses me.
my analytical textbook uses the equations below ...
