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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.

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.

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Linear sigma models and integrable systems

I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow, I faced difficulties in penetrating the literature... I'd highly appreciate any help with the following question: My aim is to relate…
user5831
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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 are we sure that no other, independent on $E$,…
Alexey Sokolik
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Constants of motion vs. integrals of motion vs. first integrals

Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of them, $2N-1$ are initial positions and velocity.…
user17116
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How do physicists use solutions to the Yang-Baxter Equation?

As a mathematician working the area of representation of Quantum groups, I am constantly thinking about solutions of the Yang-Baxter equation. In particular, trigonometric solutions. Often research grants in this area cite this as an "application"…
BBischof
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What is a good introduction to integrable models in physics?

I would be interested in a good mathematician-friendly introduction to integrable models in physics, either a book or expository article. Related MathOverflow question: what-is-an-integrable-system.
Phira
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What is precisely a Yangian symmetry?

The terms Yangian and Yangian symmetry appear in a list of physical problems (spin chains, Hubbard model, ABJM theory, $\mathcal{N}= 4$ super Yang-Mills in $d=4$, $\mathcal{N}= 8$ SUGRA in $d=4$), seem to be linked to (super)conformal symmetries and…
Trimok
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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 ask the speaker what precisely they mean by it.…
knzhou
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Topology of phase space

Context: From Liouville's integrability theorem we know that: If a system with $n$ degrees of freedom exhibits at least $n$ globally defined integrals of motion (i.e. first integrals), where all such conserved variables are in Poisson involution…
user929304
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Integrable vs. Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities ($n$ being the number of degrees of freedom), or $n$ whose Poisson brackets with each other are zero. The way I understand it, these…
Ronak M Soni
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Is there an analog to the Runge-Lenz vector for a 3D spherically symmetric harmonic potential?

The Runge-Lenz vector is an "extra" conserved quantity for Keplerian $\frac{1}{r}$ potentials, which is in addition to the usual energy and angular momentum conservation present in all central force potentials. I suppose it is not a coincidence…
user2640461
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Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie algebra) to prove Symplectic reduction theorem (on…
chhan92
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How to prove that a Hamiltonian system is *not* Liouville integrable?

To show that a system is Liouville integrable, we just need to find $n$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$. But how to prove that such a set of functions do not exist? For example, how to do this for the three-body problem?
Jiang-min Zhang
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What exactly are the 12 conserved quantities in the Two-Body Problem?

The Two-Body problem consists of 6 2nd-order differential equations \begin{equation} \ddot{\mathbf{r}}_1 = \frac{1}{m_1}\ \mathbf{F_g} \\ \ddot{\mathbf{r}}_2 = -\ \frac{1}{m_2}\ \mathbf{F_g} \end{equation} where $\mathbf{F_g}$ is the gravitational…
Matías Cerioni
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Chaos and integrability in classical mechanics

An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any examples of non-integrable systems that are not…
AngusTheMan
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Non-linear systems in classical mechanics

In general, what is meant by non-linear system in classical mechanics? Does it always concern the differential equations one ends up with (any examples would be greatly appreciated)? If so, is it considered as non-linear because of: Higher powers…
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