All Questions
64
questions
-1
votes
0
answers
80
views
Is there a straightforward simplified proof of energy conservation from time translation symmetry?
Electric charge conservation is easily proven from electric potential gauge symmetry, as follows:
The potential energy of an electric charge is proportional to the electric potential at its location.
...
3
votes
1
answer
82
views
Does quasi-symmetry preserve the solution of the equation of motion?
In some field theory textbooks, such as the CFT Yellow Book (P40), the authors claim that a theory has a certain symmetry, which means that the action of the theory does not change under the symmetry ...
- 51
3
votes
5
answers
939
views
What is the point of knowing symmetries, conservation quantities of a system?
I think this kind of question has been asked, but i couldn’t find it.
Well i have already know things like symmetries, conserved quantities and Noether’s theorem, as well as their role in particle ...
- 311
5
votes
1
answer
593
views
Does the Hamiltonian formalism yield more Noether charges than the Lagrangian formalism?
In Lagrangian formalism, we consider point transformations $Q_i=Q_i(q,t)$ because the Euler-Lagrange equation is covariant only under these transformations. Point transformations do not explicitly ...
- 145
1
vote
0
answers
54
views
Doubt Regarding Noether's theorem for time-dependent systems
I'm having problems showing Noether's theorem when the lagrangian is time dependent. I'm trying to do it not using infinitesimal transformations from the beginning, but continuous transformations of a ...
2
votes
1
answer
72
views
Some doubts about action symmetry
We know that Symmetry of the Lagrangian ($\delta L = 0$) always yields some conservation law.
Now, if $\delta L \neq 0$, that doesn't mean we won't have conservation law, because if we can show action ...
- 525
1
vote
1
answer
137
views
In a simple case of a particle in a uniform gravitational field, do we have translation invariance or not?
Consider a system where a particle is placed in a uniform gravitational field $\vec{F} = -mg\,\vec{e}_{z}$. The dynamics of this are clearly invariant under translations. When we take $z\rightarrow z+...
- 8,660
1
vote
0
answers
27
views
Analytical mechanics: Noether charge as a generator with variating fields to time direction [duplicate]
Summary
I want to clarify how can I prove the fact that "the Noether charge generates the corresponding transformation" when the infinitesimal transformation of the fields contain the ...
- 353
2
votes
1
answer
293
views
Geometrical intuition for Noether's Theorem
I have been reading some questions about the relation between Noether's Theorem and Lie Algebras and I wanted to get some intuition on it, but I didn't find what I really wanted. Also, the majority of ...
- 154
4
votes
0
answers
170
views
Noether's Theorem in non-conservative systems
In most books on classical mechanics, Noether's Theorem is only formulated in conservative systems with an action principle. Therefore I was wondering if it is possible to also do that in non-...
- 470
5
votes
2
answers
775
views
Intuition behind the definition of Continuous Symmetry of a Lagrangian (Proof of Noether's Theorem)
Suppose there is a one-parameter family of continuous transformations that maps co-ordinates $q(t)\rightarrow Q(s,t)$ where the $s$ is the continuous parameter. Also, for when $s=0$ the transformation ...
- 2,958
0
votes
2
answers
85
views
How did Noether use the total time derivation to get her conservation of energy? [duplicate]
I was informed by @hft that by combining the total time derivation:
$$\frac{dL}{dt} = \frac{\partial L}{\partial x}\dot{x} +
\frac{\partial L}{\partial \dot{x}}\ddot{x} +
\frac{\partial L}{\partial t}$...
- 625
-1
votes
2
answers
624
views
Conservation theorem for cyclic coordinates in the Lagrangian
Suppose $q_1,q_2,...,q_j,..,q_n$ are the generalized coordinates of a system.
$q_j$ is not there in the Lagrangian (it is cyclic).
Then $\frac{\partial L}{\partial\dot q_j}=constant$
In Goldstein, it ...
- 293
2
votes
3
answers
480
views
Lagrangian first integral
I want to extremize $$\int dt \frac{\sqrt{\dot x ^2 + \dot y ^2}}{y}.$$
I have thought that, since the Lagrangian $L(y, \dot y, \dot x)$ is $t$ dependent only implicitly, that i could use the fact ...
- 980
2
votes
3
answers
191
views
Noether‘s theorem: Why can function be dependent of $\dot{q}$?
We define a continuous symmetry in Lagrangian mechanics as follows:
$$\delta L\overset{!}{=}\epsilon\frac{\mathrm{d}}{\mathrm{d} t} f(q,\dot{q}, t)$$
Where $\epsilon\in\mathbb{R}$ is a parameter in ...
- 425
1
vote
0
answers
70
views
Symmetry of a time-dependent Lagrangian
How do I get the group of symmetries and the constant of motion of $L=\frac{\dot{x}^2}{2}m+V(x+ct)$ where c is a constant?
When I tried to solve it, it was to look for shifts in $x$ and $ct$ under ...
- 11
4
votes
3
answers
270
views
Newtonian vs Lagrangian symmetry
Suppose we have a ball of mass $m$ in the Earth's gravitational field ($g=const.$). Equation of motion reads as:
$$
ma = -mg
$$
From here we can conclude that we have translational symmetry of the ...
- 1,795
1
vote
1
answer
156
views
Number of conservation laws
I saw a discussion about the relation of symmetries of Lagrangian and conservation laws on a textbook of analytical mechanics. A part that was counterintuitive to me was that all the discussion was ...
- 63
0
votes
0
answers
71
views
Conservations for time or space translational invariance, why $\delta L\vert_{\text{time trans}}\neq 0$? but $\delta L\vert_{\text{space trans}}= 0$?
To summarize my question first,
Given a classical mechanics Lagrangian,
$L=L(x(t), \dot{x}(t); t)$,
Why the conservation law for time $t$-translational invariant system, under time variance $\delta ...
- 4,336
0
votes
0
answers
112
views
How conserved quantities lead to equations of motion in Lagrangian mechanics
In a classical mechanics exercise, we were asked to derive a system of ODEs from conserved quantities. As we know from Lagrangian mechanics, Euler-Lagrange equations lead to equations of motion.
I ...
- 297
2
votes
1
answer
183
views
Symmetry Condition in Noether's Theorem
Suppose $q = \{q_1,\cdots, q_i\}$ is a coordinate system for Lagrangian $L(q,\dot{q},t)$. In this text by David Morin, on page 16 in chapter 6, it states that a symmetry is a transformation of the ...
- 106
0
votes
1
answer
223
views
0
votes
0
answers
51
views
Why does the conserved quantity generate the transformation in quantum mechanics? [duplicate]
If we have a lagrangian with a symmetry then we get a conserved quantity: $$Q=Q(p,q)$$ which is a function of the conjugate momentum and the coordinates.
If we move over to quantum mechanics then we ...
- 1,933
1
vote
4
answers
597
views
Problem with Noether Theorem to prove that energy is conserved
Suppose an action $S = \int _{t_1}^{t_2} L(q(t),\dot{q}(t))$ that is invariant under an infinitesimal constant time translation $t \longrightarrow t' = t + \epsilon$, of course with $\epsilon = ...
- 1,290
0
votes
1
answer
164
views
Why is there a relationship between symmetries and conservation laws?
I am reading through my professor's notes and I am unsure as to what the intimate relationship between the symmetry property of a physical system and the conservation laws of energy, momentum, and ...
- 65
2
votes
0
answers
76
views
Conflict of domain and endpoints in Noether's theorem and energy conservation
In the derivation of energy conservation, there is the transformation $q(t)\rightarrow q'(t)=q(t+\epsilon)$, whose end points are kind of fuzzy. The original path $q(t)$ is only defined from $t_1$ to $...
2
votes
0
answers
186
views
Conserved quantities of a simple Lagrangian
Suppose the following Lagrangian with 2 degrees of freedom:
$$L = \frac{3}{2}\dot{q}^2_{1} \ + \frac{3}{2}\dot{q}^2_{2} \ + \dot{q}_{1}\dot{q}_{2}$$
My aim is to find all the conserved quantities of ...
0
votes
1
answer
50
views
Physical meaning of constants (momenta?) generated by Noether's theorem via an ${\rm SO}(3)$-action
Let $\Bbb R^3$ be our configuration space. Consider the Lagrangian $L\colon T\Bbb R^3 \cong \Bbb R^6 \to \Bbb R$ given by$$L(x,y,z,\dot{x},\dot{y},\dot{z}) = \frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^3) ...
- 545
2
votes
0
answers
425
views
Is there any reason that Landau and Lifshitz don't discuss Noether's theorem in their mechanics book? [closed]
I'm currently working my way through the Course volume one. Unless I've completely missed it, the authors omit any discussion of Noether's theorem, instead deriving various conservation laws on a case ...
- 2,157
1
vote
1
answer
163
views
Free Fall Conservation of Momentum
So I looked at the invariance of the Lagrangian under the Gallilei Transformations.
So for the free fall we have the Lagrangian
$$L = \frac{m}{2}\dot{z}^2 -mgz$$
Then I applied the transformation
$$x\...
2
votes
1
answer
321
views
Proof of Noether's theorem: How to deal with transformations in time?
I was following the proof of Noether's theorem in Lemos - Analytical Mechanics, page 73.
He considers a full infinitesimal transformation:
$$t'=t+\epsilon X(q(t),t),$$
$$q'(t')=q(t)+\epsilon\Psi(q(t),...
- 17.8k
0
votes
1
answer
2k
views
Getting a Conserved Quantity from a Lagrangian [duplicate]
So I've been messing around with the implications of Noether's theorem, and though I conceptually get what it's saying, I'm having a hard time actually using it to retrieve a conserved quantity from a ...
- 167
1
vote
1
answer
80
views
How do we define the quantity $Q$, in the conservation of energy? And what does it rely on?
Noether's theorem to me explains how a certain defined quantity (Q) is conserved (locally) in time due to the time translation symmetry, and to be more specific; if we had a ball that is placed in a ...
2
votes
1
answer
793
views
Problem using Noether's theorem in time-dependent lagrangian
I have some problems calculating the conserved quantity for a lagrangian of the
form
$$
L = \frac{1}{2}m\dot{q}^2 - f(t) a q,
$$
because I found the general problem too abstract, I tried at
first ...
- 173
3
votes
3
answers
286
views
Does Noether's theorem apply to constrained system?
The Lagrangian of a constrained system will be
$$L-\lambda_1f_1-\lambda_2f_2-...\lambda_kf_k.$$
If a transformation will not affect the constrained Lagrangian, the there is some corresponding ...
- 133
0
votes
1
answer
158
views
Question about the concepts of Noether charge and Noether current
I read that a noether current occurs when the lagrangian assume vector values. Well, what are noether current and noether charge in comparison to elementary classical mechanics notions of Noether's ...
- 3,085
1
vote
1
answer
134
views
Total energy in rheonomic systems
I'm reading Lanczos Variational Principles of Mechanics p.124, and following a discussion of how for scleronomic systems we get
$$\sum_{i=1}^{n} p_i\dot q_i - L = const.\tag{53.12}$$
For rheonomic ...
- 187
-2
votes
1
answer
151
views
Conservation laws for weird Lagrangian? [closed]
I am asked to find the conserved quantities for the following Lagrangian for a three-particle system in three dimensions
$$L = \left[\sum_{i=1}^{3} \frac{1}{2} m_i \left(|\dot{\bf{r}}|^2 - \omega^2 ...
- 620
3
votes
2
answers
464
views
Relationship between vector field, generator & scalar field in Noether's theorem
I wonder "which quantity" is conserved in relation to a specific symmetry.
I guess it is in some meaning simply the generator (in the context of Lie theory) of the symmetry, as it is true for angular ...
- 1,409
1
vote
1
answer
81
views
Showing time translation and a general rotation are symmetries of a given Lagrangian
Given a "free particle" Lagrangian:
$$
L=\frac{m}{2}\left(\frac{dq}{dt}\right)^2,
$$
(a) Show that $t \rightarrow t'= t+s$ is a symmetry of $L$.
(b) Show that $q \rightarrow q'= Rq$ s a ...
- 43
5
votes
1
answer
2k
views
Is there an "invariant" quantity for the classical Lagrangian?
$$
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)
$$
This ...
- 1,669
33
votes
3
answers
6k
views
Why is Noether's theorem important?
I am just starting to wrap my head around analytical mechanics, so this question might sound weird or trivial to some of you.
In class I have been introduced to Noether's theorem, which states that ...
- 490
0
votes
1
answer
277
views
What is the logic that leads to conservation of energy from time invariance? [duplicate]
I have read different accounts of time invariance leading to the conservation of energy, but have not encountered the specific logical explanation for it. Can someone provide it?
6
votes
1
answer
749
views
Proving Noether's theorem in classical mechanics
I'm trying to prove Noether's theorem in the context of (point-particle) classical mechanics, however, I'm a bit unsure on a few things.
To keep things as simple as possible I'm only considering the ...
- 3,207
1
vote
1
answer
651
views
From Noether's Theorem, is it true that the law of conservation of energy can be proved? [duplicate]
So what I understand is that the law of conservation of energy, likes Newton's law of motion, can't be proved. However, by Noether's Theorem, if there is a time symmetry, the energy is conserved. It ...
- 277
4
votes
2
answers
466
views
Confusion about symmetry and conservation
I think I am misunderstanding the concept of symmetry in Lagrangian mechanics or maybe I am misunderstanding the content of Noether's theorem. Let me elaborate:
Suppose $L(q,\dot q,t)$ is the ...
- 56
3
votes
3
answers
979
views
Noether's Theorem and Lagrangian symmetries
If I make the transformation
$$ x_i \rightarrow x_i' = x_i + \delta x_i,$$
I find that the Lagrangian $L = L(x_i,\dot{x_i})$ transforms as
$$ L \rightarrow L' = L + \frac{\partial L}{\partial x_i} \...
- 3,986
1
vote
1
answer
67
views
Noethers Symmetries for a system in different cases
If the lagrangian is
$$L=\frac{m}{2}\left( \dot x_1^2+ \dot x_2^2\right)−b(x_1−x_2)^2+a( \dot x_1x_2− \dot x_2x_1).$$
What are the Noether symmetries of the system and the corresponding conserved ...
- 21
2
votes
1
answer
388
views
Noether's Theorem: form of infinitesimal transformation
Noether's theorem states that if the functional $J$ is an extremal and invariant under infinitesimal transformation,
$$ t' = t+ \epsilon \tau + ...,\tag{1}$$
$$ q^{\mu'} = q^{\mu} + \epsilon \zeta^{\...
- 328
7
votes
3
answers
11k
views
Constants of motion from a Lagrangian
If I have a Lagrangian (made up equation in this case):
$$L = \frac{1}{2}mr^2\dot\theta + \frac{1}{4}mg\ddot\theta \, ,$$
can I immediately conclude that the total energy is constant because $\...
