All Questions
43
questions
8
votes
1
answer
2k
views
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(...
- 227
1
vote
0
answers
58
views
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 ...
1
vote
0
answers
52
views
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 ...
- 301
1
vote
1
answer
49
views
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) + ...
- 525
3
votes
2
answers
391
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. ...
- 860
0
votes
0
answers
80
views
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, ...
- 675
0
votes
2
answers
85
views
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}$...
- 625
6
votes
3
answers
190
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 ...
- 1,640
2
votes
2
answers
363
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 ...
- 155
1
vote
1
answer
1k
views
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 ...
- 337
2
votes
1
answer
797
views
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 ...
- 173
4
votes
2
answers
227
views
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 ...
- 34
0
votes
2
answers
441
views
Total energy in double pendulum system
Given the following double pendulum system as I outline in the picture attached, how can I use the total energy of the system to derive the equations of motion (assuming angles are small of course)?
I ...
- 553
2
votes
1
answer
198
views
Lagrange with Higher Derivatives (Ostrogradsky instability) [duplicate]
In class our teacher told us that, if a Lagrangian contain $\ddot{q_i}$ (i.e., $L(q_i, \dot{q_i}, \ddot{q_i}, t)$) the energy will be unbounded from below and it can take any lower values (in other ...
- 3,122
1
vote
3
answers
659
views
Is minimizing the action same as minimizing the energy?
When we differentiate the total energy with respect to the time and set it to zero (make it stationary), we get an expression as similar to what we get while we minimize action. Also putting the time ...
- 111