# Questions tagged [integrable-systems]

Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated to the invariants of the foliation span the tangent distribution: there exists a maximal set of Poisson-commuting invariants in phase space. May be used more broadly for systems possessing simple analytic solutions.

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### Integrability of a constraint when $x$ and $y$ depend on $t$ [closed]

I'm working through problem 6 in chapter 1 in Goldstein's classical mechanics book. I've reduced it to asking, if $x$ and $y$ are coordinates and function of time $t$, whether the differential ...
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### Does the functional form of a perturbative Hamiltonian indicate how many nonzero terms are in energy corrections?

Suppose we add a perturbative Hamiltonian to a quantum system. In principle, we can compute high order corrections to the energy levels with perturbation theory. However, I've come across some systems ...
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### Help with computations of first return time in an integrable system

I have example of an integrable system $(J,H):M\rightarrow \mathbb{R}^2$ where $J$ generates an $S^1$-action for which I'm trying to compute the action-angle coordinates. I have done all the steps but ...
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### Generalised hydrodynamics and the Dirac delta potential

Generalised Hydrodynamics is a theory of hydrodynamics for Quantum integrable systems. Those system are integrable in the sense that one can find an infinite number of conserved charges i.e. ...
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### Solubility of integrable systems and the classical XXZ model

I've been learning about integrability in the Hamiltonian sense, and trying to wrap my mind around the analytic power afforded by integrability, both in quantum and classical systems. My goal with ...
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### Does Poisson Distribution means the system is chaotic?

The Berry-Tabor Conjecture says that for classically integrable systems, the corresponding quantum systems obey the Poisson distribution for their energy-level spacing. But generally, the integrable ...
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### Is a free rigid body in 3D space an integrable system? [duplicate]

I am trying to find three integrable systems with 6 degrees of freedom using the Liouville–Arnold theorem. That means that a set of integrals of motion that correspond to a conserved quantity for ...
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### Do all members of the affine Toda field theories have fermion duals?

I know that the sine-Gordon model is $S$-dual to the massive Thirring model. Since sine-Gordon is a special case of affine Toda theory I was wondering if this extends to the generality of the models?
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### Yang-Baxter equation for an $S$ matrix depending on total momentum

I have a system where the two-particle scattering matrix $S_{12}(p_1,p_2)$ depends on the momentum difference $p_1-p_2$, and also on the total momentum $P=p_1+p_2$ in some non-trivial way. One can use ...
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In Shankar's QM book Chapter 10 pg. 274, it was said that quantum mechanically, the separability of the hamiltonian $$H=H_1(x_1, p_1)+H_2(x_2,p_2)$$ leads to the factorization of the wave function: $$\... • 4,235 1 vote 0 answers 100 views ### Integrability of one-dimensional system of motion? How can I prove that every one-dimensional system is integrable (meaning that there is a constant of motion)? It is clear that if H does not depend explicitly on time then H is indeed a constant ... 2 votes 1 answer 230 views ### Confusion about the action variable definition Suppose we have an integrable system consisting of a 2n-dimensional phase space M together with n independent functions f_{1\leq j \leq n } in involution. Suppose the level set$$M_f = \{ (p,q)...
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In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented:  \frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{...