All Questions
12
questions
28
votes
2
answers
9k
views
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
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
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
9
votes
3
answers
3k
views
Noether's theorem and time-dependent Lagrangians
Noether's theorem says that if the following transformation is a symmetry of the Lagrangian
$$t \to t + \epsilon T$$
$$q \to q + \epsilon Q.$$
Then the following quantity is conserved
$$\left( \...
- 6,435
4
votes
1
answer
609
views
Trivial conserved Noether's current with second derivatives
I'm considering a symmetry transformation on a Lagrangian
$$ \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
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
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
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
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
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
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
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