All Questions
84
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
13
votes
2
answers
408
views
Anticommutation of variation $\delta$ and differential $d$
In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is ...
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
9
votes
1
answer
1k
views
How to find the Lagrangian of this system?
I am trying to find the Lagrangian $L$ of a system I am studying. The equations of motion is:
$$\left\{
\begin{array}{c l}
r \ddot{\phi} + 2\dot{r} \dot{\phi}+k(r) \cdot r \dot{r} \dot{\phi} = ...
- 253
8
votes
2
answers
5k
views
Why are generalized positions and generalized velocities considered as independent of each other? [duplicate]
I'm confused how
$$\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
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:
$$ \sum \left\{ \left[ \frac{d}{dt} \left( \frac{\partial T}{\partial \dot q_j} \right) - \frac{\partial T}{\partial q_j} \right] - Q_j \right\} \delta ...
- 1,043
6
votes
1
answer
2k
views
How can dissipative/friction terms be incorporated into a Lagrangian? [duplicate]
I'm trying to find a suitable Lagrangian for a damped harmonic oscillator, a system that satisfies the following equation of motion:
$$m \ddot{x} + \gamma \dot{x} + \frac{d\phi}{dx} = 0.$$
What I ...
- 91
5
votes
3
answers
2k
views
Lagrangian gauge invariance $L'=L+\frac{df(q,t)}{dt}$
So, I have to prove directly (e.g. by substitution) that if a path satisfies the Euler-Lagrange equations for the Lagrangian $L$ it does so for $$L'=L+\frac{df(q,t)}{dt}.$$ Let me tell you what I have ...
- 153
5
votes
4
answers
5k
views
Lagrangian mechanics and time derivative on general coordinates
I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
- 2,383
5
votes
1
answer
712
views
Confusion in derivation of Euler-Lagrange equations
Here's a screenshot of derivation of Euler-Lagrange from feynman lecture https://www.feynmanlectures.caltech.edu/II_19.html
My doubt is in the last paragraph. I get that $\eta = 0$ at both ends, but ...
- 301
5
votes
2
answers
610
views
Principle of stationary action vs Euler-Lagrange Equation
I am a bit confused as to what I should use to derive the equations of motions from the lagrange equation.
Suppose I have a lagrange function:
$$L(x(t), \dot{x}(t)) = \frac{1}{2}m\dot{x}^2-\frac{1}{...
- 714
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
4
votes
3
answers
364
views
Is it possible for the Action $S$ to *not* have a stationary point?
So the path of an object in configuration space is given by Hamilton's principle, which states that the path which the particle travels on is the one on which the action is stationary:
$$\delta S = \...
- 686
4
votes
4
answers
260
views
Variation of a function
I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
- 309