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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Constants of motion [duplicate]

For any system performing any kind of motion with $n$ degrees of freedom, are $2n-1$ integrals of motion and also $2n$ constants of motion always present? If yes, then is there always a symmetry for ...
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Time derivative of path ordered monodromy matrix

I am currently gettting familiar with integrably systems and came the following statement in my literature: $U=U(x,\lambda,t)$ some matrix (Lax component) we define $$T(\lambda,t) = \mathcal{P} \exp \...
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131 views

The planar limit, self-duality and their relation to two dimensions

In the lecture notes by Beisert on integrability, it is stated that integrability is a property mainly in two-dimensional field theories, with some higher-dimensional examples. As higher-dimensional ...
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WZW model - affine Kac-Moody current algebra from Quantum Group exchange algebra

In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model. ...
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30 views

XXZ chain exact ground state energy

I would like to know whether you are aware of a reference where the analytical expression of the ground state energy of the XXZ model is presented, if such formula exists (probably from a Bethe Ansatz ...
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What is an example of an integrable system with arbitrary coupling constants?

It is discussed here what an integrable system is: https://mathoverflow.net/questions/6379/what-is-an-integrable-system. Does there exists an integrable system (for example in the form of a 1D-Quantum ...
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Are partial derivatives in the context of Action-Angle variables different from partial derivatives of functions?

Let's say I have a system with two degrees of freedom and I can find two independent action variables. One action variable is total energy expression, such as is often used in classical mechanics. $$...
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41 views

Conserved Quantities and Integrability in the $N$-Body Problem

Under my understanding of integrability, a system with $2n$-dimensional phase space is integrable when there are at least $n$ constants of motion satisfying some conditions (e.g., they are in ...
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Grand-canonical partition function for an integrable system

When speaking about the grand-canonical ensemble of a statistical system, one usually works with a case, when there are several conserved quantities - total number of particles $N$, angular momentum $...
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Completely Integrable Frustrated Lattice Systems

The Toda lattice is a prime example of a lattice system that is completely integrable, in the sense that it admits a Lax pair, https://doi.org/10.1143/PTP.51.703, making it easy to find soliton ...
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Itzykson Zuber Quantum Field Theory: meaning of integrable system

Here is a part of the book Quantum Field Theory by Itzykson and Zuber: I have two questions: what does the author mean that equation (1-30) form and integrable system, and why? what is the ...
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Non-Integrable models in 1+1D

Is it possible to have a non-integrable system in (1+1)D in Classical Physics? For some reason, I get the intuition that there shouldn't be any such systems. What if we consider (1+1)D systems in ...
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The physical meaning of maximal non-integrability of the contact structure

So, basically integrability is equivalent to the existence of an integral manifold of the distribution and I guess, the integral manifold is like a plane of motion where state moves in physical sense. ...
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Doubt on Lax formulation of Korteweg–de Vries equation

The Korteweg–de Vries equation is given by: $$\frac{\partial u(x,t)}{\partial t}-6u\frac{\partial u(x,t)}{\partial x}+\frac{\partial^3 u(x,t)}{\partial x^3}=0$$ This equation can be formulated using ...
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Definition of 'Antichiral' in the Thirring Model

This is my first post on this website so please excuse any poor formatting. In the lecture series on quantum integrable models, much attention is focused towards the Thirring model, $$\mathscr{L} = \...
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Question about a 2D Harmonic Oscillator with incommensurate frequencies and Integrability

In Classical Dynamics by José & Saletan [section 4.2.2] they give the example of a 2D Harmonic Oscillator whose equations of motion are \begin{equation} \ddot{x}_i+\omega_i^2x_i=0 \ \ \ \ \ \text{...
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Does the Hamilton-Jacobi equation imply that there are always $N$ conserved quantities for any system with $N$ degrees of freedom? [duplicate]

I'm reviewing the Hamilton-Jacobi equation because I'm working on a research project about Kerr black holes and the geodesics of particles gravitating them (This is not really relevant to the question,...
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Conditions of separation of variable of wave function

In quantum mechanics as far as I know when the hamiltonian operator can be written as $$H = H_1 + H_2$$ then the wave function can be written as $$\psi_1 \cdot \psi_2$$ but as we get further in ...
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Reference for Non-Linear Water Waves

In class, my professor just mentioned that some finite-amplitude water waves were satisfied by the KdV equation. Is there some reference which shows how to derive this from $1st$ principles, and also ...
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Canonically conjugate variable of rapidity [closed]

Does the rapidity $\theta \in \mathbb{R}$ have a canonically conjugate variable? More specifically, for some smooth function $f \in \mathcal{S}(\mathbb{R})$, by Plancherel's theorem we have (up to ...
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130 views

Action-angle variables for anharmonic oscillator

I have an equation of potential given: $$U = U_0\tan^2( \alpha(t)q)$$ I need to find a motion rules for that potential in terms of action-angle variables. Using the fact that Hamiltonian is equal to ...
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Physical meaning of third derivative with respect to position

I currently on a numerical solver for the KdV equation which reads $$ u_t + uu_x = u_{xxx} $$ I was wondering the physical sense of this third derivative with respect to $x$. I know that the $uu_x$ ...
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Delaunay variables

I've read a little bit about Delaunay variables, but I can't understand what they are good for. Do they make calculations easier? What is the advantage of using them? Where can I read a bit more about ...
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Why is $T\overline{T}$ deformation so exciting?

I keep an ear to high energy physics discussions, and one of the things I've heard a lot about recently in these channels is the TTbar deformation (stylized $T\overline{T}$)$^1$. Wikipedia is lacking ...
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Quantity conserved for the 3D spherically symmetric harmonic potential $V(r)=\alpha r^2$ [duplicate]

I know that in the case of the Kepler problem there is a quantity (other than energy, momentum,...) conserved which is the Runge-Lenz vector. Is there also an "exotic" quantity conserved for a 2-Body ...
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Lax pairs, Nonlinear Schrodinger Equation

Briefly: I have two Lax pairs in matrix form and using compatibility condition I have found a nonlinear partial differential equation system. I have searched this system very much seems the nonlinear ...
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Understanding the Staeckel conditions for separability of the Hamilton-Jacobi equation

I'm trying to understand the so-called Staeckel conditions, specifically the fifth one, for separability of the Hamilton-Jacobi equations, as described in Goldstein's Classical Mechanics (3:rd edition)...
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Spectral form factor for integrable systems

The spectral form factor is defined as the Fourier transform of the autocorrelation function of the energy density $$K(t) = \frac{\int d E_1 dE_2 \langle \rho(E_1) \rho(E_2)\rangle e^{i(E_1 - E_2)t}}{...
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Infinite number of conserved charges for the Sine-Gordon Lagrangian

I recently came across a paper of Witten that talks about the S-matrix of the supersymmetric non-linear sigma model. In the beginning part of the paper, he mentions that theories like the non-linear ...
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392 views

What do physicists mean by an “integrable system”?

The notion of "integrability" is everywhere in physics these days. It's a hot topic in high energy theory, atomic physics, and condensed matter. I hear the word at least once a week, and every time, I ...
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Intuition behind focusing vs defocusing in integrable systems like NLS, KdV, mKdV

The following are examples of integrable systems arising from the AKNS system (check out AKNS paper here and a short Wikipedia description) Non-Linear Schrodinger equation Korteweg-de Vries equation ...
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Lax Pairs In Integrability

I am working through Dr. Beiserts notes (https://people.phys.ethz.ch/~nbeisert/lectures/IntHS16-Notes.pdf) and have difficulty obtaining the second step in (2.9): $$\{{\rm tr}L^{k},{\rm tr}L^{\ell}\} ...
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What happens to Hubbard chain when perturbation theory blows up? Singular energy for complex interaction

Consider the spinful Hubbard chain: $$ H = - t \sum_{i,\sigma} \left( c^\dagger_{i,\sigma} c^{\vphantom \dagger}_{i+1,\sigma} + h.c. \right) + U \sum_n \left( n_{i,\uparrow} - \frac{1}{2} \right)\left(...
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Quantum integrable models with no classical integrable counterpart

I am looking for examples for quantum integrable systems that have no classical integrable limit.
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Integrability condition of perturbations of Wess-Zumino-Witten (WZW) models

When one tries to analyze the renormalization group of marginal perturbations of Wess-Zumino-Witten (WZW) model in 1+1d, only those "integrable perturbations" can be computed analytically. I wonder ...
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235 views

Couple of non-interacting, non-integrable Hamiltonian systems

I have two Hamiltonians, $H(p_1, q_1, \dots, p_n, q_n)$ and $H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m)$. They are non-interacting, indeed they have different variables. Moreover, I know that they are both ...
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Liouville's integrability theorem: action-angle variables

For classical dynamical systems, let $I_{\alpha}$ stand for independent constants of motion which commute with each other. 'Remark 11.12' on pg 443 of Fasano-Marmi's 'Analytical Mechanics' suggest ...
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Troubles with Haldane Shastry Spin Chain

I'm actually reading the article of Shastry "Exact solution of an S= 1/2 Heisenberg Antiferromagnetic Chain with Long-rnaged interactions", Phys. Rev. Lett. 60, 639 (1988)" The articles ...
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108 views

Finding period from action-angle variable in one dimensional potential [closed]

I want to calculate the period from the action-angle variable for a particle in a one dimensional potential $V = V_0 \tan^2(q \pi/2a)$. After doing some algebra I get $$I = \frac{\sqrt{2mE}}{2\pi} \...
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2answers
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Role of AdS/CFT correspondence in the context of integrability

I was wondering how the AdS/CFT correspondence fits in the context of integrability. As I understand, the AdS/CFT correspondence postulates a duality between gravity theories and CFT's. If one theory ...
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123 views

Integrability of a non-integrable quantum spin model at critical point

Is it right, that non-integrable quantum spin models in one dimension become integrable at their critical points? Or do they stay nonintegrable at the critical point also? Are there any examples known?...
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566 views

Hamiltonian with one constant of motion (besides the Hamiltonian itself)

The background of my question is a well known fact: a Hamiltonian system with $n$ degrees of freedom with $n$ constants of motion is integrable. My question is about the case in which there are only ...
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92 views

Are powers of the harmonic oscillator semiclassically exact?

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
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Which Hamiltonian systems are intrisically linear?

What physical properties has a dynamical system whose equation of motion are linear? When does it exist a change of coordinates which turn the equation of motions in a linear system? My teacher says ...
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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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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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63 views

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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226 views

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 ...