All Questions
Tagged with classical-mechanics noethers-theorem
117
questions
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
8
votes
4
answers
624
views
Connection between Noether's Theorem and classical definitions of energy / momentum
In classical mechanics, change in momentum $\Delta \mathbf p$ and change in kinetic energy $\Delta T$ of a particle are defined as follows in terms of the net force acting on the particle $\mathbf F_\...
- 1,826
4
votes
1
answer
827
views
Can anyone provide a simple, inuitive explanation for Noether's Theorem? [duplicate]
I recently came across this theorem for the first time. As I understand it, what she showed was that conservation 'laws' are often simply an artifact of symmetry or invariance.
For example, the ...
- 206
1
vote
1
answer
404
views
What does lowercase-delta mean in Noether's first theorem?
Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' ...
- 4,509
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
1
vote
0
answers
482
views
Connection between Kepler Problem and Harmonic Oscillator
Background. Take the Kepler Lagrangian as
$L^K = \frac{1}{2}\dot{q}_i\dot{q}_i + \frac{k}{q}$,
and the Lagrangian for the isotropic harmonic oscillator as
$L^H = \frac{1}{2}\dot{q}_i\dot{q}_i - \...
- 21
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?
3
votes
3
answers
1k
views
Is it sloppy to say that the isotropy of space leads to conservation of angular momentum?
Sometimes it is said that the isotropy of space leads to conservation of angular momentum. But the derivation of conservation of angular momentum follows not from isotropy of space but that of the ...
- 26.8k
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