# What is Integrability: Definition and 66 Discussions

In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space.
Three features are often referred to as characterizing integrable systems:
the existence of a maximal set of conserved quantities (the usual defining property of complete integrability)
the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability)
the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability)Integrable systems may be seen as very different in qualitative character from more generic dynamical systems,
which are more typically chaotic systems. The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over sufficiently large time.
Complete integrability is thus a nongeneric property of dynamical systems. Nevertheless, many systems studied in physics are completely integrable, in particular, in the Hamiltonian sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center (e.g., the sun) or two. Other elementary examples include the motion of a rigid body about its center of mass (the Euler top) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the Lagrange top).
The modern theory of integrable systems was revived with the numerical discovery of solitons by Martin Kruskal and Norman Zabusky in 1965, which led to the inverse scattering transform method in 1967. It was realized that there are completely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (Korteweg–de Vries equation), the Kerr effect in optical fibres, described by the nonlinear Schrödinger equation, and certain integrable many-body systems, such as the Toda lattice.
In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the Lagrangian foliation), and if the flows are complete and the energy level set is compact, this implies the Liouville-Arnold theorem; i.e., the existence of action-angle variables. General dynamical systems have no such conserved quantities; in the case of autonomous Hamiltonian systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic.
A key ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution) if, locally, it has a foliation by maximal integral manifolds. But integrability, in the sense of dynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds.
Integrable systems do not necessarily have solutions that can be expressed in closed form or in terms of special functions; in the present sense, integrability is a property of the geometry or topology of the system's solutions in phase space.

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1. ### I Manifold hypersurface foliation and Frobenius theorem

Hi, starting from this thread, I'd like to clarify some mathematical aspects related to the notion of hypersurface orthogonality condition for a congruence. Let's start from a congruence filling the entire manifold (e.g. spacetime). The condition to be hypersurface orthogonal basically means...
2. ### I Riemann integrability and uniform convergence

Was reading the Reimann integrals chapter of Understanding Analysis by Stephen Abbott and got stuck on exercise 7.2.5. In the solutions they went from having |f-f_n|<epsilon/3(a-b) to having |M_k-N_k|<epsilon/3(a-b), but I’m confused how did they do this. We know that fn uniformly converges to...
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4. ### POTW Local Integrability of a Maximal Function

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12. ### I Riemann integrability with a discontinuity

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13. ### Checking for integrability on a half-open interval

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14. ### Admissions Role of AdS/CFT correspondence in the context of integrability

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15. ### Finding Riemann Integrability for f(x) on [0,1]

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16. ### MHB Prove Integrability of f(x)| 0 to 1 Inequality

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17. ### A Are two models with the same S matrix directly related by their parameters?

Hello, I was wondering if two models show the same S matrix by a direct relation between their parameters, does that necessarily mean that both models are exactly equivalent? My idea is that this is true, but would like to know about a solid argument about it if possible, thank you!
18. ### I How does the limit comparison test for integrability go?

Hi everybody! I have another question about integrability, especially about the limit comparison test. The script my teacher wrote states: (roughly translated from German) Limit test: Let -∞ < a < b ≤ ∞ and the functions f: [a,b) → [0,∞) and f: [a,b) → (0,∞) be proper integrable for any c ∈...
19. ### Riemann's Integrability Condition

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20. ### MHB Riemann Criterion for Integrability - Stoll: Theorem 6.17

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21. ### Proving Riemann Integrability of g∘f for Linear Functions

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22. ### Integrability of a differential condition

I'm reading "The variational principles of mechanics", written by C. Lanczos and he said that, if one have the condition dq_3 = B_1 dq_1 + B_2 dq_2 and one want to know if there is a finite relation between the q_i, on account the given condition, one must have the condition \frac{\partial...
23. ### Is Zero Outer Content Sufficient for Function Integrability?

Homework Statement 1. Suppose that ##f = 0## at all points of a rectangle ##R## except on a set ##D## of outer content zero, where ##f \geq 0##. If ##f## is bounded, prove that ##f## is integrable on ##R## and ##\int \int f dA = 0##. 2. Now suppose ##f## is an integrable function on a...
24. ### Discovering Liouville Integrability in Classical Mechanics

Hi Let a classical particle with unit mass subjected to a radial potential V and moving in a plane. The Hamiltonian is written using polar coordinates (r,\phi) H(r,\phi) = \frac{1}{2}(\dot{r}^2+r^2\dot{\phi}^2) - V(r) I consider the angular momentum modulus C=r^2\dot{\phi}, and I...
25. ### Integrability Proof: Showing h(x) = 0

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26. ### Riemann Integrability of Composition

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27. ### Integrability of f(x) on [0,1]

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28. ### Exploring Integrability and Integrable Systems in Physics and Mathematics

The title is self-explanatory. What is it meant in the physics and maths community by the words integrability and integrable system?
29. ### Does continuity prove integrability?

Hi, Homework Statement I am now asked to prove that f: [0,1]->[0,1] defined thus f(0)=0 and f(x)=1/10n for every 1/2n+1<x<1/2n for natural n, is integrable. Homework Equations The Attempt at a Solution Would it suffice to show that f is continuous? I.e. that lim x->0 f(x) =...
30. ### Uniform integrability under continuous functions

Let X be a uniform integrable function, and g be a continuous function. Is is true that g(X) is UI? I don't think g(X) is UI, but I have trouble finding counter examples. Thanks.
31. ### Integrability implies continuity at a point

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32. ### Integrability of Sinusoidal Function on [-1, 1]: Finding L(f, P) and U(f, P)

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33. ### Riemann Integrability of Thomae's Function

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34. ### Integrability of Monotonic Functions on Closed Intervals Explained

The book is saying if f is monotonic on a closed interval, then f is integrable on the closed interval. Or basically if it is increasing or decreasing on the interval it is integrable on that interval This makes sense, however this theorem seems to obvious because obviously if a function...
35. ### Riemann Integrability Question

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36. ### Continuity at a point implies integrability around point?

If a function f is continuous at a point p, must there be some closed interval [a,b] including p such that f is integrable on the [a,b]? As a definition of integrable I'm using the one provided by Spivak: f is integrable on [a,b] if and only if for every e>0 there is a partition P of [a,b]...
37. ### Integrability, basic measure theory: seeking help with confusing result

The canonical example of a function that is not Riemann integrable is the function f: [0,1] to R, such that f(x)=1 if x is rational and f(x)=0 if x is irrational ( i know some texts put this the other way around, but bear with me because i can reference at least one text that does not). Hence...
38. ### Proving integrability of a composition of functions

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39. ### Integrability and Lipschitz continuity

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40. ### Is the Lebesgue Integral of sin x / x from 0 to infinity Nonexistent?

Hello all, can someone please direct me towards an argument proving the Lebesgue integral from 0 to infinity of sin x / x does not exist? Many thanks
41. ### Riemann Integrability, Linear Transformations

Homework Statement If f,g are Riemann integrable on [a,b], then for c,d real numbers, (let I denote the integral from a to b) I (cf + dg) = c I (f) + d I (g) Homework Equations The Attempt at a Solution I have the proofs for c I(f) = I (cf) and I (f+g) = I (f)...
42. ### Fourier transform of f'(x), lebesgue integrability

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43. ### Is f(x) = 1 if x is rational, 0 if x is irrational Riemann integrable on [0,1]?

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44. ### Proving integrability of a strange function

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45. ### Compositions of function and integrability (is this right?)

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46. ### Momentum Operators and the Schwartz Integrability Condition

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47. ### Proving Integrability on [0,1]: A Convergence Analysis

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48. ### Riemann integrability of composite functions

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49. ### Hamiltonian systems, integrability, chaos and MATH

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50. ### Can a Bounded Function on a Rectangle be Integrable over Q?

Homework Statement Let Q=I\times I (I=[0,1]) be a rectangle in R^2. Find a real function f:Q\to R such that the iterated integrals \int_{x\in I} \int_{y\in I} f(x,y) \; and \int_{y\in I} \int_{x\in I} f(x,y) exists, but f is not integrable over Q. Edit: f is bounded Homework...