All Questions
Tagged with galilean-relativity coordinate-systems
34 questions
6
votes
4
answers
132
views
How do we interpret measurements of Mercury's position?
When scientists measured the position of Mercury in the 18th century, they interpreted the results assuming a Euclidean background, because they did not know general relativity. So they measured $r$ ...
- 61
0
votes
0
answers
46
views
Galilean transformation vs boost matrices
I'm confused about the difference between a Galilean transformation and boost with reference to their matrices. I was given four statements (listed below) but I'm not sure what I should be looking for ...
- 1
1
vote
0
answers
29
views
Generalizing the Galilean law of addition of velocities using the Lorentz transformation [closed]
I am reading about how to generalize the Galilean law of addition of velocities using the Lorentz transformation, but I am confused about one step.
Here, I have the following equations for Lorentz ...
- 63
4
votes
4
answers
356
views
Equation of Motion Invariance in Galilean Mechanics
Consider a particle moving freely, where $\vec{r}(t)$ is the position of the particle. Suppose I move into a frame with
$$\vec{r}' =\vec{r} + \epsilon \vec{F}(\vec{r}, t)\tag{1},$$ where $\epsilon$ ...
- 283
1
vote
1
answer
42
views
Coordinate Transformation using a Matrix
Consider two inertial and resting frames $K$ and $G$. The only difference between the two frames is that the axes of $G$ has been rotated with $\theta$ with respect to $K$. $G$ is not constantly ...
- 193
1
vote
1
answer
83
views
Inertial coordinate systems being invariant under time translation in Newton's Principle of Detrimancy
I have the same question posted as Newton's equation under time translation except I am not seeking the physical justification of the first claim but rather the mathematical justification of the ...
- 27
0
votes
2
answers
353
views
Why are the transformations from the Galilean transformations affine?
In Arnold's Mathematical Methods of Classical Mechanics, he says on page 6 the following are Galilean transformations on the Galilean coordinate space $\mathbb{R} \times \mathbb{R}^3$ where $\mathbb{R}...
- 27
0
votes
1
answer
127
views
Invariance of continuity equation for Galilei transformations
I want to prove that the continuity equation for fluids, $$\dfrac {\partial \rho}{\partial t} + \nabla \cdot (\rho \textbf{u}) = 0$$ is invariant by Galilei transformations. My attempt:
Using index ...
- 115
1
vote
0
answers
90
views
Galilean invariance of Burgers Equation [closed]
I think the following statement is true: if $u$ solves the burgers equation (ie $u$ solves $$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0$$ then so does $$u^c = u(x-ct,t)+c.$$ I'...
- 123
0
votes
1
answer
52
views
Galilei transformation of mass flux
Is it possible to perform a Galilei transformation of a flux without additional information?
Say we consider a flux $q = \rho v$ that can be written as the product of density $\rho$ and a velocity ...
- 3,265
0
votes
0
answers
162
views
Formulating Conservation of Energy in Galilean Spacetime
Some background to my question (Galilean spacetime).
The notion of Galilean spacetime is defined at the beginning of Arnold's book on Classical Mechanics. It is a mathematical structure that captures ...
- 404
0
votes
2
answers
76
views
Galilean's principle implies independence of time and dependence on relative distance
Suppose a system of particles $q_1,\ldots,q_N$ of masses $m_1,\ldots,m_N$ that follow the equations of motion
$$m_j\ddot{q}_j=f_j(q_k,\dot{q}_k)$$
in an inertial frame and satisfy the Galilean ...
- 101
1
vote
1
answer
76
views
Is this the correct way to transform a trajectory between Galilean frames?
Consider the general Galilean transformation in one spatial dimension:
$$(x, t) \mapsto (x', t') = (x + vt + a, t + b).\tag{1}$$
I want to use it to transform trajectories $x(t)$ in a frame $S$ to the ...
- 1,245
1
vote
1
answer
423
views
Interpretation: Galilean Transformation of Force Laws
so my books says
The transformation that allows us to go from one inertial frame $O$
with coordinates $x_i$ to another inertial frame $O'$ with coordinates
$x_i'$ is the Galilean transformation. If ...
- 237
1
vote
1
answer
479
views
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} \...
- 51
2
votes
2
answers
521
views
Do Galilean (Euclidean) space transformations implies that time is absolute?
I recently read a paper where it says "if space is universally Euclidean, then time is universal" and I don't understand some key points about the implication.
To put in context, the author ...
- 625
2
votes
1
answer
579
views
What is the difference between boost and translation in Galilean Transformation?
This is probably a newbie question (but I guess is what I am right now) but I can't understand de difference between Galilean Boost and Galilean Translation.
I thought a boost was something like an S'...
- 43
1
vote
1
answer
417
views
Transformation law of momentum under Galilean transformation
I'm reading the article On the Galilean Covariance of Classical Mechanics (pdf link here), in which the authors want to establish the transformation rule for momentum, assuming only that $\vec{F}=d\...
- 1,071
2
votes
0
answers
128
views
Invariance of the Schoedinger equation for the Galilean transformation [closed]
Show that the schroedinger e is covariant under the galilean transformation :
$\overrightarrow{r'}=\overrightarrow{r}-\overrightarrow{V}t$
iff the wave fucntion transforms like:
$$\psi^\prime=e^{\left(...
- 189
0
votes
1
answer
63
views
How to properly use Galilean transformation in order to move between inertial frames?
I'm having trouble understanding how to use Galilean transform when moving between coordinates. e.g consider the following problem: a mass attached to a spring inside a moving cart - how can I ...
- 325
0
votes
0
answers
810
views
Is time an invariant of Galilean transformation?
Is time an invariant of Galilean transformation? By saying that I mean if there is a quantity analogous to spacetime interval in Lorentz transformation. What is the geometry of "Galilean spacetime"?
- 165
2
votes
1
answer
837
views
Galilean transformation and differentiation
Given $x=x’-vt$ and $t=t’$, why is $\frac{\partial t}{\partial x’}=0$ instead of $1/v$? $t$ seems to depend on $x’$ because if $t$ changes, $x’$ changes. Also, in this problem, $dx=dx’$ as well, but I ...
- 341
2
votes
2
answers
255
views
Unlike rotation, why a $3\times 3$ translation matrix cannot be written in 3D? or can it be?
The effect of rotation in 3d on a vector, $\vec{r}=x\hat{x}=y\hat{y}+z\hat{z}$ is given in the form a matrix product:$$\vec{r}\to O\vec{r}$$ where $O$ is a $3\times3$ proper orthogonal matrix. Can we ...
- 12.3k
1
vote
1
answer
113
views
The absoluteness of time intervals in Newtonian mechanics: how is this input used?
One of the assumptions of Newtonian mechanics is that "time is absolute". Absolute, as I understand, implies that it is the same for all observers. But it's not quite true because if Tom's watch is ...
- 27.2k
1
vote
4
answers
3k
views
Velocity of light in Galilean transformation
What is the velocity of light in Galilean transformation? Is it infinity?
user172266
0
votes
1
answer
470
views
Galilean Transformation
The transformation between inertial systems are given by affine transformations of $\mathbb{R}^{1+3}$. These are given by $t'=\lambda t+\vec{c}^\top \vec{x} +a$ and $\vec{x}'=\vec{v}t+M\vec{x}+\vec{b}$...
- 117
1
vote
1
answer
1k
views
Transformation of the operators $\mathbf\nabla$ and $\partial/\partial t$ under Galilean transformation
I want to know how are the transformations of the operators $\mathbf\nabla$ and $\partial/\partial t$ when the transformation of the Galilean relativity is applied.
This is what I've tried:
Galilean ...
- 454
3
votes
2
answers
387
views
Is relativity of simultaneity just a convention?
Lorentz transformations are well known to imply
time dilation,
length contraction, and
relativity of simultaneity.
This is prominently featured in any course on Special Relativity (SR), e.g. in ...
user175150
0
votes
1
answer
2k
views
Galilean Invariance of material derivative
I'm reading some fluid dynamics notes which are talking about a Galilean boost of the form:
$$x'=x-vt, \qquad t'=t$$
The notes immediately claim from this that the material derivative $$\frac{D}{Dt}...
1
vote
1
answer
110
views
Shouldn't work be the same in all coordinates?
We know that the work done by a force $\mathbf{F}$, along a path $\mathbf{x}$, is given by:
\begin{equation} W = \mathbf{F}^T \cdot \mathbf{x}
\end{equation}
However, suppose that i apply some change ...
- 501
5
votes
4
answers
11k
views
Are vectors truly independent of coordinate systems?
I have been told to think of vectors as existing independent of a coordinate system. This means that the magnitude of a vector should be independent of any coordinate system we choose. Galilean ...
- 4,054
1
vote
1
answer
752
views
Coordinate Transformation in Classical Mechanics
The coordinates in one inertial frame are represented by $(x,t)$. Under coordinate transformation, the coordinates in another inertial frame can be represented by $f(x(t),t)$. It can be shown that the ...
user115657
2
votes
3
answers
2k
views
Prove that the spacetime interval is not invariant under Galilean transformations [closed]
The spacetime interval $(\Delta s)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - c^2(\Delta t)^2$ is invariant under the Lorentz transformation and this isn't the case for the Galilean ...
- 231
2
votes
0
answers
443
views
Question about Origins in Galilean transformation
I'm just learning about relativity, and every equation I see for a galilean transformation of frame $S'$ (moving with uniform velocity in the $x$-direction with respect to frame $S$) is $x'=x-vt$, $y'=...
- 729