All Questions
44
questions
3
votes
0
answers
89
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could a force with corolios and centrifugal terms be written as a potential gradient? [duplicate]
I have an exam in classical mechanics next week, so I came across this problem which I did not fully understand nor any of my colleagues (it was a bonus problem in an old exam) I just want some hint ...
- 111
1
vote
1
answer
651
views
From Noether's Theorem, is it true that the law of conservation of energy can be proved? [duplicate]
So what I understand is that the law of conservation of energy, likes Newton's law of motion, can't be proved. However, by Noether's Theorem, if there is a time symmetry, the energy is conserved. It ...
- 277
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
1
vote
1
answer
1k
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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
1
vote
1
answer
193
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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
1
answer
5k
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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
0
votes
1
answer
958
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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
3
votes
0
answers
222
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Does the additivity property of Integrals of motion and Lagrangians valid in all situations?
I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
- 345
3
votes
2
answers
5k
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Conservation of energy when the Lagrangian includes a potential function
When proving that the homogeneity of time leads to the conservation of energy,
(This is the proof from Landau for the case when there is no field present.)
(Uses the Einstein's summation convention)...
- 163
4
votes
1
answer
576
views
Sufficient conditions for the energy to be not conserved?
I'm almost embarrased to ask this question because I thought I was by now very confident with classical mechanics.
Someone has stated that given a mechanical system with a Lagrangian $L$ s.t. $\frac{...
- 909
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
9
votes
3
answers
3k
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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
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
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 ...
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