Complex lamellar vector field: Difference between revisions

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The '''Nekhoroshev estimates''' are an important result in the theory of [[Hamiltonian system]]s concerning the long-time stability of solutions of integrable systems under a small perturbation of the Hamiltonian. The first paper on the subject was written by [[Nikolay Nekhoroshev]] in 1971.<ref>{{cite journal| title=Behavior of Hamiltonian systems close to integrable | journal=Functional Analysis and Its Applications | year=1971 | first=Nikolay N. | last=Nekhoroshev | authorlink=Nikolay Nekhoroshev | volume=5 | issue=4 | pages=338&ndash;339 | id= | doi=10.1007/BF01086753}}</ref>
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The theorem complements both the [[KAM theorem]] and the phenomenon of instability for nearly integrable Hamiltonian systems, sometimes called Arnold Diffusion, in the following way:  The KAM Theorem tells us that ''many'' solutions to nearly integrable Hamiltonian systems persist under a perturbation for ''all'' time, while, as [[Vladimir Arnold]] first demonstrated in 1964,<ref>{{cite journal| title=Instability of dynamical systems with several degrees of freedom | journal=[[Soviet Mathematics]] | year=1964 | first=Vladimir I. | last=Arnold | authorlink=Vladimir Arnold | volume=5 | issue= | pages=581&ndash;585 | id=}}</ref> some solutions do not stay close to their integrable counterparts for all time.  The Nekhoroshev estimates tell us that, nonetheless, ''all'' solutions stay close to their integrable counterparts for an ''exponentially long time''.  Thus, they restrict how quickly solutions can become unstable.
 
==Statement==
Let <math> H(I) + \epsilon h(I, \theta) </math> be a nearly integrable <math>n</math> degree-of-freedom Hamiltonian, where <math> (I, \theta) </math> are the [[action-angle variables]]. Ignoring the technical assumptions and details in the statement, Nekhoroshev estimates assert that:
 
: <math> |I(t) - I(0)| < \varepsilon^{1/(2n)} </math>  
 
for 
 
: <math> |t|< \exp\left({c\left({ 1\over\varepsilon}\right)^{1/(2n)}}\right) </math>
 
where <math> c </math> is a complicated constant.  For a precise, refined statement of the theorem see.<ref>{{cite journal| title=On Nekhoroshev's estimate for quasi-convex Hamiltonians | journal=[[Mathematische Zeitschrift]] | year=1993 | first=Jürgen | last=Pöschel | volume=213 | issue=1 | pages=187&ndash;216 | url=http://www.poschel.de/pbl/nek-1.pdf | format=PDF | accessdate=2010-08-02 | doi=10.1007/BF03025718}}</ref>
 
==See also==
*[[Arnold diffusion]]
*[[KAM theorem]]
 
==References==
<references/>
 
[[Category:Dynamical systems]]

Latest revision as of 06:42, 5 May 2014

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