All Questions
15
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
8
votes
2
answers
5k
views
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,
$$\frac{\partial\dot{\...
- 1,977
9
votes
1
answer
930
views
Mathematics of the Virtual Displacement
So I'm pretty certain this question has been asked to death here, but I still can't find a good explanation of a very particular aspect of the virtual displacements in physics.
Background
For ...
- 221
3
votes
3
answers
2k
views
Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative?
I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows:
\begin{align}
\dfrac{\delta [f(g(x,\dot{x}))h(x,\dot{x})] } {\delta x}
=
\left(
\...
- 2,107
0
votes
2
answers
90
views
Help with deriving the Euler-Lagrange equation (evaluating at $ \varepsilon = 0$ before solving partials)
I am using wiki here to help me understand the deriving of the euler-lagrage equations
How do we get from:
\begin{equation}
\left.\frac{dJ_\varepsilon}{d\varepsilon}\right|_{\varepsilon = 0} = \int_a^...
- 283
0
votes
1
answer
452
views
Least action principle : is $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $ always true?
(Just some recalls)
We have an action on which we want to apply Least action principle.
$$ S=\int_{t_i}^{t_f} L(q,\dot{q},t)dt$$
We assume that $t \mapsto q(t)$ is the function that will extremise ...
- 1,450
6
votes
1
answer
2k
views
How can dissipative/friction terms be incorporated into a Lagrangian? [duplicate]
I'm trying to find a suitable Lagrangian for a damped harmonic oscillator, a system that satisfies the following equation of motion:
$$m \ddot{x} + \gamma \dot{x} + \frac{d\phi}{dx} = 0.$$
What I ...
- 91
6
votes
1
answer
2k
views
Why does the partial derivative with respect to $x$ of a function depending only on $\dot{x}$ vanish? [duplicate]
In Classical Mechanics by Goldstein it says:
$$ \sum \left\{ \left[ \frac{d}{dt} \left( \frac{\partial T}{\partial \dot q_j} \right) - \frac{\partial T}{\partial q_j} \right] - Q_j \right\} \delta ...
- 1,043
5
votes
3
answers
2k
views
Lagrangian gauge invariance $L'=L+\frac{df(q,t)}{dt}$
So, I have to prove directly (e.g. by substitution) that if a path satisfies the Euler-Lagrange equations for the Lagrangian $L$ it does so for $$L'=L+\frac{df(q,t)}{dt}.$$ Let me tell you what I have ...
- 153
5
votes
4
answers
5k
views
Lagrangian mechanics and time derivative on general coordinates
I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
- 2,383
5
votes
3
answers
2k
views
Time dependence of the Lagrangian of a free particle?
I am working through Landau's book on Classical Mechanics. I understand the logic and physics of isotropy and homogeneity of space-time behind the derivation of the Lagrangian for a free particle, but ...
- 345
3
votes
2
answers
1k
views
Why can we consider the endpoint fixed in the derivation of the Euler-Lagrange equation in mechanics?
In mechanics, we obtain the equations of motion (Euler-Lagrange equations) via Hamilton's principle by considering stationary points of the action
$$ S = \int_{t_i}^{t_f} L ~ dt $$
where we have $L=T-...
- 133
1
vote
4
answers
640
views
Question about Principle of least action - Landau
I have this conceptual question: In Landau's book of classical mechanics, about the principle of least action, it's written:
$$\left. \delta S =\frac{\partial L}{\partial v} \delta q \right\rvert_{...
user153036
0
votes
1
answer
440
views
Confusing with the equation $(2.4)$ and $(2.5)$ of Landau and Lifshitz, Mechanics, Chapter 1, The principle of Least Action
I'm a 12th Grader and I'm interested in Lagrangian Mechanics and having a bit of knowledge about the Newtonian Mechanics. So, I found a book of Landau and Lifshitz's Mechanics and started reading from ...
user208739