0
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0answers
30 views

Proof of Arnold-Liouville's Thm: movement in angular coordinates conditionally periodic

I'm reading Arnold's book Mathematical Methods in Classical Mechanics and got stuck on the proof of Liouville's theorem on integrable systems. The proof finishes with Problem 11: Show that the motion ...
0
votes
0answers
36 views

Geodesic of symplectic manifold

Can I define geodesic on a manifold without Riemann structure? To be more specific, how can I define geodesic at symplectic manifold? Let's just look at simple case with symplectic form as ...
4
votes
2answers
109 views

A problem about symplectic manifolds in Arnold's book

There is a problem in Arnold's Mathematical Methods of Classical Mechanics which says that: Show that the map $A: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ sending $(p, q) \rightarrow (P(p,q), ...
2
votes
0answers
85 views

Geometric Cauchy Problem

I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes. We are in the following setting: let $Q$ be a manifold (of dimension ...
5
votes
1answer
126 views

Physical Meaning of Symplectic Vector Fields

The mathematics of symplectic (as well as Hamiltonian) vector fields is something that has been quite clear to me for some time, but recently I have been thinking much more about what certain ...
0
votes
1answer
456 views

Showing Jacobi identity for Poisson Bracket

We were given the following problem: show that $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$ where $[A,[B,C]]$ et cetera are Poisson brackets. As I understand it this is a poisson bracket (where ...
3
votes
1answer
78 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
1
vote
1answer
81 views

Will this set of functions form coordinates on a symplectic manifold?

Consider a symplectic manifold $(M, \omega)$. Let us define a concept of a complete set of observables: A set of functions $f_i : M \to \mathbb R$ form a complete set of observables if any ...