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anthony2005
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The title is self-explanatory. What is it meant in the physics and maths community by the words integrability and integrable system?
...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.
... the deterministic nature of these systems does not make them predictable.[
No, integrability means: can a given relationship between derivatives be integrated to yield a relationship between functions. For example, given the systemSo, is it correct to state in general that: "an integrable system is a system which thanks to certain properties its dynamics is exactly solvable" ?
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: ∂^{2}f/∂x∂y = ∂^{2}f/∂y∂x,
that is, ∂F/∂y = ∂G/∂x.
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.
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.
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.
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.
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.