All Questions
Tagged with classical-mechanics lagrangian-formalism
1,466
questions
2
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1
answer
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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
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votes
1
answer
43
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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}=-\...
- 838
7
votes
3
answers
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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 ...
- 7,592
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1
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In Lagrangian mechanics, do we need to filter out impossible solutions after solving?
The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
- 50.2k
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1
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57
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IPhO2014 T1 with Lagrange multipliers
I'm trying to solve IPhO2014 Problem 1 with the method of Lagrange multipliers and I'm facing some problems.
I choose the origin to be at the center of the cylinder, and use three coordinates: $(x, y)...
- 13
1
vote
3
answers
65
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Oscillating inverted hemisphere Lagrangian mechanics problem
I am trying to solve a hw problem on Lagrangian mechanics. Here is the problem:
The main issue I am having is setting up the kinetic energy. I don't understand whether the hemisphere has both ...
- 11
0
votes
0
answers
88
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Deeper explanation for Principle of Stationary Action [duplicate]
The Principle of Stationary Action (sometimes called Principle of Least Action and other names) is successfully applied in a wide variety of fields in physics. It can often can be be used to derive ...
- 16.9k
-2
votes
1
answer
108
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Why the choice of Configuration Space in Hamilton's Principle is $(q, \dot{q}, t)$? [closed]
In most physics books I've read, such as Goldstein's Classical Mechanics, the explanation of Hamilton's principle took into consideration the equation (1) known as Action:
$$\displaystyle I = \int_{...
- 113
3
votes
3
answers
130
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Is there a proof that a physical system with a *stationary* action principle cannot always be modelled by a *least* action principle?
I'm aware that with Lagrangian mechanics, the path of the system is one that makes the action stationary. I've also read that its possible to find a choice of Lagrangian such that minimization is ...
- 50.2k
1
vote
0
answers
52
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Units for the Calculus of Variations [duplicate]
Just a quick question regarding the units for a quantity. I just started reading a QFT textbook, and it starts out with a little bit of Calculus of Variations. Specifically, there is a result that ...
0
votes
1
answer
69
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Changing variables from $\dot{q}$ to $p$ in Lagrangian instead of Legendre Transformation
This question is motivated by a perceived incompleteness in the responses to this question, which asks why we can't just substitute $\dot{q}(p)$ into $L(q,\dot{q})$ to convert it to $L(q,p)$, which ...
- 7,398
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vote
4
answers
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Directly integrating the Lagrangian for a simple harmonic oscillator
I've just started studying Lagrangian mechanics and am wrestling with the concept of "action". In the case of a simple harmonic oscillator where $x(t)$ is the position of the mass, I ...
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-1
votes
2
answers
126
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Why do we multiply the Euler-Lagrange equations by negative one?
As I've learned classical mechanics from different sources, I've seen both
$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_k} \right) - \frac{\partial L}{\partial q_k} = 0,$$
and
$$\frac{\...
- 183
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votes
0
answers
46
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Arc length between configurations in the "mass distance"
In classical Lagrangian mechanics, the mass $M$ is a Riemannian metric on the configuration space $Q$.
Does the "arc length" of a path $\gamma : [0, 1] \to Q$,
$$
\int_0^1 {\lVert{\gamma'(t)}...
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-2
votes
1
answer
72
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How would one use Lagrangian mechanics to obtain equations of motion for a conical pendulum? [closed]
I've defined the origin as the center of rotation for the particle on the pendulum. Then:
$$ x = r\cos{\theta} $$ $$ y = r\sin{\theta} $$ $$z = 0$$
From here, the potential energy is $V = 0$ since $z =...
- 9
1
vote
0
answers
83
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Holonomic constraints as a limit of the motion under potential
In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$...
- 133
1
vote
1
answer
33
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Regarding varying the coefficients of constraints in the pre-extended Hamiltonian in Dirac method
consider the following variational principle:
when we vary $p$ and $q$ independently to find the equations of motion, why aren't we explicitly varying the Coeff $u$ which are clearly functions of $p$ ...
4
votes
3
answers
152
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Analyzing uniform circular motion with Lagrangian mechanics
Consider swinging a ball around a center via uniform circular motion. The centripetal acceleration is provided by the tension of a rope. Now, is this force a constraint force? If it is, since it is ...
- 143
1
vote
1
answer
65
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Constraint Force and Lagrangian multiplier
I can't seem to find my mistake in this problem and I think it stems from not understanding how to correctly form constraints and the meaning behind the Lagrangian multiplier.
So first of all I ...
- 11
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votes
0
answers
90
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Classification of equilibrium configurations for particles subject to elastic force constrained on a circle
I am interested in classifying all the possible equilibrium configurations for an arrangement of $l$ equal point particles $P_1, P_2, . . . , P_l$ $(l > 2)$ on a circle of radius $R$ and centre $O$....
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2
votes
1
answer
173
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Definition of generalized force in Lagrangian formalism
In some texts (e.g. Taylor's Classical Mechanics), the generalized force is defined to be (I'll simplify to one particle in one dimension for ease of notation): $Q \equiv \frac{\partial{L}}{\partial{q}...
- 7,398
0
votes
1
answer
76
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Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
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votes
3
answers
630
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Is Principle of Least Action a first principle? [closed]
It is on the basis of Principle of Least Action, that Lagrangian mechanics is built upon, and is responsible for light travelling in a straight line.
Is its the classical equivalent of Schrodinger's ...
0
votes
1
answer
53
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Trouble with Lagrangian and Newtonian mechanics [closed]
I'm a pure mathematician and I was doing some physics for fun. I was trying to obtain the equations of motion of a particle moving along a curve $y(x)$ under the effect of gravitational force which ...
-3
votes
1
answer
58
views
A least principle action and human behaviour [closed]
Does anyone have an idea how the authors of this paper https://www.nature.com/articles/s41598-021-81722-6#additional-information
solved the equation 10? Thank you in advance
- 13
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votes
0
answers
50
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Is there a Lorentz invariant action for a free multi-particle system?
I want to write down a Lorentz-invariant action of free multi-particle systems.
I know that a Lorentz-invariant action for each particle might be expressed as
$$
S[\vec{r}]=\int dt L(\vec{r}(t),\dot{\...
- 145
1
vote
1
answer
66
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Landau/Lifshitz action as a function of coordinates [duplicate]
In Landau/Lifshitz' "Mechanics", $\S43$, 3ed, the authors consider the action of a mechanical system as a function of its final time $t$ and its final position $q$. They consider paths ...
- 349
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0
answers
53
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Motion around stable circular orbit
Hello I am to solve whether it is possible for body of mass $m$ to move around stable circular orbit in potentials: ${V_{1} = \large\frac{-|\kappa|}{r^5}}$ and ${V_{2} = \large\frac{-|\kappa|}{r^{\...
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1
answer
49
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Definition of generalized momenta in analytical mechanics
I've seen mainly two definitions of generalized momenta, $p_k$, and I wasn't sure which one is always true/ the correct one:
$$p_k\equiv\dfrac{\partial\mathcal T}{\partial \dot q_k}\text{ and }p_k\...
- 296
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0
answers
53
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Constrained Lagrangian simulation. Fails depending on constraint definition [closed]
Introduction
I want to simulate a robotic leg that has a closed kinematic chain. The analytical equation i derived are compared with a simulink multibody model.
Initially, my simulation failed, and ...
