All Questions
64
questions
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 $\...
8
votes
4
answers
2k
views
Noether's theorem for space translational symmetry
Imagine a ramp potential of the form $U(x) = a*x + b$ in 1D space. This corresponds to a constant force field over $x$. If I do a classical mechanics experiment with a particle, the particle behaves ...
- 1,222
1
vote
1
answer
193
views
How is it possible to vary time without affect the coordinates or their derivatives?
In the context of Noether's theorem , the Hamiltonian is the constant of motion associated with the time-translational invariance of the Lagrangian. Time-translational invariance is equivalent to the ...
- 3,093
3
votes
2
answers
821
views
Is there something more to Noether's theorem?
From the definition of Lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space. Is the reverse true? Are Lagrangian mechanics ...
- 4,256
9
votes
2
answers
604
views
How general are Noether's theorem in classical mechanics?
I'm going through the derivations of Noether's theorems and I have several criticisms as to how they are presented in popular sources (note that I'm only referring to classical mechanics here and not ...
- 1,619
4
votes
2
answers
635
views
Derivation of law of inertia from Lagrangian method (Landau)
I'm reading Landau's Book.
He tries to conclude the law of inertia from the Lagrange equations.
For that, he argues (by nice suppositions about space and time), that the lagrangian must depend only ...
- 285
4
votes
1
answer
609
views
Trivial conserved Noether's current with second derivatives
I'm considering a symmetry transformation on a Lagrangian
$$ \delta A = \int L(q +\delta q, \dot{q} + \delta \dot{q} , \ddot{q} + \delta \ddot{q}) dt $$
the general variation takes the form
$$ \...
- 2,299
0
votes
1
answer
958
views
Showing time-invariance of Lagrangian with time-displacement operator
I am trying to show that the time-invariance of the Lagrangian of a simple one-particle system implies energy conservation for that system. The first step is, well, to show that the Lagrangian is time-...
- 5,371
1
vote
1
answer
705
views
Given potentials, how does one find conserved quantities using Noether's theorem?
I've been asked to find the conserved quantities of the following 3D potentials:
$U(\vec{r}) = U(x^2)$,
$U(\vec{r}) = U(x^2 + y^2)$ and
$U(\vec{r}) = U(x^2 + y^2 + z^2)$.
For the first one, ...
- 11
2
votes
1
answer
333
views
Lagrangian under time transformation
Given a Lagrangian $$L(q,\dot{q},t)=\sum_{ij}a_{ij}(q)\dot{q}_i\dot{q}_j-V(q_1,q_2,\cdots,q_f)$$show that under a time transformation $t=\lambda T$ ($\lambda$ = constant), the invariance of $\int_1^...
- 592
30
votes
6
answers
8k
views
Noether Theorem and Energy conservation in classical mechanics
I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
- 10.1k