All Questions
7
questions
148
votes
8
answers
18k
views
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
25
votes
2
answers
2k
views
Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are ...
- 281
57
votes
7
answers
9k
views
Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,826
0
votes
1
answer
452
views
Least action principle : is $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $ always true?
(Just some recalls)
We have an action on which we want to apply Least action principle.
$$ S=\int_{t_i}^{t_f} L(q,\dot{q},t)dt$$
We assume that $t \mapsto q(t)$ is the function that will extremise ...
- 1,450
4
votes
2
answers
1k
views
Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
2
votes
1
answer
442
views
What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,523
1
vote
2
answers
398
views
Total time derivatives and partial coordinate derivatives in classical mechanics
This may be more of a math question, but I am trying to prove that for a function $f(q,\dot{q},t)$
$$\frac{d}{dt}\frac{∂f}{∂\dot{q}}=\frac{∂}{∂\dot{q}}\frac{df}{dt}−\frac{∂f}{∂q}.\tag{1}$$
As part of ...
