All Questions
Tagged with integrable-systems sg.symplectic-geometry
31 questions
3
votes
1
answer
429
views
Integrability of Schroedinger's equation
Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...
- 191
5
votes
0
answers
149
views
Deformation quantization of an integrable system
What is known about lifting n Poisson commuting functions on a 2n-dimensional symplectic manifolds (say R^2n) to Moyal-Weyl commuting functions?
- 401
1
vote
0
answers
71
views
Is there a relation between symplectic toric orbifolds and semi-toric systems?
So recently I have been studying semi-toric systems which are a generalization of toric symplectic manifolds and allow for the presence of focus-focus fibers. These were proved to be classified by $5$ ...
- 791
1
vote
0
answers
39
views
Question on the proof of doing a nodal trade, almost-toric fibrations
I am trying to understand the details of the proof of lemma $6.3$ of the following notes https://arxiv.org/pdf/math/0210033.pdf, which give us specific conditions of when we can swap a neighborhood of ...
- 791
1
vote
0
answers
66
views
Doing a nodal trade in a semi-toric system
Recently I have been studying semi-toric systems and almost toric fibrations. For the purpose of semi-toric fibrations I have been reading these notes https://arxiv.org/pdf/math/0210033.pdf. ...
- 791
5
votes
1
answer
237
views
Intuition for almost periodic solution and Poincaré recurrence theorem
I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer.
Suppose that we have a PDE that admit a solution $u$ that can be ...
- 93
3
votes
0
answers
74
views
Coordinates for quasiperiodic motion after reconstruction
Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...
- 141
3
votes
0
answers
199
views
Integrable systems and Lagrangian fibrations
It is known that every integrable system gives rise to a Lagrangian fibration via action-angle variables. My question is how to tell if a given Lagrangian fibration is an integrable system, that is ...
6
votes
0
answers
170
views
Introduction to the Adler-van Moerbeke theory
Is there a good introduction to the Adler-van Moerbeke theory of solving completely integrable systems by linearizing the flow on the Jacobian of an algebraic curve, for someone with a background in ...
8
votes
0
answers
285
views
Connection between integrable systems and group actions
An integrable system can be defined as a symplectic manifold together with the maxiumum possible number of Poisson commuting functions on the manifold which are almost everywhere independent. By the ...
- 979
23
votes
4
answers
3k
views
What is an "integrable hierarchy"? (to a mathematician)
This is one of those "what is an $X$?" questions so let me apologize in advance.
By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so ...
- 7,789
4
votes
0
answers
152
views
Integrable systems with Fano phase space?
What are some known examples of finite-dimensional integrable systems with symplectic Fano phase space?
Here by integrable system we mean a symplectic manifold $(X,
\omega)$ of dimension $2n$ with $...
user74900
2
votes
0
answers
118
views
Embeddings of the configuration space into the phase space of integrable systems
As always, I'm not sure if I'm about to ask a very stupid question, and I apologise if that is the case.
Most systems from physics come from classical Hamiltonians, defined on the phase space of ...
- 979
5
votes
1
answer
839
views
Why is every Hamiltonian system locally integrable?
It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional ...
- 654
7
votes
0
answers
144
views
Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold
Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will ...
- 4,787
2
votes
1
answer
148
views
multiplicity free actions - Guillemin&Sternbergy collective integrability
In this post I already ask a similar question.
Assume $M$ is a symplectic manifold of dimension $2n$. Assume $G$ is a Liegroup, $\mathfrak{g}$ be the Liealgebra and $\mathfrak{g^*}$ the corresponding ...
- 501
1
vote
1
answer
331
views
Lagrangian foliation
Let $(M,\omega)$ be a sympletic manifold and $\{ \cdot, \cdot \}$ the corresponding Poisson-bracket. Assuming $M$ is completely integrable w.r.t $f=f_1$, so we find $n = \frac{1}{2}\dim M$ functions $...
- 501
2
votes
0
answers
159
views
Pulled back foliation is completely integrable
There is a question that arises, while I'm trying to understand Guillemin & Sternbergs paper "On collective complete integrability according to the method of Thimm".
Assume $M$ is a symplectic ...
- 501
10
votes
1
answer
188
views
Sign problem in a Calogero-Moser system: proof of integrability?
Everyone of us had sometimes this awful feeling that some sign is lost in a calculation and that this sign is perturbing some fundamental understanding of what is going on. I feel the same has ...
- 1,143
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
2
votes
0
answers
329
views
moduli space of meromorphic $G$-Higgs bundles
I want to clarify with some topics in moduli space of semistable $G$-Higgs bundles on curve $X$ (genus $g$ is large enough) of fixing topological type $d \in \pi_1(G)$. Simpson's construction gives us ...
- 171
2
votes
0
answers
165
views
Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom
Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation
$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$
can ...
- 1,038
4
votes
1
answer
395
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
1
vote
1
answer
103
views
Reduction along an Orbit for C.-M. systems
I am having trouble in understanding the section of this paper http://www-math.mit.edu/~etingof/zlecnew.pdf
where the author introduces the Calogero-Moser system as the reduction of a manifold $M$ on ...
- 121
9
votes
1
answer
502
views
Detecting Monodromy in Integrable Systems
Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know ...
- 401
8
votes
1
answer
1k
views
is the geodesic flow on Hyperbolic Plane completely integrable?
I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and $H:M\...
- 345
1
vote
1
answer
826
views
About the geometry of completely integrable systems
During a conversation I heard an assertion that I found at least dubious for the lack of adeguate hypothesis, but I am not able to imagine a counterexample, even if it is probably obvious to some of ...
- 4,306
11
votes
1
answer
713
views
Weakest condition for an integrable, almost-symplectic manifold?
I was recently speaking with someone who works in Computational Chemistry and they mentioned that in a lot of the computational simulations they do, they have systems that are not symplectic but still ...
- 515
10
votes
3
answers
864
views
Is the 'massive' Calogero-Moser system still integrable?
Background
The (rational) Calogero-Moser system is the dynamical system which describes the evolution of $n$ particles on the line $\mathbb{C}$ which repel each other with force proportional to the ...
- 13k
12
votes
3
answers
1k
views
Equations for Integrable Systems
So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further ...
- 16k
22
votes
2
answers
3k
views
What is the relationship between integrable systems and toric degenerations?
Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system?
An integrable system ...
- 4,949