All Questions
44
questions
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0
answers
78
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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.
...
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
59
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
53
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
50
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
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. ...
- 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, ...
- 695
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
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 ...
- 1,640
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 ...
- 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 ...
- 347
2
votes
1
answer
803
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
202
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
660
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
13
votes
3
answers
2k
views
In a Lagrangian, why can't we replace kinetic energy by total energy minus potential energy?
TL;DR: Why can't we write $\mathcal{L} = E - 2V$ where $E = T + V = $ Total Energy?
Let us consider the case of a particle in a gravitational field starting from rest.
Initially, Kinetic energy $T$ is ...
- 255
1
vote
0
answers
81
views
What is the physical interpretation of a Lagrangian with $\dot{x}^4$?
Among the exercises in the first chapter of Goldstein's book "Classical Mechanics", it appears the lagrangian
$$
L\left(x,\frac{dx}{dt}\right) = \frac{m^2}{12}\left(\frac{dx}{dt}\right)^4 + m\left(\...
- 1,211
1
vote
1
answer
297
views
Landau Classical Mechanics - Disintegration of particles
I am reading Landau's Classical Mechanics book and I'm having trouble understanding the concept of "internal energy". In Landau's words:
the energy of a mechanical system which is at rest as a whole ...
- 129
1
vote
4
answers
597
views
Problem with Noether Theorem to prove that energy is conserved
Suppose an action $S = \int _{t_1}^{t_2} L(q(t),\dot{q}(t))$ that is invariant under an infinitesimal constant time translation $t \longrightarrow t' = t + \epsilon$, of course with $\epsilon = ...
- 1,290
2
votes
2
answers
370
views
Conservation of total energy for a system with holonomic constraints
Consider a system with generalized coordinates $u_1, u_2$ and $u_3$ such that $u_1$ and $u_2$ are dependent through the following holonomic constraint
\begin{equation}
G(u_1, u_2)=0.
\end{equation}
It ...
- 23
0
votes
1
answer
2k
views
Getting a Conserved Quantity from a Lagrangian [duplicate]
So I've been messing around with the implications of Noether's theorem, and though I conceptually get what it's saying, I'm having a hard time actually using it to retrieve a conserved quantity from a ...
- 167
1
vote
1
answer
80
views
How do we define the quantity $Q$, in the conservation of energy? And what does it rely on?
Noether's theorem to me explains how a certain defined quantity (Q) is conserved (locally) in time due to the time translation symmetry, and to be more specific; if we had a ball that is placed in a ...
1
vote
1
answer
108
views
Lagrange Equation - Basics
The basic equation of Lagrange is given by,
$$\frac{\mathrm d}{\mathrm dt} \frac{\partial L}{\partial \dot{q_j}} - \frac{\partial L}{\partial q_j} = Q_j \tag{1}$$
where $T$ is the kinetic energy, $V$ ...
- 17
0
votes
1
answer
529
views
Total work zero along the virtual displacement
I'm having some trouble understanding virtual work and displacement, especially a particular section of Goldstein. I'll use an example to explain my difficulty, but I realize this might be the product ...
- 2,849
2
votes
1
answer
125
views
Potential energy and conservation law
I'm preparing for my masters entrance exam on pure mathematics (thought some problems are devoted to classical/lagrangian mechanics). I would be grateful to clarify some basics regarding the ...
- 123
1
vote
1
answer
134
views
Total energy in rheonomic systems
I'm reading Lanczos Variational Principles of Mechanics p.124, and following a discussion of how for scleronomic systems we get
$$\sum_{i=1}^{n} p_i\dot q_i - L = const.\tag{53.12}$$
For rheonomic ...
- 187
0
votes
1
answer
555
views
Is the relation between Hamilton's and Lagrange's equations the same as that between conservation of energy and the equations of motion?
Conservation of energy is, usually, a $\textbf{first order}$ non linear differential equation, generally written as
$$
\frac{m\dot{q}^2}{2} +V(q) = cte.
$$
Taking the derivative yields the usual ...
- 132
1
vote
2
answers
1k
views
Show the total energy is conserved
If the Lagrangian does not depend explicitly on time, then the quantity $E$ given by
$$E := p\dot{x} - L \tag{1}$$
is conserved.
I'm really confused. Normally the total energy is given by $$E = T ...
- 135
0
votes
1
answer
277
views
What is the logic that leads to conservation of energy from time invariance? [duplicate]
I have read different accounts of time invariance leading to the conservation of energy, but have not encountered the specific logical explanation for it. Can someone provide it?
