All Questions
44
questions
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Question about Problem $12$ in Chapter $11$ from Kibble & Berkshire's book
I write again the problem for convinience:
A rigid rod of length $2a$ is suspended by two light, inextensible strings of length $l$ joining its ends to supports also a distance $2a$ apart and level ...
8
votes
1
answer
2k
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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(...
- 227
1
vote
0
answers
58
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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 ...
1
vote
1
answer
49
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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) + ...
- 525
2
votes
1
answer
154
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Differentiation of the on-shell action with respect to time
From the on-shell action, we derive the following two:
$\frac{\partial S}{\partial t_1} = H(t_1)$,
$\frac{\partial S}{\partial t_2} = -H(t_2)$,
where $H = vp - L$ is the energy function.
I have two ...
- 525
1
vote
4
answers
568
views
What is the difference between total energy and the Lagrangian energy function?
I am primarily looking for the difference in definitions to see how they differ. Given a Lagrangian $L(q_{j}, \dot{q}_{j}, t)$ of a system of finitely many particles, we may define (using Einstein ...
- 8,640
1
vote
1
answer
78
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Meaning of 2 kinetic energy terms in the equations
I have this problem (The two rods will be called links. Link 1 has length $a_1$ while link 2 has length $a_2$. The distance of the center of mass of each link to their respective joint is $l_i$):
And ...
1
vote
2
answers
76
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Conservation of kinetic energy and external forces
In Goldstein's "Classical Mechanics", at page 360 below eq. (8.84) it is stated that:
"If, further, there are no external forces on the system (monogenic and holonomic), ..., then $T$ ...
- 402
1
vote
3
answers
134
views
Euler-Lagrange and Conservation of the Hamiltonian giving two different Equations of Motion
Consider the following Lagrangian:
$$L=mR\left[\frac{1}{2}R\left(\dot{\theta}^2+\omega^{2}\sin^{2}\theta\right)+g\cos\theta\right],$$
with an associated Hamiltonian
$$H=mR\left[\frac{1}{2}R\left(\dot{\...
1
vote
1
answer
88
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Energy of a system executing forced oscillations
In L&L's textbook of Mechanics (Vol. 1 of the Course in Theoretical Physics) $\S 22$ Forced oscillations, one finds the following statement:
\begin{equation}
\xi = \dot{x} + i \omega x, \tag{22.9}...
- 39
0
votes
0
answers
675
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Lagrangian intuition [duplicate]
I am new to lagrangian mechanics and it just baffles me the idea of subtracting potential energy from kinetic energy. Why don't we use kinetic energy alone and the least action path (between two ...
- 105
1
vote
1
answer
103
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Why Lagrangian is $L=\frac{1}{2}mv^2$ and not $mv^2$ for a free particle in an intertial frame? Both are proportional to the square of velocity
Landau writes the Lagrangian of a free particle in a second inertial frame as $$L(v'^{2})=L(v^2)+\frac{\partial L}{\partial v^2}2\textbf{v}\cdot{\epsilon},$$ and then it's written that the Lagrangian ...
- 41
0
votes
1
answer
479
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Central force motion and angular cyclic coordinates
(Goldstein 3rd edition pg 72)
After reducing two-body problem to one-body problem
We now restrict ourselves to conservative central forces, where the potential is $V(r)$ function of $r$ only, so that ...
- 1,260
7
votes
2
answers
1k
views
Example in motivation for Lagrangian formalism
I started reading Quantum Field Theory for the Gifted Amateur by Lancaster & Blundell, and I have a conceptual question regarding their motivation of the Lagrangian formalism. They start by ...
- 203
3
votes
1
answer
537
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Doubt in the expression of Lagrangian of a system [duplicate]
There is a problem given in Goldstein's Classical Mechanics Chapter-1 as
20. A particle of mass $\,m\,$ moves in one dimension such that it has the Lagrangian
\begin{equation}
L\boldsymbol{=}\...
- 436