All Questions
44
questions
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?
