All Questions
64
questions
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
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
9k
views
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
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