All Questions
Tagged with classical-mechanics lagrangian-formalism
1,466
questions
31
votes
4
answers
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How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
- 955
10
votes
3
answers
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Is there a valid Lagrangian formulation for all classical systems?
Can one use the Lagrangian formalism for all classical systems, i.e. systems with a set of trajectories $\vec{x}_i(t)$ describing paths?
On the wikipedia page of Lagrangian mechanics, there is an ...
- 8,523
3
votes
1
answer
924
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Origins of the principle of least time in classical mechanics
Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least ...
- 1,352
1
vote
1
answer
1k
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Is the number of independent constants of a system equal to the number of degree of freedom of it?
Maybe the question is not very clear myself since I am not a physics major.But can you help me make this question clearer and then give me some comments on it?
I got that this holds in gravitional ...
- 131
2
votes
1
answer
1k
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Principle of Least Action; Newton's 2nd Law of Motion
This question is based on the description of Longair in his book "Theoretical Concepts in Physics".
He starts by giving some provisions:
Conservative force field
Fixed times $t_1$ and $t_2$
Object ...
- 23
3
votes
1
answer
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Problems that Lagranges equations of the 1st kind can solve whereas the 2nd kind can't?
Can anyone give examples of mechanics problems which can be solved by Lagrange equations of the first kind, but not the second kind?
- 3,108
9
votes
2
answers
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How does the canonical momentum $p_i\equiv\frac{\partial L}{\partial\dot q_i}$ transform under a coordinates change $\mathbf q\to\mathbf Q$?
The canonical momentum is defined as
$$p_{i} = \frac {\partial L}{\partial \dot{q_{i}}}, $$
where $L$ is the Lagrangian.
So actually how does $p_{i}$ transform in one coordinate system $\textbf{q}$ to ...
- 91
130
votes
10
answers
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Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
- 8,540
7
votes
1
answer
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What variables does the action $S$ depend on?
Action is defined as,
$$S ~=~ \int L(q, q', t) dt,$$
but my question is what variables does $S$ depend on?
Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$?
In Wikipedia I've ...
- 173
5
votes
3
answers
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D'Alembert's Principle: Where does $-Q_j$ come from?
This is a follow-up question to D'Alembert's Principle and the term containing the reversed effective force.
From the second term of Eq. (1.45)
$$\begin{align*}
\sum_i{\dot{\mathbf{p}}_i \...
- 1,483
8
votes
2
answers
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Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time
My question is in reference to Landau's Vol. 1 Classical Mechanics. On Page 6, the starting paragraph of Article no. 4, these lines are given:
If an inertial frame $К$ is moving with an ...
user4235
4
votes
2
answers
1k
views
Invariance and forms of the Lagrangian
I have been reading the 1st chapter of Landau & Lifshitz Mechanics, and due to its concise style been facing a few problems. I hope you can help me out here somehow.
Does the "homogeneity of ...
- 93
8
votes
2
answers
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Why are generalized positions and generalized velocities considered as independent of each other? [duplicate]
I'm confused how
$$\dot{\mathbf{r}}_{j}=\sum_{k}\frac{\partial\mathbf{r}_{j}}{\partial q_{k}}\dot{q}_k+\frac{\partial\mathbf{r}_{j}}{\partial t}$$
leads to the relation,
$$\frac{\partial\dot{\...
- 1,977
28
votes
2
answers
9k
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Invariance of Lagrangian in Noether's theorem
Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$.
However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
- 2,890
76
votes
7
answers
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What is the difference between Newtonian and Lagrangian mechanics in a nutshell?
What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the ...
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29
votes
3
answers
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Are there examples in classical mechanics where D'Alembert's principle fails?
D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero.
This is obviously true for the ...
- 5,318
3
votes
3
answers
484
views
Are there measurable effects to scaling the action by a constant?
Classically, we obtain the equations of motion by finding a path which has an action that is stationary with respect to small changes in the path. That is the path for which:
$\delta S =0$
Scaling ...
- 2,135
5
votes
2
answers
2k
views
How the Lagrangian of classical system can be derived from basic assumptions?
It is well known that the Lagrangian of a classical free particle equal to kinetic energy. This statement can be derived from some basic assumptions about the symmetries of the space-time. Is there ...
- 111
96
votes
4
answers
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Physical meaning of Legendre transformation
I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
- 1,227
2
votes
1
answer
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Does Action in Classical Mechanics have a Interpretation? [duplicate]
Possible Duplicate:
Hamilton's Principle
The Lagrangian formulation of Classical Mechanics seem to suggest strongly that "action" is more than a mathematical trick. I suspect strongly that it ...
- 139
3
votes
1
answer
1k
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Lagrangian density of linear elastic solid
I need the general expression for the lagrangian density of a linear elastic solid. I haven't been able to find this anywhere. Thanks.
becko
49
votes
8
answers
15k
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Classical mechanics without coordinates book
I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
4
votes
1
answer
1k
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Distinguishing mechanical systems from general dynamical systems
In the following let a "mechanical system" be a system of $n$ spatial objects moving in physical space.
Consider you are given a function $q:\mathbb{R} \rightarrow \mathcal{M}^n$ with $\mathcal{M}$ a ...
148
votes
8
answers
18k
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Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
- 2,155
7
votes
3
answers
4k
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Constraint force on a rod
I really hope someone will take a quick look at the following, I would just love to better understand it...
This exercise is from Arnold's "Mathematical Methods of Classical Mechanics", p. 97 in the ...
- 2,426
29
votes
9
answers
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Book about classical mechanics
I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...