All Questions
Tagged with classical-mechanics lagrangian-formalism
1,464
questions
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0
answers
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Getting an opposite sign for the centrifugal potential energy in the effective potential [duplicate]
Consider a system whose Lagrangian is
$$L = \frac12 \mu\left( \dot r^2 + r^2 \dot\theta^2 \right) -U(r) $$
By the Euler-Lagrange equation,
$$\frac{\partial L}{\partial\theta}=\frac{d}{dt}\frac{\...
- 115
0
votes
0
answers
31
views
Centrifugal Governor Question [closed]
I've been working through Hand and Finch's Analytical Mechanics and have just attempted this question:
My attempt at a solution is as follows:
First, find the kinetic energy of the two masses $m$ by ...
2
votes
2
answers
161
views
QFT introduction: From point mechanics to the continuum
In any introductory quantum field theory course, one gets introduced with the modification of the classical Lagrangian and the conjugate momentum to the field theory lagrangian (density) and conjugate ...
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6
votes
3
answers
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
-3
votes
2
answers
76
views
Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
In Lagrangian mechanics the momentum is defined as:
$$p=\frac{\partial \mathcal{L}}{\partial \dot q}$$
Also we can define it as:
$$p=\frac{\partial S}{\partial q}$$
where $S$ is Hamilton's principal ...
- 443
0
votes
0
answers
26
views
Prerequisites for studying Lev Landau Mechanics vol. 1 [closed]
Lev Landau Mechanics vol. 1 dives directly into Lagrangians and Hamiltonians. What do you think are the prerequisites in order to study and grasp it?
1
vote
2
answers
106
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Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?
This is from Analytical Mechanics by Louis Hand et al. The proof is about Maupertuis' principle. The author seems to say that Hamilton's principle allow a path to have both a process of time forward ...
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0
votes
1
answer
86
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Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates
I am solving problem 19 of ch 10 of Goldstein mechanics. The problem is:
A three-dimensional harmonic oscillator has the force constant k1 in the x- and y- directions and k3 in the z-direction. Using ...
- 13
1
vote
1
answer
61
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Vanishing virtual work done by non-holonomic constraints
I was reading classical mechanics by NC Rana. I was reading a topic on vanishing virtual work done due to constraint forces. How do you prove that the virtual work done by non-holonomic constraint ...
- 41
0
votes
2
answers
79
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Generalized momentum
I am studying Hamiltonian Mechanics and I was questioning about some laws of conservation:
in an isolate system, the Lagrangian $\mathcal{L}=\mathcal{L}(q,\dot q)$ is a function of the generalized ...
- 59
1
vote
1
answer
53
views
Confusing Goldstein Statement about Magnitude of the Lagrangian
On page 345 of Goldstein's Classical Mechanics 3rd Ed., he writes:
...the Hamiltonian is dependent both in magnitude and in functional form upon the initial choice of generalized coordinates. For the ...
- 7,388
3
votes
1
answer
54
views
Understanding gauge in Lagrangian mechanics [duplicate]
I know given a Lagrangian $\mathcal{L}$ satisfying the Euler-Lagrange equations, then the Lagrangian $\mathcal{L}'=\mathcal{L}+\frac{d}{dt}f(q,t)$ is also a solution of said equations. Nonetheless, I ...
- 1,619
0
votes
2
answers
60
views
Extending the Lagrangian of a double pendulum to systems with more complex shapes
The total kinetic energy of a double pendulum can be calculated as follows:
$$L = \frac{1}{2} (m_1 + m_2) {l_1}^2 \dot{\theta_1}^2 + \frac{1}{2} m_2 {l_2}^2 \dot{\theta_2}^2 + m_2 l_1 l_2 \dot{\...
0
votes
1
answer
65
views
How to describe the dynamics of a magnetic monopole charge in the external EM field using a Lagrangian in terms of the EM potentials?
The equation of motion of a magnetic charge in the fixed external electromagnetic field $\mathbf{E},\mathbf{B}$ is
$$
\frac{d}{dt}(\gamma m \mathbf{v})=q_m(\mathbf{B}-\mathbf{v}\times\mathbf{E}),
$$
...
- 21
2
votes
1
answer
50
views
Trying to derive relativistic dispersion formula
If we define conserved quantities of motion as constants arising from continuous symmetries of the system (Lagrangian), why does the following argument not give the correct result?
Let $\gamma: I \to ...
- 498
1
vote
2
answers
127
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On Landaus&Lifshitz's derivation of the lagrangian of a free particle [duplicate]
I'm reading the first pages of Landaus&Lifshitz's Mechanics tome. I'm looking for some clarification on the derivation of the Lagrange function for the mechanical system composed of a single free ...
0
votes
1
answer
25
views
Subsequent motion (time evolution) of angled dipoles in electric field
Suppose we have a system of two dipoles, each with dipole moment $\mathbf{p}=2aq$ each aligned at angles $\theta$ and $-\theta$ with the horizontal. I’m thinking of an angle bracket shape, essentially....
- 73
0
votes
2
answers
49
views
How did the boundary term vanish in deriving equation of motion from Lagrangian? [closed]
I was deriving the equation of motion from Lagrangian, by using the principle of least action. Usually, at this point in derivation,
$$\int dt \frac{\partial L}{\partial \dot{q}} \frac{\partial}{\...
1
vote
1
answer
72
views
Analogy of Euler-Lagrange-equation and Continuity equation
It seems to me that there is a link between the continuity equation
$$\nabla\rho u + \frac{\partial \rho}{\partial t} = 0$$
and the Euler-Lagrange equation for Lagrangian mechanics
$$\nabla_q L - \...
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2
votes
0
answers
72
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Why can't we treat the Lagrangian as a function of the generalized positions and momenta and vary that? [duplicate]
Some background: In Lagrangian mechanics, to obtain the EL equations, one varies the action (I will be dropping the time dependence since I don't think it's relevant) $$S[q^i(t)] = \int dt \, L(q^i, \...
- 93
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votes
1
answer
56
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Question about Problem $12$ in Chapter $11$ from Kibble & Berkshire's book
I write again the problem for convinience:
A rigid rod of length $2a$ is suspended by two light, inextensible strings of length $l$ joining its ends to supports also a distance $2a$ apart and level ...
0
votes
0
answers
30
views
"Pseudo-Potential" for acceleration in the $x$ plane?
First Post!
My study group for classical mechanics using Taylor and Thornton and Marion and I found this problem while trying to study for our final. The set up has a particle in a tube that is ...
5
votes
0
answers
58
views
Group theoretical approach to conservation laws in classical mechanics
I'm doing some major procrastination instead of studying for my exam, but I wanted to share my thought just to confirm if I'm right.
Suppose that the action, $S(\mathcal{L})$ forms the basis of a ...
- 362
2
votes
1
answer
77
views
What is the benefit of Hamiltonian formalism to promote ($q,\dot{q},t$) from Lagrangian to ($q,p,t$) despite getting the same EOM finally? [duplicate]
Hamiltonian formalism follows
$$H(q,p,t)=\sum_i\dot{q_i}p_i-L(q_i,\dot{q}_i,t) $$ and $$\dot{p}=-\frac{\partial H}{\partial q}, \dot{q}=\frac{\partial H}{\partial p} $$
but finally these will get the ...
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0
votes
0
answers
19
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Precise Definition of Degrees of Freedom [duplicate]
I am taking Analytical Mechanics and while reading Goldstein's and LL something bothered me: can I say that a degree of freedom is an independent (generalized) coordinate?
What bothers me is that we ...
user400821
1
vote
1
answer
102
views
Perturbation of central field potential
i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field.It is easy to write Lagrangian function:
$$ L = \frac{m}{2}(\dot{\rho}^2 + \rho^2\dot{\phi}^2) + ...
3
votes
5
answers
937
views
What is the point of knowing symmetries, conservation quantities of a system?
I think this kind of question has been asked, but i couldn’t find it.
Well i have already know things like symmetries, conserved quantities and Noether’s theorem, as well as their role in particle ...
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2
votes
1
answer
86
views
Why does a free theory's action have to be quadratic?
From my naive understanding of the symmetry principle, in inertial frames the space is uniform and homogeneous, so the action must not depend explicitly on coordinates (or fields). Thus the action ...
- 195
0
votes
1
answer
43
views
What is the physical significance of this generalised potential?
Consider a generalised potential of the form $U=-f\vec{v}\cdot\vec{r}$ where $f$ is a constant. This potential should not contribute any internal forces between particles as
\begin{equation}
\vec{F}=-\...
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7
votes
3
answers
2k
views
Something fishy with canonical momentum fixed at boundary in classical action
There's something fishy that I don't get clearly with the action principle of classical mechanics, and the endpoints that need to be fixed (boundary conditions). Please, take note that I'm not ...
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