Questions tagged [integrable-systems]
116 questions
14
votes
1 answer
$\tau$ structure of the sixth Painlevé equation
I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation.
I read that this equation had a $\tau$-structure, so we can determine a $\tau$ function associated…
12
votes
1 answer
On the Liouville-Arnold theorem
A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to find a transformation which leads to action-angle…
ablagi
- 125
11
votes
0 answers
Frobenius Condition for Singular Integrable Distributions
A smooth "singular" distribution $D\subseteq TM$ on an $n$-dimensional manifold $M$ is integrable if it is tangent to immersed submanifolds $N_\alpha$ that are disjoint and cover $M$. If dim$D=k$ constant, then the distribution is regular and the…
Marlo
- 1,223
9
votes
0 answers
Understanding the notation when finding action-angle coordinates
I'm trying to learn the basics of KAM theory and I wanted to first get to grips with Liouville integrability for Hamiltonian systems but I'm getting rather confused by the notation which seems to be universally assumed.
Suppose we have a time…
Theo Diamantakis
- 1,444
8
votes
1 answer
Is there any relation between Galois solvability and integrability of hamiltonian systems?
Galois theory provides a method and formalism to study solutions of polynomial equations and solvability.
Dynamical Hamiltonian systems have a somewhat similar concept of integrability.
Since many connections or reductions exist between differential…
Nikos M.
- 2,326
7
votes
1 answer
Commuting vector fields with common first integrals
Let $M$ be a $n-$dimensional smooth manifold and $X\in\chi(M)$ be a smooth vector field defined on it. Let $f_1,...,f_{n-2}:M\rightarrow \mathbb{R}$ be functionally independent first integrals of $X$, i.e.
$$ L_{X}f_i=0,\quad \forall i=1,2,...,n-2.…
Dadeslam
- 876
- 8
- 22
7
votes
0 answers
Action-angle variables in non-compact level sets
I am currently studying the Arnold-Liouville theorem, more precisely the construction of action-angle coordinates. I am following mainly the books "Physics for mathematicians I: Mechanics" by Spivak and "Mathematical Methods of Classical Mechanics"…
G. Gallego
- 671
5
votes
1 answer
What is the relationship between these two definitions of generating functions?
I'm doing my bachelor's thesis on Integrable Hamiltonian Systems, and one important part of the thesis will be proving the Liouville theorem. For this theorem I'm using the book by Arnold "Mathematical Methods of Classical Mechanics", but up to this…
francisg
- 51
4
votes
1 answer
How to show that Yang-Baxter equation is the same as braid equation?
The quantum Yang-Baxter equation is $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$. The braid equation is $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$. It is said that these two equations are equivalent. How to prove this? Thank you very much.
LJR
- 14,870
4
votes
0 answers
Unifying three avatars of the integral: the antiderivative, the pairing and the conserved quantity
In the first semester of calculus two types of integrals are taught to students: definite integral and indefinite integral (a.k.a. antiderivative). They are proven to be related through the fundamental theorem of calculus.
Later, both these notions…
Andrey Dmitrenko
- 143
4
votes
0 answers
Proving an Identity involving sums related to the $Z(N)$-Ising model
Background:
I am trying to construct meromorphic functions satisfying a number of axioms, so-called form factors which are important objects in integrable quantum models, following this paper. Building blocks for these form factors are functions…
Cream
- 412
4
votes
0 answers
Flow on a Torus is Transitive iff it is Incommensurate
Consider the system
$$
\dot{\mathbf{x}}(t)=\mathbf{a}
$$
where $t\in\mathbb{R},\mathbf{x}\in\mathbb{R}^n,$ and $\mathbf{a}\in\mathbb{R}^n.$
It is well known that the flow on the n-torus $\mathbb{T}^n$ generated by…
garserdt216
- 601
4
votes
0 answers
Vector fields generating a transformation
It would be great if someone can explain to me what the following means:
Vector fields $V_i, i=1,2,3$ generate 3 single-parameter groups of transformations in $\mathbb R$ -- $$\tilde x =x\exp(\alpha);$$$$\tilde x = x+\beta;$$$$ \tilde x = {x\over…
Henry
- 63
3
votes
0 answers
Reference Request for differential ideals of Pfaffian forms on jet bundles
My setting is the following:
Given two families of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1...n-m$), define
$$\eta^i := \sum_{j=1}^m a^i_j\omega^j + \sum_{j=1}^{n-m} b^i_j\mathrm dF^j.$$
Roughly speaking, I am…
cknoll
- 457
3
votes
1 answer
Non-integrable systems
If a Hamiltonian system in $\mathbb{R}^{2n}$ has $n$ suitable first integrals, then it is called an integrable system, and the Arnold-Liouville theorem tells us all sorts of nice things about the system: In particular, if a flow is compact then the…
J. M. F. Tsang
- 157
- 8