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.
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...
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...
Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1-form ##\omega##, namely that ##d\omega = \omega \wedge \alpha## is actually equivalent to the existence of smooth maps ##f## and ##g## such that ##\omega = fdg##.
I found this About...
Let ##f## be a measurable function supported on some ball ##B = B(x,\rho)\subset \mathbb{R}^n##. Show that if ##f \cdot \log(2 + |f|) ## is integrable over ##B##, then the same is true for the Hardy-Littlewood maximal function ##Mf : y \mapsto \sup_{0 < r < \infty}|B(y,r)|^{-1} \int_{B(y,r)}...
Apologies for potentially being imprecise and clunky, but I'm trying understand integrability of the following Hamiltonian
$$H(x,p)=\langle p,f(x) \rangle$$
on a 2n dimensional vector space
$$T^{\ast}\mathcal{M} =\mathbb{R}^{2n}.$$
Clearly this is just the 1-form $$\theta_{(x,p)} =...
Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows:
$$
S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$
A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...
We show that there is a partition s.t. the upper sum and the lower sum of ##f## w.r.t. this partition converge onto one another.
Let ##\epsilon>0##.
Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a...
Suppose we have an infinite dimensional Hilbert-like space but that is incomplete, such as if a subspace isomorphic to ##\mathbb{R}## had countably many discontinuities and we extended it to an isomorphism of ##\mathbb{R}^{\infty}##. Is there a measure of integrating along any closed subset of...
I am trying to evaluate an integral with unknown variables ##a, b, c## in Mathematica, but I am not sure why it takes so long for it to give an output, so I just decided to cancel the running. The integral is given by,
##\int_0^1 dy \frac{ y^2 (1 - b^3 y^3)^{1/2} }{ (1 - a^4 c^2 y^4)^{1/2} }##
We have a function ##f: [a,b] \mapsto \mathbb R## (correct me if I'm wrong but the range ##\mathbb R## implies that ##f## is bounded). We have a partition ##P= \{x_0, x_1 , x_2 \cdots x_n \}## such that for any open interval ##(x_{i-1}, x_i)## we have
$$
f(x) =g(x)
$$
(##g:[a,b] \mapsto \mathbb...
Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if
$$ sup \{L (f,P) : \text{P belongs to the set of...
So, I know that a function is integrable on an interval [a,b] if
##U(f,P_n)-L(f,P_n)<\epsilon ##
So I find ##U(f,P_n## and ##L(f,P_n##
##L(f,P_n)=5(3-\frac{1}{n}-0)+5(3+\frac{1}{n}-(3-\frac{1}{n}))+7(4-(3+\frac{1}{n}))=22-\frac{2}{n} ##...
For a closed interval ##[a,b]## I have learned that ##U(f,P)-L(f,P)=\frac{(f(b)-f(a))\cdot(b-a)}{N}## where ##N## is the number of subintervals of ##[a,b]## (if ##f## is monotonically decreasing, change the numerator of the fraction to ##f(a)-f(b)##). However, if the interval is half-open, then...
I have a masters degree. I studied general relativity and quantum field theory. I was interested in applying to PhD programs for AdS/CFT. I was wondering how integrability fits in the context of AdS/CFT. As I understand, the AdS/CFT correspondence postulates a duality between gravity theories...
Homework Statement
Find a such that f is Riemann integrable on [0,1], where:
##f = x^acos(1/x)##, x>0 and f(0) = 0
Homework EquationsThe Attempt at a Solution
I found at previous points a such that f is continuous, bounded and derivable, but I am not sure how to use that (as all these...
Prove that the function
f(x) = 1+x, 0 \le x \le 1, x rational
f(x) = 1-x, 0 \le x \le 1, x irrational (they are one function, I just don't know how to use the LATEX code properly)
is not integrable on [0,1]
I don't know where to start, I tried to evalute the lower and upper Riemann sum but it...
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!
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 ∈...
Homework Statement
Here is a link to the proof I am reading: https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf
Homework EquationsThe Attempt at a Solution
The proof to which I am referring can be found on pages 8-9. At the top of page 9, the author makes an assertion which I endeavored to...
I am reading Manfred Stoll's book: Introduction to Real Analysis.
I need help with Stoll's proof of Riemann's Criterion for Integrability - Stoll: Theorem 6.17
Stoll's statement of this theorem and its proof reads as follows:
https://www.physicsforums.com/attachments/3941In the above proof we...
Homework Statement
Prove or give a counter example of the following statement:
If f: [a,b] \to [c,d] is linear and g:[c,d] \to \mathbb{R} is Riemann integrable then g \circ f is Riemann integrable
Homework EquationsThe Attempt at a Solution
I'm going to attempt to prove the statement is...
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...
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...
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...
Homework Statement Let h(x) = 0 for all x in [a,b] except for on a set of measure zero. Show that if \int_a^b h(x) \, dx exists it equals 0.
We are given the hint that a subset of a set of measure zero also has measure 0.
Homework Equations
We've discussed the Lebesgue integrability...
Homework Statement
Let ψ(x) = x sin 1/x for 0 < x ≤ 1 and ψ(0) = 0.
(a) If f : [-1,1] → ℝ is Riemann integrable, prove that f \circ ψ is Riemann integrable.
(b) What happens for ψ*(x) = √x sin 1/x?
Homework Equations
I've proven that if ψ : [c,d] → [a,b] is continuous and for every set...
Homework Statement
Let f(x) be deﬁned on [0,1] by
f(x) = 1 if x is rational
f(x) = 0 if x is irrational.
Is f integrable on [0,1]? You may use the fact that between any two rational numbers
there exists an irrational number, and between any two irrational numbers there exists
a...
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) =...
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.
Homework Statement
If f is integrable on [a,b], prove that there exists an infinite number of points in [a,b] such that f is continuous at those points.
Homework Equations
I'm using Spivak's Calculus. There are two criteria for integrability that could be used in this proof (obviously...
The problem states:
Decide if the following function is integrable on [-1, 1]
f(x)=\left\{{sin(\frac1{x^2})\;\text{if}\;x\in[-1,\;0)\cup(0,\;1]\atop a\;\text{if}\;x=0}
where a is the grade, from 1 to 10, you want to give the lecturer in this course
What I don't understand is how to find L(f...
Homework Statement
Show the Thomae's function f : [0,1] → ℝ which is defined by f(x) = \begin{cases} \frac{1}{n}, & \text{if $x = \frac{m}{n}$, where $m, n \in \mathbb{N}$ and are relatively prime} \\ 0, & \text{otherwise} \end{cases} is Riemann integrable.
Homework Equations
Thm: If fn...
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...
This isn't a homework question. My adviser has me studying basic analysis and has lately pushed me towards the following question:
"Let f be any continuous function. Can we prove that there exists a SEQUENCE of step functions that converges UNIFORMLY to f?"
I have noticed this idea is...
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]...
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...
Homework Statement
Show that if f is an integrable function on [a,b] then g(x) which is defined to be sin(f(x)) is also integrable
Homework Equations
The Attempt at a Solution
I started off by trying to show that since f is integrable it has an Upper sum and a Lower sum where...
(I've been lighting this board up recently; sorry about that. I've been thinking about a lot of things, and my professors all generally have better things to do or are out of town.)
Is there an easy way to show that if f is Lipschitz (on all of \mathbb R), then
\int_{-\infty}^\infty f^2(x)...
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)...
a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous.
b) show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0.
i am a little stuck on part b). i am trying to use the dominated convergence theorem but i am a bit confused on what...
Homework Statement
Let A={1/n, n =natural number}
f: [0,1] -> Reals
f(x) = {1, x in E, 0 otherwise
Prove f is riemann integrable on [0,1]
Homework Equations
The Attempt at a Solution
Not quite sure, but I think supf = 1 and inf f= 0 no matter what partition you take, then...
Homework Statement
Hi guys. I'm really struggling with this problem. Any help is welcomed.
Suppose I have a function f(y) = \intg(x)/(x^2) on the set [(y/2)^(1/2), \infty]. g(x) is known to be integrable over all of R.
I want to show that f is integrable over [0,\infty], and that the...
Compositions of functions and integrability (is this right?)
Homework Statement We know that if f is integrable and g is continuous then g\circf is integrable. Show to this is not necessarily true for piecewise continuity. We are given the hint to use a ruler function and characteristic...
Hi All,
When computing the commutator \left[x,p_{y}\right], I eventually arrived (as expected) at \hbar^{2}\left(\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) - \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)\right) and I realized that, as correct...
This is a question I have been struggling with for some days now, but have not been able to answer.
Suppose gn are nonnegative and integrable on [0, 1], and that gn \rightarrow g almost everywhere.
Further suppose that for all \epsilon > 0, \exists \delta > 0 such that for all A \subset...
Hi, I'm stuck on this problem here about composite function, help is appreciated:
Let g : [a,b] -> [c,d] be Riemann integrable on [c,d] and f : [c,d] -> R is Riemann integrable on [c,d]. Prove that f o g is Riemann integrable on [a,b] if either f or g is a step function
I was able to solve...
Hi there,
My objective is to study Hamiltonian systems, integrable and non integrable systems, where there will be chaos, etc. I have a general idea of everything.. the destroyed tori, the symplectic structure of hamilton's equations, etc. But nothing is very clear to me! And the most...
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...