All Questions
44
questions
0
votes
1
answer
58
views
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
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
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
2
votes
1
answer
156
views
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
580
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
views
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
77
views
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
136
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
$$H=mR\left[\frac{1}{2}R\left(\dot{\...
1
vote
1
answer
88
views
Energy of a system executing forced oscillations
In L&L's textbook of Mechanics (Vol. 1 of the Course in Theoretical Physics) $\S 22$ Forced oscillations, one finds the following statement:
\begin{equation}
\xi = \dot{x} + i \omega x, \tag{22.9}...
- 39
0
votes
0
answers
689
views
Lagrangian intuition [duplicate]
I am new to lagrangian mechanics and it just baffles me the idea of subtracting potential energy from kinetic energy. Why don't we use kinetic energy alone and the least action path (between two ...
- 105
1
vote
1
answer
103
views
Why Lagrangian is $L=\frac{1}{2}mv^2$ and not $mv^2$ for a free particle in an intertial frame? Both are proportional to the square of velocity
Landau writes the Lagrangian of a free particle in a second inertial frame as $$L(v'^{2})=L(v^2)+\frac{\partial L}{\partial v^2}2\textbf{v}\cdot{\epsilon},$$ and then it's written that the Lagrangian ...
- 41
0
votes
1
answer
482
views
Central force motion and angular cyclic coordinates
(Goldstein 3rd edition pg 72)
After reducing two-body problem to one-body problem
We now restrict ourselves to conservative central forces, where the potential is $V(r)$ function of $r$ only, so that ...
- 1,270
7
votes
2
answers
2k
views
Example in motivation for Lagrangian formalism
I started reading Quantum Field Theory for the Gifted Amateur by Lancaster & Blundell, and I have a conceptual question regarding their motivation of the Lagrangian formalism. They start by ...
- 203
3
votes
1
answer
537
views
Doubt in the expression of Lagrangian of a system [duplicate]
There is a problem given in Goldstein's Classical Mechanics Chapter-1 as
20. A particle of mass $\,m\,$ moves in one dimension such that it has the Lagrangian
\begin{equation}
L\boldsymbol{=}\...
- 436
1
vote
1
answer
64
views
How is kinetic energy $T$ given by $T=\dfrac{1}{2}\sum_{i}p_{i}\dot{q_{i}}$ in Hamiltonian and Lagrangian mechanics?
Im going through a website teaching Hamiltonian mechanics and I know the below
$$-\dot{p}_{i}=\dfrac{\partial H}{\partial q_{i}} \tag{14.3.12}$$
$$\dot{q}_{i}=\dfrac{\partial H}{\partial p_{i}} \tag{...
- 1,270
3
votes
1
answer
113
views
Gram-Schmidt Orthogonalisation for scalars
I'm reading Chapter 11 (Normal Modes) of Classical Mechanics (5th ed.) by Berkshire and Kibble and came across this on pg. 253:
The kinetic energy in terms in terms of the generalised coordinates is ...
- 1,047
0
votes
1
answer
237
views
Why is the energy function not always equal to total energy? [duplicate]
Why is the energy function $h = \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $ not always equal to total energy $E = T + V$? Here $T$ is Kinetic Energy and $V$ is Potential Energy. I've read ...
- 75
1
vote
0
answers
129
views
Difference between eigenvalues of the potential energy Hessian vs. "generalized" eigenvalues with respect to a kinetic energy "metric"
Simple version
Consider if we have a Lagrangian defined by
$$L(q,\dot{q}) = \frac{1}{2} g_{ij}(q) \dot{q}^i \dot{q}^j - U(q) \tag{1a}$$
where the potential energy $U(q)$ has a single minimum at $q=0$ (...
- 3,710
12
votes
2
answers
2k
views
Can the Lagrangian be written as a function of ONLY time?
The lagrangian is always phrased as $L(t,q,\dot{q})$.
If you magically knew the equations $q(t)$ and $\dot{q}(t)$, could the Lagrangian ever be written only as a function of time?
Take freefall for ...
- 484
2
votes
1
answer
145
views
Is it possible to derive the equations of motion from the energy of a system alone, without knowing canonical coordinates or the Lagrangian?
Is it possible to derive the equations of motion from the energy of a system alone, without knowing canonical coordinates or the Lagrangian?
I'm confused about which parts of the fundamental ...
- 21
3
votes
1
answer
1k
views
Why is total kinetic energy always equal to the sum of rotational and translational kinetic energies?
My derivation is as follows.
The total KE, $T_r$ for a rigid object purely rotating about an axis with angular velocity $\bf{ω}$ and with the $i$th particle rotating with velocity $ \textbf{v}_{(rot)...
user86425
3
votes
4
answers
660
views
Misunderstanding in deriving Newton’s law from Euler-Lagrange equation
When deriving Newton’s law from Euler-Lagrange equation for a particle, the Lagrangian is defined as the kinetic energy minus the potential energy, but the problem is that the kinetic energy is ...
- 31
0
votes
0
answers
69
views
When the hamiltonian isn't equal to energy? [duplicate]
I have the following hamiltonian:
$$H = \frac{p_1^2}{2}+\frac{(p_2-k\;q_2)^2}{2} ,\qquad k\in\mathbb{R}.$$
I know that the hamiltonian isnt explicitly dependent on time so $H$ is a motion ...
0
votes
1
answer
109
views
von Neumann on the Hamiltonian
I'm reading von Neumann's book on QM and I'm slightly confused by a simple point. He writes,
"The energy is a given function of the coordinates and their time derivatives: $E = L(q_1,..,q_k;\dot{q}...
- 101
0
votes
1
answer
147
views
Kinetic energy in Lagragian mechanics [duplicate]
In my classical mechanics class they asked why the kinetic energy for an holonomic mechanical system has the homogeneous quadratic form.
Of course for a autonomous standard system( system that is ...
- 419
1
vote
2
answers
69
views
Given the Lagrangian of a system, is there a way to extract the total energy?
If an object of mass $m$ is under the action of a conservative force and there are no constraints on the system, can $E=K+U$ be obtained? If yes, I am more interested if the answer could be ...
- 11
0
votes
1
answer
59
views
Does $L=T-V$ still hold when $L$ is NOT time-dependent?
I am aware that the Lagrangian $L=T-V$ where $T$ is the kinetic energy and $V$ is the potential energy when $L$ depends on, for example, $r, \dot{r}, t$. My question is, does this still hold when the ...
- 3
2
votes
1
answer
1k
views
Difference between the energy and the Hamiltonian in a specific example
The problem is the following:
Consider a particle of mass $m$ confined in a long and thin hollow pipe, which rotates in the $xy$ plane with constant angular velocity $\omega$. The rotation axis ...
- 1,045
2
votes
0
answers
104
views
Does the principle of least action assume the kinetic energy law as an axiom? [closed]
I should probably wait till I finish the complete derivation before asking this. but maybe it will help me to know ahead of time. I like to always know what the axiom's are before I start a topic and ...
user86411