Integrabilitäts

Integrabilitäts is a German term that translates to "integrability" in English. It is a concept primarily used in mathematics and physics to describe the extent to which a system, particularly a dynamical system, can be solved analytically or is well-behaved in terms of its behavior. A system is considered integrable if its equations of motion can be solved exactly, often by finding a sufficient number of conserved quantities or constants of motion.

In Hamiltonian mechanics, a system is integrable if it has $n$ degrees of freedom and possesses $n$ independent integrals of motion in involution. These integrals allow for a complete description and prediction of the system's future state. This is in contrast to non-integrable systems, which often exhibit chaotic behavior and are not analytically solvable. The study of integrability is crucial for understanding the long-term dynamics of physical systems, from celestial mechanics to quantum field theory. The transition from integrable to non-integrable behavior is a fundamental topic in chaos theory.