Editing Hamiltonian system (section)

== Hamiltonian chaos ==
Certain Hamiltonian systems exhibit [[Chaos theory|chaotic behavior]]. When the evolution of a Hamiltonian system is highly sensitive to initial conditions, and the motion appears random and erratic, the system is said to exhibit Hamiltonian chaos.

=== Origins ===
The concept of chaos in Hamiltonian systems has its roots in the works of [[Henri Poincaré]], who in the late 19th century made pioneering contributions to the understanding of the [[three-body problem]] in [[celestial mechanics]]. Poincaré showed that even a simple [[Newton's law of universal gravitation|gravitational system]] of three bodies could exhibit complex behavior that could not be predicted over the long term. His work is considered to be one of the earliest explorations of chaotic behavior in [[Physics|physical systems]].<ref name="Poincare2">Poincaré, Henri. "New Methods of Celestial Mechanics." (1892)</ref>

=== Characteristics ===
Hamiltonian chaos is characterized by the following features:<ref name="ott" />

'''Sensitivity to Initial Conditions''': A hallmark of chaotic systems, small differences in initial conditions can lead to vastly different trajectories. This is known as the butterfly effect.<ref>{{Cite journal |last=Lorenz |first=Edward N. |date=1963-03-01 |title=Deterministic Nonperiodic Flow |url=https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=20 |issue=2 |pages=130–141 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |issn=0022-4928|doi-access=free }}</ref>

'''Mixing''': Over time, the phases of the system become uniformly distributed in phase space.<ref>{{Cite book |last=Kornfel'd |first=Isaak P. |title=Ergodic Theory |last2=Fomin |first2=Sergej V. |last3=Sinaj |first3=Jakov G. |date=1982 |publisher=Springer |isbn=978-1-4615-6929-9 |series=Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |location=New York, NY Heidelberg Berlin}}</ref>

'''Recurrence''': Though unpredictable, the system eventually revisits states that are arbitrarily close to its initial state, known as [[Poincaré recurrence theorem|Poincaré recurrence]].

Hamiltonian chaos is also associated with the presence of ''chaotic invariants'' such as the [[Lyapunov exponent]] and [[Kolmogorov-Sinai entropy]], which quantify the rate at which nearby trajectories diverge and the complexity of the system, respectively.<ref name="ott" />

=== Applications ===
Hamiltonian chaos is prevalent in many areas of physics, particularly in classical mechanics and statistical mechanics. For instance, in [[plasma physics]], the behavior of charged particles in a magnetic field can exhibit Hamiltonian chaos, which has implications for [[nuclear fusion]] and [[Astrophysical plasma|astrophysical plasmas]]. Moreover, in [[quantum mechanics]], Hamiltonian chaos is studied through [[quantum chaos]], which seeks to understand the quantum analogs of classical chaotic behavior. Hamiltonian chaos also plays a role in [[astrophysics]], where it is used to study the dynamics of [[star clusters]] and the stability of [[Galaxy|galactic]] structures.<ref>{{Citation |last=Regev |first=Oded |title=Astrophysics, Chaos and Complexity in |date=2009 |url=https://doi.org/10.1007/978-0-387-30440-3_26 |work=Encyclopedia of Complexity and Systems Science |pages=381–399 |editor-last=Meyers |editor-first=Robert A. |access-date=2023-06-25 |place=New York, NY |publisher=Springer |language=en |doi=10.1007/978-0-387-30440-3_26 |isbn=978-0-387-30440-3|url-access=subscription }}</ref>