All Questions
Tagged with galilean-relativity lagrangian-formalism
20 questions
2
votes
1
answer
159
views
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
1
vote
0
answers
58
views
Independence of Lagrange function from time and position
In Landau & Lifshitz "Mechanics", it is said that from the time/space homogeneity Lagrange function is independent from time/position. I always thought that homogeneity means that motion ...
- 39
4
votes
4
answers
357
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
2
votes
1
answer
247
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Action of free particle is invariant under Galilean transformation / Transformation of derivative
I want to show that the action of a free particle is invariant under a Galilean transformation
$$
(t,\vec{x})\rightarrow (t+a,R\vec{x}+\vec{v}t+b)=(t^\prime, \vec{x}^\prime) \quad\text{where}\quad R\...
- 405
4
votes
3
answers
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
0
votes
0
answers
183
views
Lagrangian invariance under Galilean transform and conservation law of linear momentum
I'm currently taking a course of analytical mechanics. We've learned about the invariance of the Lagrangian under change of coordinates, and showed that we get the same Lagrangian for free particle ...
- 325
5
votes
2
answers
274
views
Do total derivatives have anything to do with central extensions?
I recently got interested in the Galilean group and its central extension and found a paper "Quantization on a Lie group: Higher-order Polarizations" by Aldaya, Guerrero and Marmo.
Before asking my ...
- 11.8k
0
votes
0
answers
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
1
vote
1
answer
326
views
Galilei Invariance and Newton Third Law
Let's say we have a system of two point particles that can interact with each other by forces that are position and velocity dependent. The forces might or might not be derivable from a generalized ...
- 458
1
vote
1
answer
182
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Lagrangian of free particle - classical case
I have a question, more related to a mathematical aspect of physics, which seems I am not understanding very well.
So, by applying Galilean transformation between two reference frames, which move at ...
- 19
3
votes
3
answers
1k
views
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
3
votes
1
answer
2k
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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
5
votes
1
answer
2k
views
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)
$$
This ...
- 1,708
3
votes
1
answer
1k
views
Derive the Lagrangian that yields the free Schrödinger's equation from Galileian Invariance
The Lagrangian Density $$L(\Psi, \Psi^*)=i \hbar \dot{\Psi} \Psi^* + \frac{\hbar^2}{2m} \Psi \Delta \Psi^*$$ will yield the schroedinger equations for $\Psi$ and $\Psi^*$. Can we derive this ...
- 6,970
2
votes
0
answers
56
views
Why should the potential of a non-relativistic isolated system be velocity independent?
The lagrangian function of an non-relativistic isolated system of point masses is
$$L=\sum_i\frac{m_i}{2}\dot{\vec r}_i^2-V,$$
where the potential function $V$ represents all interactions.
If we ...
- 18k
0
votes
0
answers
459
views
Galileo principle (from Landau Lifshitz to derive free particle Lagrangian)
I am reading the Landau & Lifshitz on mechanics to understand how we find the free particle Lagrangian, and there are some things that I don't understand.
First, he defines an inertial frame as ...
- 1,560
1
vote
2
answers
606
views
Expansion in $\epsilon$ and $v^2$ dependence of the Lagrangian - Landau & Lifshitz's Mechanics [duplicate]
On page 4 of Landau & Lifshitz's Mechanics they say
$$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right).$$ Expanding this expression in powers of $\...
- 11
2
votes
2
answers
855
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Mechanics Landau Galilean Principle
I started reading Landau's Mechanics book and was having some trouble understanding the Galilean Relativity Principle.
What does Landau mean by saying space to be homogenous and isotropic and time is ...
- 1,586
37
votes
3
answers
26k
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Deriving the Lagrangian for a free particle
I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away.
Proving that a free ...
- 473
30
votes
5
answers
9k
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Galilean invariance of Lagrangian for non-relativistic free point particle?
In QFT, the Lagrangian density is explicitly constructed to be Lorentz-invariant from the beginning. However the Lagrangian
$$L = \frac{1}{2} mv^2$$
for a non-relativistic free point particle is ...
- 4,156