All Questions
9 questions
0
votes
1
answer
160
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Reference for action-angle coordinates [closed]
Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !
- 97
2
votes
0
answers
195
views
How to check conditions for Liouville-Arnold theorem? [closed]
Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem:
Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...
- 679
10
votes
0
answers
658
views
Determinant as a Hamiltonian
Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...
- 366
7
votes
0
answers
479
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Question about theorem in Arnold's book on action-angles variables
I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question)
If you don't have the book or ...
- 259
3
votes
1
answer
355
views
Local symplectomorphisms become global ones?
It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on.
Now, I stumbled over the question whether in some particular cases, as I will describe below, ...
- 259
2
votes
0
answers
1k
views
Proof of Arnold-Liouville theorem in classical mechanics [closed]
I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...
- 181
4
votes
1
answer
396
views
Weinstein's local classification of Lagrangian foliations
In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...
- 783
20
votes
4
answers
3k
views
What is the role of contact geometry in the hamiltonian mechanics?
Let us assume someone is interested in the study of Hamiltonian mechanics.
What are good examples to illustrate him of the usefulness of contact geometry in this context?
On one hand the Hamiltonian ...
- 4,306
8
votes
1
answer
2k
views
Calculating the geodesic equation for a particular set of phase-space coordinates
Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...
- 8,532