All Questions
12
questions
49
votes
8
answers
15k
views
Classical mechanics without coordinates book
I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
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
7
votes
1
answer
3k
views
Proof that total derivative is the only function that can be added to Lagrangian without changing the EOM
So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much ...
- 560
5
votes
2
answers
2k
views
Confusion with Virtual Displacement
I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
- 348
25
votes
3
answers
29k
views
Constructing Lagrangian from the Hamiltonian
Given the Lagrangian $L$ for a system, we can construct the Hamiltonian $H$ using the definition $H=\sum\limits_{i}p_i\dot{q}_i-L$ where $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Therefore, to ...
- 26.8k
25
votes
1
answer
3k
views
What's the physical intuition for symplectic structures?
I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector field....
- 1,043
11
votes
2
answers
1k
views
Is the Legendre transformation a unique choice in analytical mechanics?
Consider a Lagrangian $L(q_i, \dot{q_i}, t) = T - V$, for kinetic energy $T$ and generalized potential $V$, on a set of $n$ independent generalized coordinates $\{q_i\}$. Assuming the system is ...
- 1,281
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
3
votes
3
answers
3k
views
Virtual displacement and generalized coordinates
I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem.
The system is composed by $n$ points with ...
- 4,654
7
votes
1
answer
841
views
Rigorous version of field Lagrangian
In Classical Mechanics the configuration of a system can be characterized by some point $s\in \mathbb{R}^n$ for some $n$. In particular, if it's a system of $k$ particles then $n = 3k$ and if there ...
- 36.4k
4
votes
2
answers
870
views
Higher order covariant Lagrangian
I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at Klein-...
- 1,105
2
votes
1
answer
143
views
Arnold's holonomic constraints being limits of potential energy
The following quote comes from Arnold's "Mathematical methods in mechanics" book:
"We consider potential energy $U_N = Nq_2^2 + U_0(q_1, q_2) $, depending
on parameter $N$ (which we ...
- 399