All Questions
239
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
130
votes
10
answers
41k
views
Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
- 8,540
48
votes
5
answers
4k
views
Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the Euler-...
- 1,350
41
votes
7
answers
11k
views
Is there a proof from the first principle that the Lagrangian $L = T - V$?
Is there a proof from the first principle that for the Lagrangian $L$,
$$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$
in classical mechanics? Assume that Cartesian coordinates are used. ...
- 761
31
votes
4
answers
6k
views
How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
- 955
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
24
votes
4
answers
4k
views
Confusion regarding the principle of least action in Landau & Lifshitz "The Classical Theory of Fields"
Edit: The previous title didn't really ask the same thing as the question (sorry about that), so I've changed it. To clarify, I understand that the action isn't always a minimum. My questions are in ...
- 28.3k
23
votes
3
answers
2k
views
What makes a Lagrangian a Lagrangian?
I just wanted to know what the characteristic property of a Lagrangian is?
How do you see without referring to Newtonian Mechanics that it has to be $L=T-V$?
People constructed a Lagrangian in ...
- 1,880
18
votes
7
answers
2k
views
When/why does the principle of least action plus boundary conditions not uniquely specify a path?
A few months ago I was telling high school students about Fermat's principle.
You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, ...
- 103k
18
votes
4
answers
3k
views
D'Alembert's Principle: Necessity of virtual displacements
Why is the d'Alembert's Principle
$$\sum_{i} ( {F}_{i} - m_i \bf{a}_i )\cdot \delta \bf r_i = 0$$
stated in terms of "virtual" displacements instead of actual displacements?
Why is it so necessary ...
- 1,487
17
votes
1
answer
8k
views
How do non-conservative forces affect Lagrange equations?
If we have a system and we know all the degrees of freedom, we can find the Lagrangian of the dynamical system. What happens if we apply some non-conservative forces in the system? I mean how to deal ...
user58143
16
votes
5
answers
6k
views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
- 1,263
14
votes
6
answers
2k
views
Does universal speed limit of information contradict the ability of a particle to pick a trajectory using Principle of Least Action?
I'm doing some self reading on Lagrangian Mechanics and Special Relavivity. The following are two statements that seem to be taken as absolute fundamentals and yet I'm unable to reconcile one with the ...
user297575
14
votes
4
answers
2k
views
Connection between different kinds of "Lagrangian"
Being a physic student I first heard the term: "Lagrangian" during a course about Lagrangian mechanics; at that time this term was defined to me in the following way:
For a classic, non ...
- 4,577
13
votes
3
answers
2k
views
In a Lagrangian, why can't we replace kinetic energy by total energy minus potential energy?
TL;DR: Why can't we write $\mathcal{L} = E - 2V$ where $E = T + V = $ Total Energy?
Let us consider the case of a particle in a gravitational field starting from rest.
Initially, Kinetic energy $T$ is ...
- 255