All Questions
84
questions
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:
$$L(x(t), \dot{x}(t)) = \frac{1}{2}m\dot{x}^2-\frac{1}{...
- 714
6
votes
1
answer
2k
views
How can dissipative/friction terms be incorporated into a Lagrangian? [duplicate]
I'm trying to find a suitable Lagrangian for a damped harmonic oscillator, a system that satisfies the following equation of motion:
$$m \ddot{x} + \gamma \dot{x} + \frac{d\phi}{dx} = 0.$$
What I ...
- 91
1
vote
2
answers
298
views
Action principle and Functional derivative in CM
I want to extremize this well known action.
$$S[\phi]=\int \mathcal{L}(\phi(t),\dot{\phi}(t)) dt $$
The result is also well known. It turns out to be E-L equation.
The Action principle states that the ...
- 107
3
votes
2
answers
302
views
Derivation of Hamilton-Jacobi equation
I am trying my own way of deriving the Hamilton Jacobi equation
$$\frac{\partial S}{\partial t} = -H \tag{1}$$
through direct variation. I think the difficulty of doing this is that the upper limit ...
- 121
0
votes
2
answers
285
views
Taylor expansion in derivation of Noether-theorem
In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by:
$$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
- 13
0
votes
2
answers
239
views
$\int (f(x+\delta x) - f(x)) dx = \int \left ( \frac{df(x)}{dx} \delta x \right) dx$
From Landau and Lifshitz's Mechanics Vol: 1
$$
\delta S= \int \limits_{t_1}^{t_2} L(q + \delta q, \dot q + \delta \dot q, t)dt - \int \limits_{t_1}^{t_2} L(q, \dot q, t)dt \tag{2.3b}$$
$$\Rightarrow ...
user208739
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
1
vote
1
answer
258
views
Feynman Lecture Principle of Least Action: Glossed over Taylor expansion?
His initial one dimensional derivation of Newton's Second Law using the Principle of Least Action, I believe is fairly concise and easy to read. However, I did get hung up on his use of the Taylor ...
- 802
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:
$$\delta S = \...
- 686
3
votes
1
answer
269
views
Schwinger's variation of the action of point particle with *both* time and position as independent variables
In Chapter 8, pages 86-87, equations (8.5)-(8.11) of Julian Schwinger et al., Classical Electrodynamics, the equations of motion for the following action principle of a point particle in an external ...
- 707
1
vote
0
answers
119
views
When and why is $\frac{d}{dt}\delta q^{i}=\delta \frac{dq^{i}}{dt}$ true? [duplicate]
Apparently my question is different from Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}δq=δ\frac{dq}{dt}$. I hadn't noticed because the answer given in the comments to this question was ...
- 2,015
1
vote
0
answers
60
views
For the Lagrangian stationary action formula does the eta function for a specific path vary the distance from the true path? [closed]
This question can apply to any variation calculus problem although it has come up in my case for the stationary action principle so I will stick to the application I am using it for.
The action is ...
user86411
2
votes
1
answer
215
views
Does the integral in the action formula regarding the principle of stationary action represent an area or a length?
I am referring to the Feynman Lectures. The second volume has the "Principle of Least Action" as one of his lectures. (See after the 2nd paragraph below figure 19-6.) Although he does not explicitly ...
user86411
2
votes
1
answer
157
views
Why isn't it important, after which coordinates the Variation of the action integral is done?
I often read,that if the lagrangian $L=p\dot{q}-H$ of a pair of coordinates in phase space $(q,p)$ and $P\dot{Q}- K $, for some new pair of coordinates $(Q,P)$ only differ by a total time derivative $...
- 23
9
votes
1
answer
1k
views
How to find the Lagrangian of this system?
I am trying to find the Lagrangian $L$ of a system I am studying. The equations of motion is:
$$\left\{
\begin{array}{c l}
r \ddot{\phi} + 2\dot{r} \dot{\phi}+k(r) \cdot r \dot{r} \dot{\phi} = ...
- 253