All Questions
25
questions
1
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0
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83
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Holonomic constraints as a limit of the motion under potential
In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$...
- 133
2
votes
2
answers
154
views
What is the most general transformation between Lagrangians which give the same equation of motion?
This question is made up from 5 (including the main titular question) very closely related questions, so I didn't bother to ask them as different/followup questions one after another. On trying to ...
- 785
4
votes
2
answers
137
views
Gauge Symmetry of the Lagrangian
My teacher told the following statement to me during office hours. Is it correct and if so, how could one go about proving it?
Given a material system subject to holonomic and smooth constraints ...
- 402
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 ...
3
votes
3
answers
704
views
Why is the Legendre transformation the correct way to change variables from $(q,\dot{q},t)\to (q,p,t)$?
I always found Legendre transformation kind of mysterious. Given a Lagrangian $L(q,\dot{q},t)$, we can define a new function, the Hamiltonian, $$H(q,p,t)=p\dot{q}(p)-L(q,\dot{q}(q,p,t),t)$$ where $p=\...
- 11.8k
0
votes
0
answers
412
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Invertibility of the Legendre Transformation
The above image shows the Legendre Transformation in the context of an introduction to the Hamiltonian formalism.
My question is in 4.6, wherein $u(x, y)$ has been defined; what is the guarantee ...
- 317
2
votes
1
answer
143
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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
1
vote
1
answer
129
views
Meaning and Origin of an Expression which Involves Virtual Displacement
As an additional point of confusion related to the answer given here:
Confusion with Virtual Displacement
I have encountered the following expression in my study of virtual displacements.
$$\delta{...
- 348
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
3
votes
1
answer
997
views
Is the phase space a linear vector space?
Is the phase space in classical mechanics a linear vector space (LVS)? If yes,, can we define operators, inner products in this space?
Edit: I have seen Liouville operator $\mathcal{L}$ in Classical ...
- 26.8k
1
vote
2
answers
86
views
Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)? [duplicate]
Consider a classical mechanical system with generalized coordinates $q_i$, $i \in \{1,\dots\,n\}$. And Lagrangian $L$. Given a path $\gamma$ (with coordinates $\gamma_i$) and two times $t_1$ and $t_2$ ...
- 1,692
3
votes
0
answers
54
views
Action for solution of general nth order differential equation [duplicate]
Suppose I want to find solution to a general nth order differential equation.
(If I am right about the logic then) one might say that the solution $y\equiv y(x)$ is that function for which the ...
- 957
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
25
votes
3
answers
29k
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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
4
votes
1
answer
388
views
How does one express a Lagrangian and Action in the language of forms?
In Lipschitzs Classical Mechanics a Lagrangian is defined as:
$L(q,q',t)$ for some trajectory $q(t)$ of a particle
And the action is defined as:
$S:=\int^a_b L(q,q',t) dt$
How does one express ...
- 13.1k
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
0
votes
0
answers
144
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Action principles and covariant equations [duplicate]
Can every physically sound differential equation, that is covariant, deterministic etc. be derived by extremising a suitable action using a suitable lagrangian, that may be arbitary. Is this a ...
- 1,578
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
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
8
votes
1
answer
264
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Lagrangian formalism and Contact Bundles
In his Applied Differential Geometry book, William Burke says the following after telling that the action should be the integral of a function $L$:
A line integral makes geometric sense only if it'...
- 36.4k
25
votes
1
answer
3k
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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
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
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
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
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 ...