All Questions
Tagged with classical-mechanics lagrangian-formalism
1,466
questions
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Poincare-Cartan form of charged particle in electromagnetic field
In the paper by Littlejohn, 1983, the canonical Hamiltonian $h_c$ of a charged particle in electromagnetic field is given by,
$$
h_c (\vec{q}, \vec{p}, t) = \frac{1}{2m} \left[ \vec{p} - \frac{e}{c} \...
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4
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260
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Variation of a function
I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
- 309
5
votes
1
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Does the Hamiltonian formalism yield more Noether charges than the Lagrangian formalism?
In Lagrangian formalism, we consider point transformations $Q_i=Q_i(q,t)$ because the Euler-Lagrange equation is covariant only under these transformations. Point transformations do not explicitly ...
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1
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Why is gravitational potential energy negative in this Lagrangian? [closed]
The question is given as follows:
From (6.109) shouldn't the Lagrangian be K(kinetic) - U(potential), but here its K + U? Unless the potential energy is negative, if so I'm struggling to come to ...
0
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1
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Euler-Lagrange confusion
Consider the action $S = \int dt \sqrt{G_{ab}(q)\dot{q}^a\dot{q}^b}.$
Now for computing the Euler-Lagrange equations, we need the time derivative of $\frac{\partial L}{\partial \dot{q}^c} = \frac{1}{\...
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47
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Equation of motion from lagrangian for an holonomic system with fixed constraints
We know that the lagrangian function of a holonomic system subject to fixed constraints has the form
$$\mathcal{L}(\mathbf{q,\dot{q}})=\frac{1}{2} \langle \mathbf{\dot{q},A(q)\dot{q}} \rangle - U(\...
1
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1
answer
54
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Sufficient condition for conservation of conjugate momentum
Is the following statement true?
If $\frac{\partial \dot{q}}{\partial q}=0$, then the conjugate momentum $p_q$ is conserved.
We know that conjugate momentum of $q$ is conserved if $\frac{\partial L}{\...
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2
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330
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Generalized vs curvilinear coordinates
I am taking the course "Analytical Mechanics" (from on will be called "AM") this semester. In our first lecture, my professor introduced the notion of generalized coordinates. As ...
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2
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How to change generalised coordinates in a Lagrangian without inverting the coordinate transformation?
Given a Lagrangian using the standard cartesian coordinates.
$$ \mathcal{L} = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - \frac{1}{2}k(x^2 + y^2) $$
How to move to the hyperbolic coordinates given as
$$2 x ...
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49
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Vibration of a continuous uniform chain and the normal modes
The question is:
A vertically hanged chain with the upper end attached to a fixed point. I try to find the normal modes under the small $\theta$ condition.
Consider the mass $\mathrm{d}m$ with ...
0
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107
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Euler-Lagrange equations with constraints
Show that if there are $M$ independent constraints $\phi_m(x_\mu,p_\mu)$ there are $M$ of the $\ddot{x}_i$'s that the Euler-Lagrange equations cannot be solved for.
Attempt of solution:
Assume that ...
- 357
2
votes
1
answer
175
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Doubts about Noether's theorem derivation
Assume you have an action:
$S[q] = \int L(q, \dot q, t)$ (i.e $q$ is a function of time). (1) Then you do a transformation on $q(t)$ such as $\sigma(q(t), a)$ where $a$ is infinetisemal and this ...
2
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Physical model described by modify Helmholtz equation
The wave equation $\partial_t^2u=c\Delta u$ is usually handled through a time-harmonic ansatz, which reduces it to Helmholtz equation $\Delta u+\omega^2u=0$.
I'm interested in the following modified ...
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-2
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2
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On the physical meaning of functionals and the interpretation of their output numbers
I am studying about functionals, and while looking for some examples of functionals in physics, I have run into this handout .
Here are two questions of mine.
1- This handout starts as follows (the ...
user386360
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1
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80
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Lagrangian and Hamiltonian Mechanics: Conjugate Momentum
I am a physics undergraduate student currently taking a classical mechanics course, and I am not able to understand what conjugate/canonical momentum is (physically). It is sometimes equal to the ...
2
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3
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148
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Derivation of Hamiltonian by constraining $L(q, v, t)$ with $v = \dot{q}$
I am trying to reconstruct a derivation that I encountered a while ago somewhere on the internet, in order to build some intuition both for $H$ and $L$ in classical mechanics, and for the operation of ...
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Doubt Regarding Noether's theorem for time-dependent systems
I'm having problems showing Noether's theorem when the lagrangian is time dependent. I'm trying to do it not using infinitesimal transformations from the beginning, but continuous transformations of a ...
1
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0
answers
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Is the invariance of the Lagrangian under some transformation equivalent to the covariance of the motion equation? [duplicate]
Take the Lagrangian $L=\frac{1}{2}m{{\left( \frac{{\rm{d}}}{{\rm{d}}t}x \right)}^{2}}-\frac{1}{2}k{{x}^{2}}$, for example.
The equation of motion of this system should be given by $m\frac{{{{\rm{d}}}^{...
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1
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Is $F=-\nabla V$ a form of the least action principle? [closed]
Only for conservative systems, of course.
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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(...
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2
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Is there a way to express the collisionless boltzmann equation in terms of positions, velocities, times, without the distribution function?
Suppose I have data that represents a field of positions and velocities. If the distribution function (DF) for the data is $f(x,v,t)$, I know that the DF must obey
$$\frac{\partial f}{\partial t} + \...
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Even considering angular momentum conservation, is Newtonian mechanics the same as Lagrange mechanics? [duplicate]
I heard that Lagrange mechanics can be derived from Newtonian mechanics, and Newtonian mechanics can be derived from Lagrange mechanics. I've heard many times that they have equal explanatory power. ...
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What is $2\dot{r}\dot{\theta}$? [duplicate]
I am in a classical mechanics class and we are currently working on Lagrangians. I notice that very often when working with an angular velocity the formula $2\dot{r}\dot{\theta}$ comes up. What is the ...
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1
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320
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Obtaining Euler-Lagrange equations for a mass attached to a spring, connected to a pendulum via a pulley [closed]
I'm trying to setup the Lagrangian for the following system, I'm quite confident this is correct, but I would like a second pair of eyes to analyse my solution. Here is the problem at hand
A block ...
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2
answers
186
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Equations of motion for coupled harmonic oscillators
We just started QFT, and I'm following our professor's notes but there is a passage I do not understand. We are speaking about a system of $N$ coupled harmonic oscillators $y_j(t)$ for $j = 1, ..., N$ ...
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1
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111
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What is an application of Nielsen's Form of the Lagrange's equations?
In chapter 1 problem 7 of the 3rd edition of Goldstein the reader is asked to prove that the Nielsen's form of the Lagrange's equations is equivalent to Lagrange's equations.
Lagrange's equations:
$$\...
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2
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338
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Constraint equation for an elastic pendulum
I would like to know if you can help me determine the restraining force for an elastic pendulum. The problem is the following
A particle of mass $m$ is suspended by a massless spring of length $L$. ...
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1
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85
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Hamiltonian analysis of relational $N$-Particle Dynamics
I am following "A Shape Dynamics Tutorial, Flavio Mercati" (https://arxiv.org/abs/1409.0105), and have problems understanding the hamiltonian formulation of $N$-particle dynamics as sketched ...
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2
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107
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Help in understanding this derivation of Lagrange Equations in Non-Holonomic case
Whittaker, Analytical dynamics pg 215
I don't understand how we get the final equations relating $Q_r$ with $\lambda$ given the conditions above?
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Independence of generalized coordinates in the derivation of Lagrange equations from d'Alembert's Principle
I am confused by this remark in the derivation of Lagrange equations from d'Alembert's principle in Goldstein:
I am not comfortable that I understand why, at this late stage of the derivation, they ...
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