All Questions
8
questions
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
17
votes
2
answers
7k
views
Lagrangian of an effective potential
If there is a system, described by an Lagrangian $\mathcal{L}$ of the form
$$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$
where $T$ is the kinetic ...
- 377
10
votes
3
answers
4k
views
Is there a valid Lagrangian formulation for all classical systems?
Can one use the Lagrangian formalism for all classical systems, i.e. systems with a set of trajectories $\vec{x}_i(t)$ describing paths?
On the wikipedia page of Lagrangian mechanics, there is an ...
- 8,523
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( \...
- 6,425
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
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
3
votes
1
answer
5k
views
Is the Hamiltonian conserved or not?
The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
- 219
1
vote
1
answer
1k
views
What are the assumptions behind the Lagrangian derivation of energy?
What are the assumptions behind the Lagrangian derivation of energy? I understand that we're searching for a function $L$ that describes a set of physics so that solving the energy minimization ...
- 133