All Questions
Tagged with classical-mechanics differentiation
14 questions
154
votes
9
answers
19k
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 ...
- 2,235
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
15
votes
2
answers
4k
views
Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?
Why is the following equation true?
$$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$$
where $\mathbf{v}_i$ is velocity, $\mathbf{r}_i$ is the ...
- 1,483
8
votes
1
answer
712
views
When motion begins, do objects go through an infinite number of position derivatives?
This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
- 83
57
votes
7
answers
10k
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 ...
- 1,883
0
votes
1
answer
483
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 ...
- 1,560
4
votes
2
answers
994
views
Why do we consider potential energy function $U(x)$ differentiable?
Recently when skimming through my physics-text I encountered an interesting definition of Force
$$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$
We were taught that some functions are continuous but not ...
- 373
7
votes
2
answers
1k
views
A confusion about notation in Goldstein
On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation
$$\dfrac{d}{dt}{\bf p}_i = {\bf F}_i^{(e)...
- 37.4k
4
votes
2
answers
1k
views
Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
2
votes
1
answer
464
views
What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,703
1
vote
2
answers
461
views
Total time derivatives and partial coordinate derivatives in classical mechanics
This may be more of a math question, but I am trying to prove that for a function $f(q,\dot{q},t)$
$$\frac{d}{dt}\frac{∂f}{∂\dot{q}}=\frac{∂}{∂\dot{q}}\frac{df}{dt}−\frac{∂f}{∂q}.\tag{1}$$
As part of ...
25
votes
3
answers
3k
views
Why don't we see the covariant derivative in classical mechanics?
I am wondering why I have seen the covariant derivative for the first time in general relativity.
Starting from the point that the covariant derivative generalise the concept of derivative in curved ...
- 873
2
votes
5
answers
348
views
Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
- 69
0
votes
1
answer
1k
views
Squaring the momentum operator in QM becomes a second derivative. How?
$\frac{p^2}{2m}$ is the Kinetic energy in classical mechanics. However, the same $p^2$ becomes the second derivative $\frac{\partial ^2}{\partial x^2}$ in the Kinetic Energy operator in QM. I mean it ...
- 21