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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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Level set of Hamiltonian are the orbits?

Just a small question : If $x(t)=(p(t),q(t))$, then the position $x(t)$ of a particle is given by $$\dot p=-H_q(x(t))\quad \text{and}\quad \dot q=H_p(x(t)).$$ In particular, if $x$ solve the previous ...
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Different action-angle variables for a 2D harmonic oscillator

Consider a bidimensional harmonic oscillator. Ref. 1 says that, when the frequencies are commensurable, separating the variables in cartesian or polar coordinates leads to different action-angle ...
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28 views

If oscillatory motion is not simple (or chaotic), is it then by definition complex?

I'm trying to logically deduce or show that a specific type of motion is complex. It is two-dimensional oscillatory motion that can be expressed by coupled second order non-linear differential ...
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Algebraic Bethe Ansatz state generator problem

Given $B(\lambda)=T^0_1 (\lambda)$ the component of the monodromy matrix T that creates a state, $\lambda$ the spectral parameter and $| \Omega \rangle$ the reference ground state, In "Quantum Groups ...
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Poincare return map as area-preserving map

I'm trying to get some intuition into how the Poincare return map is area-preserving (when there are two momenta and two positions). Suppose $H=H(q_1,q_2,p_1,p_2)$, and let's suppose the system is ...
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Using action-angle variables in non-periodic system

I'm a little confused by the discussion in the last section $\S 50$ of Landau and Lifshitz's (Classical) Mechanics (1960, first English ed.). Here, they consider finite motion of a system whose ...
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Is *every* planar/2D system integrable?

Consider the generic following planar/2D system: $$\begin{cases} \frac{dx}{dt} = A(x,y)\\ \\ \frac{dy}{dt} = B(x,y), \end{cases}$$ where $A,B$ are two functions. Reading Classical Mechanics by ...
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121 views

Integrability of generalized Richardson-Hubbard model

Recently I got a bit interested in the possibility of finding spectrum of few interesting class of lattice quantum mechanical hamiltonians like Richardson's pairing hamiltonian, 1D Hubbard hamiltonian,...
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Looknig for resources on finding periodic orbit and stability on multidimensional Hamiltonian systems

I am looking for resources (books, papers, algorithms, codes) that explicitly explain the computation and analysis (using the monodromy matrix) of periodic orbits of multidimensional Hamiltonian ...
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182 views

When can action-angle variables be defined?

According to Goldstein, "We can define action-angle variables for [a separable Hamiltonian] system when the orbit equations for all of the $(q_i, p_i)$ pairs describe either closed orbits (...
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How can I say whether a Hamiltonian is integrable or not?

The transverse field Ising Hamiltonian $$ H = J\sum_{i=0}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{x}\sum_{i=0}^{N}\sigma_{i}^{x} $$ is integrable because it can be exactly solved using Jordan Wigner ...
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Conformal invariance in Toda field theories

A standard Toda field theory action will be of the shape: $$ S_{\text{TFT}} = \int d^2 x~ \Bigg( \frac{1}{2} \langle\partial_\mu \phi, \partial^\mu \phi \rangle - \frac{m^2}{\beta^2} \sum_{i=1}^r ...
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224 views

Physical intuition behind Poincaré–Bendixson theorem

The Poincaré–Bendixson theorem states that: In continuous systems, chaotic behaviour can only arise in systems that have 3 or more dimensions. What is the best way to understand this criteria ...
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Is there any approximation in which the double pendulum has an exact solution?

We know that the double pendulum isn't an integrable system, since only energy is conserved versus two degrees of freedom. My question is, does a physical approximation exist, in which the total time ...
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Link between integrability and soliton solutions

I have been doing some research on the properties and dynamics of solitons (in particular, solitons in superfluids) and several works and papers mention the link between solitonic solutions and ...
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How to define a generalised Gibbs ensemble (GGE) for classical integrable systems?

We have developed a complete description of generalised Gibbs ensemble (GGE) for quantum integrable systems, after including quasi-local charges. However, for classical integrable systems, such ...
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How to deduce the commutativity of the transfer matrix in algebraic bethe ansatz (and integrable systems)?

This is supposed to be a one-line derivation, but how to deduce the commutativity of the transfer matrix (in algebraic bethe ansatz) from the RTT relation? Thank you!
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Meaning of Yang-Baxter equation for classical $r$-matrix

I'm reading this [math/9802054] paper on the structure of the phase space of Chern-Simons TQFT. I'm stuck at the definition of the classical $r$-matrix, which goes as follows: This might sound dumb, ...
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What is the relationship between the integrability of a quantum many-body system and thermalization?

If a quantum many-body system is integrable, does it imply the system would always thermalized or many-body localized?
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169 views

R-matrix for spin chains

In algebraic Bethe ansatz procedure, one of the central objects is the R-matrix satisfying the Yang-Baxter equation, but all the papers/books give directly its expression without deriving it, so my ...
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28 views

Elliptic R-matrix and Yang Baxter solution for XYZ model [duplicate]

in the framework of QISM, How can i derive the R-matrix for XYZ Heisenberg model?
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29 views

$R$ matrix for XYZ spin chain [duplicate]

Trying to understand how the Algebraic Bethe Ansatz works, I'm actually reading some papers and trying to apply for XXZ or XYZ model. But my problem is that I don't know how to find the R-matrix ...
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48 views

Algebraic Bethe Ansatz and $R$-matrices [duplicate]

Trying to understand how the Algebraic Bethe Ansatz works, I'm actually reading some papers and trying to apply for XXZ or XYZ model. But my problem is that I don't know how to find the R-matrix ...
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50 views

Question about a system with all bounded orbits closed and maximal integrable

Given Hamiltonian system with $2n$-dim phase space, if there exist $k\ge n$ independent integrals of motions then we call it integrable Hamiltonian system. The largest number of independent integrals ...
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What does a non-integrable Hamiltonian system mean? [duplicate]

Is it something that cant be solved exactly? Are there any examples of such system? Is high-dimension lattice models like Ising model or Hubbard models considered as non-integrable systems?
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How are action variables linked to first integrals of a Hamiltonian?

Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one ...
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What is the point of using statistical mechanics on an integrable system?

On one hand, the ergodic hypothesis is usually justified by chaotic dynamics. On the other hand, it seems necessary to consider an integrable system in order to compute a nice, closed-form partition ...
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Reference for Bethe Ansatz solution of 1D spinless Hubbard model

I want to numerically solve 1D spinless Hubbard model using Bethe Ansatz. Can you provide me some online references for that.
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229 views

Proof of constructing Action-Angle Coordinates on Hamiltonian System

By Liouville-Arnold Theorem, we know we can construct action-angle coordinates such that the Hamiltonian system, when described in these coordinates, will have a form that is integrable by quadratures....
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Any model integrable but not separable?

In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable?
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The string hypothesis of the Bethe solutions of Heisenberg XXX model

I am studying L. Fadeev's "How Algebraic Bethe Ansatz works for integrable model". He takes Heisenberg $XXX_{1/2}$ model as an example. After obtaining the Bethe Ansatz Equations (BAE) for the roots {$...
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1answer
135 views

Quantum Integrable models and string theory

I’ve seen some recent papers on integrability by string theorists like Ed Witten or Kevin Costello but I don’t know why quantum integrable models are of interest to the string community. I would ...
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101 views

Conditions for periodic motions in classical mechanics

What are the conditions for classical motion to be periodic? In one dimension, if the motion is bounded, then it is also periodic. However, I don't think this generalizes to higher dimensions. I am ...
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Correlation functions of exactly solvable 1D quantum models

Quantum 1d spin-1/2 transverse Ising and XY models are both related to 2d classical Ising model. Are there any known simple explicit relations between correlation functions of this models? Something ...
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60 views

Maximal number of conserved quantities (classical integrability)

In these notes on page 4 the author says that if a $2d$-dimensional phase space has $d$ conserved quantities $F_{\mu}$ that Poisson commute, then $H$ can be written as a function of the $F_{\mu}$. Why ...
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254 views

Are there necessary and sufficient conditions for ergodicity?

What are the necessary and sufficient conditions (if any) for ergodicity (or non-ergodicity)? I see for instance that some integrable systems are not ergodic. For instance a linear chain of harmonic ...
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Conserved Quantities And Lagrangian Mechanics [duplicate]

By Noether's Theorem we know that any conserved quantity can be associated with a continuous symmetry. I was reading Kepler's Problem and I saw that there can be conserved quantities like Laplace-...
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71 views

XXh model for $1/2$-spin chain

I'm looking for correlation function (i.e. $\langle g \rvert \hat{S}_i^z \hat{S}_{i+n}^z\lvert g \rangle$, where $g$ stands for ground state) for given Hamiltonian($\hat{S}^z_i = \hat{c}^+_{j} \hat{c}...
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258 views

Why is it difficult to understand the phenomena of chaos in Newtonian mechanics?

This is a very simple-minded question. Why is it difficult to understand the phenomena of chaos in Newtonian mechanics and one has to turn to Hamiltonian formulation? I haven't read much about chaos ...
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KdV suggests a connection between waves in shallow water and the potential in the Schrödinger equation. What is the intuitive explanation?

The KdV equation $$v_t+\frac{1}{4}v_{xxx}-\frac{3}{2}vv_x=0$$ was originally invented to model waves in shallow water. However, it is well known that it also has applications in quantum mechanics. ...
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How can we define temperature in integrable systems?

For a thermalised system, at equilibrium, the usual quantum statistical ensembles are valid and we can define temperature at an equilibrium between system and its bath. But for integrable systems (non-...
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Example of an adiabatically perturbed integrable 2 d.o.f. Hamiltonian?

Consider the following (classical) Hamiltonian system: $H(u,v,p,q, \tau)$, where $(u,v)$ and $(p,q)$ are conjugated variables and $\tau = \epsilon t$ is a slowly varying parameter, $0 < \epsilon &...
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572 views

Integrability of quantum spin models

Based on this wiki page: https://en.wikipedia.org/wiki/Heisenberg_model_(quantum) The XXZ model is exactly Bethe ansatz solvable, but based on this paper (pape 5): https://arxiv.org/abs/1011.0380, the ...
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Why are we sure that integrals of motion don't exist in a chaotic system?

The stadium billiard is known to be a chaotic system. This means that the only integral of motion (quantity which is conserved along any trajectory of motion) is the energy $E=(p_x^2+p_y^2)/2m$. Why ...
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Is there a connection between Bertrands theorem and Chaos theory?

Bertrand's theorem states Among central force potentials with bound orbits, there are only two types of central force potentials with the property that all bound orbits are also closed orbits, ...
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170 views

Simple problem solvable with Bethe ansatz [closed]

I want some exercise for my students. Is there any simple but still non-trivial problem which can be solved with Bethe ansatz? The Heisenberg model is still too heavy.
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Liouville integrability vs. separability [duplicate]

A system is integrable if the Liouville integrability condition holds or the Hamilton-Jacobi equation is completely separable. The problem is, are the two the same actually? Namely, the one is ...
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372 views

2D harmonic oscillator having 4 constants of the motion and superintegrability

A 2D harmonic oscillator \begin{align} H=p_x^2+p_y^2+x^2+y^2 \end{align} has 4 constants of the motion: $E$ the total energy, $D$ the energy difference between coordinates, $L$ the angular momentum ...
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206 views

Nonlinear PT-symmetric model and integrability

I am coming from a maths background and these concepts are a bit new to me. I am studying an exercise paper and I am encountering the following problem. With $u_i(t)=u_i$ and of course $u_i u_i^*=|u|^...
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Finding higher order first integrals of a Hamiltonian

Suppose I have some general Hamiltonian of the form, $$ H(q,p) = \frac{1}{2} g^{\mu \nu} p_{\mu} p_{\nu} + V(q)$$ It is simple to find basic first integrals $F$ such that the Poisson bracket is zero,...