# Exploring Integrability and Integrable Systems in Physics and Mathematics

• anthony2005
In summary: Integrability is a property of a dynamical system that allows it to be solved in whole or in part. A system is said to be integrable if certain conditions are met, such as the system's derivatives being continuous and compatible with the integral law.
anthony2005
The title is self-explanatory. What is it meant in the physics and maths community by the words integrability and integrable system?

Have you seen an explanation like this:

http://en.wikipedia.org/wiki/Integration_(mathematics )

where they first discuss integrating a smooth function...

or is this what really interests you:

http://en.wikipedia.org/wiki/Integrable_system

like maybe one of the systems listed at the end of the article??

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So, is it correct to state in general that: "an integrable system is a system which thanks to certain properties its dynamics is exactly solvable" ?

You should wait for someone who is more up to date on math and current terminology than I...but I'll give you my 2 cents:

first, you posted this under Quantum Physics,so if you are looking for a specific answer, check here in the Wikipedia article:

Quantum integrable systems

that seems different from you latest post.

second, You may have to define what 'solvable' means to you because the section in Wikipedia says this:

General dynamical systems

...The distinction between integrable and nonintegrable dynamical systems thus has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in exact form.

and under the 'Chaos' link

... the deterministic nature of these systems does not make them predictable.[

So, is it correct to state in general that: "an integrable system is a system which thanks to certain properties its dynamics is exactly solvable" ?
No, integrability means: can a given relationship between derivatives be integrated to yield a relationship between functions. For example, given the system

∂f/∂x = F(x,y)
∂f/∂y = G(x,y)

does f(x,y) exist? Answer, only if an integrability condition is satisfied: ∂2f/∂x∂y = ∂2f/∂y∂x,

that is, ∂F/∂y = ∂G/∂x.

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it is more suited with classical section,integrability of system is classified according to it's holonomicity.In classical dynamics a system which is non holonomic has at least one non-integrable eqn.they look like
Ʃaidqi +atdt=0
this eqn should not be a total differential(or can be converted).there are many examples of it.One simple and particular is rolling of a sphere on a rough surface.Point of contact satisfy a non integrable relation.

Bill_K said:
No, integrability means: can a given relationship between derivatives be integrated to yield a relationship between functions. For example, given the system

∂f/∂x = F(x,y)
∂f/∂y = G(x,y)

does f(x,y) exist? Answer, only if an integrability condition is satisfied: ∂2f/∂x∂y = ∂2f/∂y∂x,

that is, ∂F/∂y = ∂G/∂x.

That's only half of the truth! Your integrability conditions are sufficient only for simply connected regions in the $(x,y)$ plane, where $F$ and $G$ are free of singularities and smoothly differentiable.

A simple but eluminating example is the potential curl
$$\vec{F}(\vec{x})=\frac{-y \vec{e}_x+x \vec{e}_y}{r^2}.$$
It's everywhere curl free, except in the origin, i.e.,
$$\partial_x F_y-\partial_y F_x=0,$$
but it does not have a unique potential in every region in the plane that contains the origin, where the singularity sits.

Indeed, integrating the vector field along any circle around the origin gives $2 \pi$.

To make the potential unique, one has to cut the plane by a ray starting from the origin. A standard choice is the negative $x$-axis. I.e., you take out the points $(x,0)$ with $x \leq 0$.

It's most easy to find the corresponding potential by introducing polar coordinates. Here, we use
$$(x,y)=r (\cos \varphi,\sin \varphi)$$
with $$\varphi \in (-\pi,\pi),$$
which automatically excludes the negative x axis. The function $\vec{F}$ then reads
$$\vec{F}=\frac{\vec{e}_{\varphi}}{r}.$$
The potential thus can be a function of only $\varphi$, and the gradient reads
$$\vec{F} \stackrel{!}{=}-\vec{\nabla} V(\varphi)=-\frac{1}{r} V'(\varphi).$$
This gives, up to a constant
$$V(\varphi)=-\varphi.$$
The potential is indeed unique everywhere except along the negative $x$ axis, along which it has a jump
$$V(\varphi \rightarrow \pi-0^+)=-\pi, \quad V(\varphi \rightarrow -\pi + 0^+)=+\pi.$$
In Cartesian Coordinates this potential is given by
$$V(\vec{x})=-\mathrm{sign} y \arccos \left (\frac{x}{\sqrt{x^2+y^2}} \right ).$$

## 1. What is integrability in physics and mathematics?

Integrability refers to the property of a system or equation being able to be solved exactly, typically through the use of integrals. This means that the system has a set of well-defined solutions that can be expressed in terms of known mathematical functions.

## 2. What are integrable systems in physics and mathematics?

Integrable systems are systems or equations that exhibit integrability. These systems are often characterized by having a sufficient number of conserved quantities that allow for exact solutions to be obtained.

## 3. What is the significance of integrability in physics and mathematics?

Integrability plays a crucial role in the study of many physical and mathematical phenomena. It allows for the exact solutions of equations, which can provide insights into the underlying dynamics and behaviors of a system. It also allows for the simplification of complex systems, making them easier to study and understand.

## 4. What are some examples of integrable systems in physics and mathematics?

Examples of integrable systems include the harmonic oscillator, the Kepler problem, the two-body problem in classical mechanics, the Ising model in statistical mechanics, and the Korteweg-de Vries equation in nonlinear dynamics. These systems have well-known exact solutions and have been extensively studied in physics and mathematics.

## 5. How is integrability studied and applied in physics and mathematics?

Integrability is studied through various mathematical techniques, such as symmetry analysis, perturbation theory, and the use of special functions. It is also applied in a wide range of fields, including classical mechanics, quantum mechanics, statistical mechanics, nonlinear dynamics, and differential geometry. The study of integrable systems has led to many important developments in theoretical physics and mathematics.

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