All Questions
44
questions
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Is there a straightforward simplified proof of energy conservation from time translation symmetry?
Electric charge conservation is easily proven from electric potential gauge symmetry, as follows:
The potential energy of an electric charge is proportional to the electric potential at its location.
...
- 12.1k
30
votes
6
answers
8k
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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 ...
- 21.9k
2
votes
2
answers
366
views
Can classical Lagrangian mechanics be obtained directly from energy conservation?
Is there a way to derive classical Lagrangian mechanics (in particular, the classical Lagrangian $L = T-V$ and the Euler-Lagrange equation), under the simple assumption that mechanical energy is ...
- 73.8k
2
votes
1
answer
803
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A particle constrained to always move on a surface whose equation is $\sigma (\textbf{r},t)=0$. Show that the particle energy is not conserved
In Goldstein's Classical mechanics question 2.22
Suppose a particle moves in space subject to a conservative potential $V(\textbf{r})$ but is constrained to always move on a surface whose equation is ...
CommunityBot
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8
votes
1
answer
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If the Lagrangian depends explicitly on time then the Hamiltonian is not conserved?
Why is the Hamiltonian not conserved when the Lagrangian has an explicit time dependence? What I mean is that it is very obvious to argue that if the Lagrangian has no an explicit time dependence $L=L(...
- 3,778
1
vote
0
answers
53
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Intuition behind energy not being conserved in Rheonomous mechanical system [closed]
firstly, this is what Rheonomous System means. So, in such a system, the kinetic energy is not exactly just a quadratic function of generalized velocities because one of the generalized coordinates ...
- 207k
1
vote
1
answer
1k
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Pendulum with Oscillatory Support - A question on Lagrangian Mechanics [closed]
Recently I have been attempting Morin's Introduction to Classical Mechanics (2008) but I got rather stuck on question 6.3 on the topic of Lagrangian Mechanics. Attached are the problem and the ...
- 207k
1
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0
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59
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Why is the conserved Lagrangian energy $E$ equal to the total energy in this example but not in a similar example? [duplicate]
I am aware that there exists duplicates to the title and have gone through the answers but it still doesn't answer my issue with a statement in the last image.
These two similar situations with slight ...
- 207k
1
vote
1
answer
50
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Potential energy with Taylor series for particle
I have been doing the following problem:
Imagine we got a particle in $U(x)$ field and we need to consider the motion of the particle near $x=a$. It says to use Taylor series for $U(x)$
$U(x) = U(a) + ...
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3
votes
2
answers
396
views
Hamiltonian conservation in different sets of generalized coordinates
In Goldstein, it says that the Hamiltonian is dependent, in functional form and magnitude, on the chosen set of generalized coordinates. In one set it might be conserved, but in another it might not. ...
- 207k
0
votes
0
answers
80
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Substituting the conservation of angular momentum into the Binet formula results in contradiction [duplicate]
Background Information
The lagrangian of a particle in a central force field $V(r)$ is
$$
L=\frac12m(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\dot\varphi^2)-V(r).
$$
The particle must move in a plane, ...
- 207k
0
votes
2
answers
85
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How did Noether use the total time derivation to get her conservation of energy? [duplicate]
I was informed by @hft that by combining the total time derivation:
$$\frac{dL}{dt} = \frac{\partial L}{\partial x}\dot{x} +
\frac{\partial L}{\partial \dot{x}}\ddot{x} +
\frac{\partial L}{\partial t}$...
- 207k
6
votes
3
answers
191
views
Modelling friction as a conservative force
Friction is usually considered as a non-conservative force, but by considering the microscopic movement of particles which produce the friction, it seems we can model friction as conservative force ...
- 49.9k
17
votes
2
answers
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Lagrangian of an effective potential
If there is a system, described by an Lagrangian $\mathcal{L}$ of the form
$$\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 ...
- 207k
4
votes
2
answers
227
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Are these two expressions for $\mathrm{d}E/\mathrm{d}t$ compatible?
This is purely recreational, but I'm eager to know the answer.
I was playing around with Hamiltonian systems whose Hamiltonian is not equal to their mechanical energy $E$.
If we split the kinetic ...
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