All Questions
Tagged with classical-mechanics noethers-theorem
117
questions
3
votes
1
answer
1k
views
Deriving $p = mv$ from translational symmetry (momentum conservation law)?
"In classical mechanics, momentum is defined as the quantity which is
conserved under global spatial translations or, alternatively, as the
generator of spatial translations."
(G.Parisi, Quantum ...
8
votes
2
answers
7k
views
Explicit time dependence of the Lagrangian and Energy Conservation
Why is energy (or in more general terms,the Hamiltonian) not conserved when the Lagrangian has an explicit time dependence?
I know that we can derive the identity:
$\frac{d \mathcal{H}}{d t} = - {\...
- 1,487
27
votes
2
answers
6k
views
Why does the classical Noether charge become the quantum symmetry generator?
It is often said that the classical charge $Q$ becomes the quantum generator $X$ after quantization. Indeed this is certainly the case for simple examples of energy and momentum. But why should this ...
- 7,936
2
votes
0
answers
198
views
Small unclarity in proof of Noether's Theorem
I'm trying to understand the proof of Noether's Theorem in my Classical Mechanics class. We formulated it as follows:
A continous symmetry is defined as a flow $\phi^{\lambda}(q(t))$ which leaves the ...
user17574
26
votes
1
answer
18k
views
Constants of motion vs. integrals of motion vs. first integrals
Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
user17116
2
votes
1
answer
133
views
Please provide the simplest example you can think of, of generators of time evolution and generalized coordinates
I was reading the Wikipedia article about Noether's theorem and this thing popped out:
Then the resultant perturbation can be written as a linear sum of the
individual types of perturbations
...
- 431
8
votes
2
answers
2k
views
Translation Invariance without Momentum Conservation?
Instead of the actual gravitational force, in which the two masses enter symmetrically, consider something like $$\vec F_{ab} = G\frac{m_a m_b^2}{|\vec r_a - \vec r_b|^2}\hat r_{ab}$$ where $\vec F_{...
- 153
5
votes
1
answer
9k
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How do you know if a coordinate is cyclic if its generalized velocity is not present in the Lagrangian?
Goldstein's Classical Mechanics says that a cyclic coordinate is one that doesn't appear in the Lagrangian of the system, even though its generalized velocity may appear in it (emphasis mine). For ...
- 2,251
7
votes
2
answers
2k
views
What's the importance of Noether's theorem in Physics
The Noether's theorem that I want to mention is the following: Noether's theorem.
I know the importance of Noether's contribution to modern algebra. Can anyone write about Noether's theorem in ...
- 173
57
votes
6
answers
19k
views
What symmetry causes the Runge-Lenz vector to be conserved?
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...
- 5,725
28
votes
2
answers
9k
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Invariance of Lagrangian in Noether's theorem
Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$.
However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
- 2,890
1
vote
0
answers
1k
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Do symmetries increase the number of conserved quantities? [closed]
Let us consider a classical mechanical system of N particles in a constant external field. We have 3N coordinates and 3N velocities, so totally 6N unknown variables. We have 6N ordinary differential ...
