All Questions
64
questions
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