All Questions
Tagged with classical-mechanics lagrangian-formalism
1,466
questions
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Are there any experiments that examine Hamilton's Principle directly?
Or can it be examined?
I 'd glad if you can share some ideas about "principles" in general.
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2
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Newton vs Lagrange's equations for a variable length pendulum
Consider a pendulum with a variable string length $l=f(\theta)$. The Lagrangian is:
$$L = \frac{m}{2}(\dot{l} ^ 2 + l^2 \dot{\theta} ^ 2) + mgl\cos\theta.$$
Using Lagrange multipliers for the ...
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1
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Virtual displacement for a block sliding down a wedge
A block slides on a frictionless wedge which rests on a smooth horizontal plane. There are two constraints in this system. One that the wedge can only move horizontally and another that the block must ...
CommunityBot
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Is there a straightforward simplified proof of energy conservation from time translation symmetry?
Electric charge conservation is easily proven from electric potential gauge symmetry, as follows:
The potential energy of an electric charge is proportional to the electric potential at its location.
...
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6
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Noether Theorem and Energy conservation in classical mechanics
I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
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35
votes
2
answers
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Lagrangian and Hamiltonian EOM with dissipative force
I am trying to write the Lagrangian and Hamiltonian for the forced Harmonic oscillator before quantizing it to get to the quantum picture. For EOM $$m\ddot{q}+\beta\dot{q}+kq=f(t),$$ I write the ...
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2
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1
answer
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Non-Holonomic constraint in rigid body dynamics
I have solved many problems on Holonomic constraint using Lagrange multiplier method but I don't know how to tackle problems on non-Holonomic constraint.
Can anyone help me with the following problem ...
CommunityBot
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3
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1
answer
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Does quasi-symmetry preserve the solution of the equation of motion?
In some field theory textbooks, such as the CFT Yellow Book (P40), the authors claim that a theory has a certain symmetry, which means that the action of the theory does not change under the symmetry ...
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What is the relationship between phase space and Jacobian in Nakahara Eq.1.15 (under this equation)?
In Nakahara's Geometry, topology and physics, under Eq.1.15 they give an equation
\begin{align*}
\det\left(\frac{\partial p_i}{\partial\dot{q}_j}\right)=\det\left(\frac{\partial^2L}{\partial\dot{q}_i\...
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3
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How to prove that $dh/dt = ∂h/∂t$ in Lagrangian mechanics?
We already know that the energy function $h(q,\dot{q},t)$ (not the Hamiltonian!) in classical mechanics follows the equation $$dh/dt = −∂L/∂t\tag{1}$$ but how can we show that $$dh/dt = ∂h/∂t\tag{2}$$ ...
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I need an explanation for the time derivative omissions when solving for the Lagrangian of a system [closed]
So I have been self-studying Landau and Lifshitz’s Mechanics for a little bit now, and I have been working through the problems, but Problem 3 is giving me some trouble. I solved the Lagrangian ...
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4
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Why can't we ascribe a (possibly velocity dependent) potential to a dissipative force?
Sorry if this is a silly question but I cant get my head around it.
- 207k
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2
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292
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Generalized forces of constraint
When using the method of Lagrange undetermined multipliers, it's assumed that the constraint generalized force, $Q_j$, is given by:
$$Q_j=\lambda \cdot \frac{\partial f}{\partial q_{j}}$$
Where $f$ is ...
CommunityBot
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1
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2
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327
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How do I check if a transformation is a point transformation?
In Lagrangian mechanics, I came across the notion of a point transformation which leaves the Lagrangian invariant. Normally it is denoted as follows.
$$Q = Q(q,t).$$
Now, unlike in the case of a ...
CommunityBot
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2
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2
answers
165
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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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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,
$$\frac{\partial L}{\partial\theta}=\frac{d}{dt}\frac{\...
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3
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$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)$$
Should ...
Buzz♦
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2
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Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?
This is from Analytical Mechanics by Louis Hand et al. The proof is about Maupertuis' principle. The author seems to say that Hamilton's principle allow a path to have both a process of time forward ...
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2
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Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
In Lagrangian mechanics the momentum is defined as:
$$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 ...
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1
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1
answer
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What is the degrees of freedom (Lagrange equation) of two connected spool rolling down two inclines?
I'm quite confused as to how to use the Lagrange equation [second type] in a system which features a spool rolling down an incline. I think this particular example is quite representative of what is ...
CommunityBot
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4
answers
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Is there an error in Susskinds' derivation of Euler-Lagrange equations?
First, I believe there is a trivial error. The second equation should have another $\Delta t$ multiplying everything on the right. It is divided out later when the equation I set equal to 0.
Given ...
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1
answer
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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 ...
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1
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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 ...
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1
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333
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How do we get Maupertuis Principle from Hamilton's Principle?
Maupertuis principle says that if we know the initial and final coordinates but not time, the total energy and the fact that energy is conserved, we can choose the "right" path from all mathematically ...
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2
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Lagrangian function for two swivelling masses attached by a spring
I am just having a hard time finding the Lagrangian for this question. There are two massless rigid rods lengths R (connected to mass M) and r (connected to mass m) which both pivot around a fixed ...
CommunityBot
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2
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82
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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 ...
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1
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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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1
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Clarifications regarding the Maupertuis/Jacobi principle
I'm slightly confused regarding the Maupertuis' Principle. I have read the Wikipedia page but the confusion is even in that derivation. So, say we have a Lagrangian described by $\textbf{q}=(q^1,...q^...
CommunityBot
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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 ...
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3
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1
answer
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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 ...
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