All Questions
Tagged with classical-mechanics lagrangian-formalism
1,464
questions
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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,
$$\frac{\partial L}{\partial\theta}=\frac{d}{dt}\frac{\...
- 115
0
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0
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31
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Centrifugal Governor Question [closed]
I've been working through Hand and Finch's Analytical Mechanics and have just attempted this question:
My attempt at a solution is as follows:
First, find the kinetic energy of the two masses $m$ by ...
2
votes
2
answers
161
views
QFT introduction: From point mechanics to the continuum
In any introductory quantum field theory course, one gets introduced with the modification of the classical Lagrangian and the conjugate momentum to the field theory lagrangian (density) and conjugate ...
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3
answers
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$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)$$
Should ...
-3
votes
2
answers
76
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Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
In Lagrangian mechanics the momentum is defined as:
$$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
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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
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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
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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 ...
- 1,629
0
votes
2
answers
60
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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
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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
$$
\frac{d}{dt}(\gamma m \mathbf{v})=q_m(\mathbf{B}-\mathbf{v}\times\mathbf{E}),
$$
...
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2
votes
1
answer
50
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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 ...
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