All Questions
Tagged with galilean-relativity classical-mechanics
56 questions
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A problem with unique decomposition of Galilean transformation
It is very well known that any Galilean transformation on $\mathbb{R}^3\times\mathbb{R}$ can be uniquely written as composition of a maps of the following type:
Uniform motion: $(x,t)\to(x+tv,t)\quad ...
- 63
2
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1
answer
159
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Proving that the Lagrangian of a free particle depends only on $|\boldsymbol{v}|^2$
The question is NOT answered by
Deriving the Lagrangian for a free particle,
as the answers therein assume the quadratic dependence, which is what
I am trying to prove. Additionally, while one of the ...
- 23
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1
answer
182
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Understand the definition of frame and inertial frame in Arnold's Galilean spacetime definition
In Arnold's Mathematical Methods of Classical Mechanics, we define the physical space time as a four dimensional affine space with associated Galilean structure. I understand this part.
Now what I'm ...
- 275
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203
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Relationship between symmetries and additive integrals of motion
I'm currently reading Landau and Lifshitz's Statistical Physics. In it, they attempt to justify that the density function only depends on the energy by arguing that the logarithm of this function is ...
- 1,000
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4
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233
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Reference frame doubts about isotropy
Landau & Lifshitz on p.5 in their "Mechanics" book states the following:
...a frame of reference can always be chosen in which space is
homogeneous and isotropic and time is homogeneous....
1
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1
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83
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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
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2
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353
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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
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3
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257
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How to show the velocity of free motion is constant in Galileo's relativity principle?
Picture below is from Landau & Lifshitz's Mechanics. How to get the red line from green line?
- 361
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1
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Implications of Galilei-Invariance on a time-independent potential
I'm trying to compute a result shown in my classical mechanics lecture on my own. Namely, consider that a system composed of $n$ particles follows a law of force
$m_k\ddot{\vec{x_k}} = \vec{F_k}(\vec{...
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Is Galilean boost actually a gauge transformation?
In elementary physics, it is well-known that the Newton's law
$$\vec{F}=m\vec{a}$$
is invariant under Galilean transformations. However, Galilean relativity is not introduced in details in ordinary ...
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Dummy variables and Galilean Invariance
I've faced a small doubt, and I was hoping someone could verify this for me.
According to Galilean transformation, consider $2$ frames - $S_1$ and $S_2$ moving relative to each other. $S_1$ is at rest,...
- 1,853
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123
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Affine space in classical mechanics and it's applicability in general relativity
In the first chapter of Arnold book of Classical Mechanics while giving Galilean structure of spacetime we're introduced to affine space. As already mentioned in answers to this question this is done &...
- 3,063
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137
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What do these Casimir invariants of the Galilean group physically represent?
There exist Casimir invariants of the Galilean group which commute with all the generators of the group. They are, of course, Galilean scalars (i.e., scalars under space and time translations, ...
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Assumptions in Galilean and Relativistic Frame Transformation
While deriving the frame transformation equations, either the Galilean Transformation or Lorentz transformation. I have seen almost all authors mentioning/assuming that if an inertial frame $\textbf{S}...
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Conservation of Energy in Collision
Consider two cars of mass $m$ travelling towards each other at velocity $v$. A bystander (mass $m_b$) on the side of the road would calculate that cars have a total kinetic energy of $mv^2$, while a ...
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Conceptual questions about Galilean relativity
I want to see if Galilean relativity is as mind-blowing as a think it is.
I'm imagining two individuals each on their own planets. Ignore gravity for now, they are attached to the planets, and there ...
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727
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What is a time-dependent symmetry in Hamiltonian mechanics?
I've read something from John Baez which I don't understand:
If we consider a single nonrelativistic free particle - in
one-dimensional space, to keep life simple - and describe its state by
its ...
- 757
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264
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The mathematical model of the Galilean transformation in Hamiltonian mechanics
In my previous question, I asked about the Galilean invariance of the Hamiltonian. I've got already two answers, probably good but I have difficulties interpreting them. Both answers write the ...
- 757
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2
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586
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When is a Hamiltonian Galilean invariant?
The Hamiltonian of a point mass connected to a fixed point by a spring in (1-dimensional) space is
$$H(x,p)=\frac{p^2}{2m}+\frac{1}{2}kx^2\tag{1}\label{eq1}$$
The Hamiltonian equations are
$$\begin{...
- 757
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1
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307
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Mathematical model for Euclidean space in classical mechanics
Even knowing some basics of maths and physics I get puzzled when I try to systematise some concepts for better understanding. One is basically on how all the mathematical concepts comprise the model ...
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123
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About the definition of world-line in Arnold's book
I am reading Arnold's Mathematical Methods of Classical Mechanics. I have some questions about the definition of world-line. The book says:
A curve in Galilean space which appears in some (and ...
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2
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146
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Galilean invariance of Newton second law with potential force
Let's consider particle in some potential $V(x)$. Newton law:
$$m \ddot x= -\partial _x V(x)$$
After Galilean transformation:
$$
x^\prime = x -vt
\;\;\;\;\;\;\;\;\;\;
t^\prime = t
$$
Form of equation ...
- 5,737
2
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2
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521
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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
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133
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Two bodies initially at rest with arbitrary interaction remain in the line that connects them [closed]
I am working through VI Arnold "Mathematical Methods of Classical Mechanics". One of the first problems, after defining Galilean structures and Newton's equation of motion, it is to prove '...
- 940
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69
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What is actually meant when it is said Scalar is invariant?
As far as i know a quantity is called invariant if it satisfies some specific transformations.
Now,Suppose a body is moving with velocity $\vec{v}$ as measured from the lab frame.Its non-relativistic ...
- 1,108
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3
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107
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What does it mean that two frames are " in a state of constant, rectilinear motion with respect to one another"?
This expression ( applied to reference frames) " being in a state of constant, rectilinear motion with respect to one another "is frequently used as self explanatory . Though I might appear as stupid, ...
user229097
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About an ambiguity that really prevents me from understanding the principle " the laws of physics are invariant in all inertial frames"
The principle is often stated as self explanatory ....
The sentence " the laws of physics are the same in all inertial frames" could mean 2 very different things ( at least from my point of view).
...
user229097
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0
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237
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Most general form of Lagrangian only with respect to Galilean invariance
Let us assume we are doing classical one point particle mechanics.
Assume that the least action principle holds. Also, assume that Lagrangian $L$ is a function only of coordinate $x$, its derivative $\...
- 409
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54
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Inertial frame definition in Rindler Introduction to STR vs Landau' & Lifshitz Mechanics
Juxtaposing Rindler's Introduction to STR (page 7) vs Landau's Mechanics (page 5) inertial frame definition,I get that rindler assumes frame moving uniformly w.r.t inertial frame as an inertial frame ...
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3
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Why the Lagrangian of a free particle cannot depend on the position or time, explicitly?
On p. 5 in $\S$3 pf the book of Mechanics by Landau & Lifshitz, it is claimed that
[...] for a free particle, the homogeneity of space and time implies
that Lagrangian cannot depend on ...
- 2,313
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293
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Relationship between the Galilei Group and the Phase Space
This question is kind of a follow up question to my last question on the need for canonical commutation relations and conjugate observables. A comment from Valter Moretti suggested that, given a ...
- 6,955
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2
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607
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Galilean transformations of velocity
If I perform a Galilean boost $$x' = x - vt \\ t'=t$$ between two frames $S$ and $S'$, observers in each frame would disagree on the velocity of a particle because
$$ \frac{dx'}{dt'} = \frac{dx}{dt} -...
- 4,054
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1k
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Understanding Galilean Structure
I’m a student with a pure math background starting to work through Arnold’s “Mathematical Methods...” and I’m struggling right of the bat with Section 1.2 on Galilean Structure. (pg 4 - 6)
So we have ...
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1
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122
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Question about Galilean time invariance
I've been reading Arnold's book on Classical Mechanics. I understand that most "classical" forces such as gravity, spring are supposed to be Galilean invariant. But what if I start a rocket, and ...
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Definition of Galilean structure in Arnold's book?
I am reading Arnold's Mathematical Methods of Classical Mechanics.
He quickly introduces the notion of Galilean structure. The universe is defined as the affine space $A^4$ and time is defined as a ...
- 951
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1
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The Lagrangian of a free particle in Landau & Lifshitz
In Landau & Lifshitz's derivation of the Lagrangian of a free particle in a galilean frame of reference one finds the following argument: the equations of motion in two galilean frames must be ...
user115153
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1
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Is there an "invariant" quantity for the classical Lagrangian?
$$
L = \sum _ { i = 1 } ^ { N } \frac { 1 } { 2 } m _ { i } \left| \dot { \vec { x } _ { i } } \right| ^ { 2 } - \sum _ { i < j } V \left( \vec { x } _ { i } - \vec { x } _ { j } \right)
$$
This ...
- 1,708
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441
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How can I interpret or mathematically formalize Maxwellian, Leibnizian, and Machian space-times?
I've been reading the book, World Enough and Space-Time, and I came across a rough list of classical space-times with varying structural significance.
Here is the same list, minus Machian Space-time,...
1
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2
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891
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Question on Galilean transformation
Let $a$ be a scalar, $D$ a rotation matrix and $b$ and $v$ are $1\times 3$-vectors.
We had the following Galiean transformation:
$(t, x(t)) \to (t + a, Dx + b + v\cdot t)$
But why is it not
$(t, x(...
user102683
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0
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Why should the potential of a non-relativistic isolated system be velocity independent?
The lagrangian function of an non-relativistic isolated system of point masses is
$$L=\sum_i\frac{m_i}{2}\dot{\vec r}_i^2-V,$$
where the potential function $V$ represents all interactions.
If we ...
- 18k
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459
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Galileo principle (from Landau Lifshitz to derive free particle Lagrangian)
I am reading the Landau & Lifshitz on mechanics to understand how we find the free particle Lagrangian, and there are some things that I don't understand.
First, he defines an inertial frame as ...
- 1,560
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2
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606
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Expansion in $\epsilon$ and $v^2$ dependence of the Lagrangian - Landau & Lifshitz's Mechanics [duplicate]
On page 4 of Landau & Lifshitz's Mechanics they say
$$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right).$$ Expanding this expression in powers of $\...
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A question concerning the Galilean invariance of Newton's laws
When proving the Galilean invariance of Newton's laws is it tacitly assumed that all equations are covariant, i.e. that they are form invariant?
For example, it is fairly trivial to show that the ...
- 3,267
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2
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572
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Why does Galilean invariance imply that particles that start rest stay on the same line?
I'm reading Arnol'd for self study. I'm struggling with this question: "Show that any system of two particles will remain on the same line that connected them at the initial moment, if they started at ...
- 169
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1
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752
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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
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564
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Derive Galilean transformation. (The meaning of the relativity)
In the book The meaning of the relativity Einstein says that in classic mechanics two postulates are previously supposed:
1.- The time is absolute.
2.- The longitude is absolute.
And this implies ...
- 353
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1
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How to prove that "all unaccelerated frames behave likely for all isolated bodies"? [closed]
Say in an unaccelerated frame "S" a "isolated body A" moves with constancy of velocity , can we predict mathematically that any other such body B will move with same velocity in that frame....
My ...
- 131
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2
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575
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Galilean relativity & the road to special relativity
Firstly, I just want to make sure that I've understood the notions of relative and absolute quantities correctly.
Elementary analysis shows that position and velocity are relative quantities. Indeed, ...
- 3,093
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1
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601
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Can someone explain intuitively how, for a Galilean universe, $A^4$ is equivalent to $\Bbb{R} \times \Bbb{R}^3$?
I am reading Arnold's book on classical mechanics. Obviously, everyone who's studied basic physics feels comfortable with $\Bbb{R} \times \Bbb{R}^3$. This is just a pair $(t,\mathbf{x})$. There are ...
- 4,231
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3
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Why Galilean spacetime is not $\mathbb{E}^4$?
In Newtonian mechanics the physical spacetime is a Galilean spacetime with an affine surjection $\pi : \mathbb{A}^4\to \mathbb{E}^1$ from affine space $\mathbb{A}^4$ to Euclidean space $\mathbb{E}^1$. ...
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