Poincare's theorem on nonintegrable system

[dikim012@gmail.com]

Poincare's nonintegrabel theorem states that there is a class of
systems which we can not find any invariant of motion except energy
and momentum. Does Poincare's theorem prevent us to devise any
mathematics to solve those system?

[Arnold Neumaier]

dikim012@gmail.com wrote:
> Poincare's nonintegrabel theorem states that there is a class of
> systems which we can not find any invariant of motion except energy
> and momentum. Does Poincare's theorem prevent us to devise any
> mathematics to solve those system?

It depends what you mean by 'solve'. Such systems are routinely solved
in an approximate way. Most of what figures under chaos belongs to
this class of systems...


Arnold Neumaier

[tessel@um.bot]

On Thu, 2 Mar 2006 dikim012@gmail.com wrote:

> Poincare's [nonintegrability] theorem states that there is a class of
> systems [for] which we can not find any invariant of motion except
> energy and momentum. Does Poincare's theorem prevent us to devise any
> mathematics to solve those system?

It implies that one cannot hope to find "solutions" in the sense in which
one would naively hope.

References, in rapidly increasing order of difficulty:

Peter Galison
Einstein's clocks, Poincar's maps : empires of time
W. W. Norton, 2003

E. Atlee Jackson
Perspectives of nonlinear dynamics
Cambridge University Press, 1989

Henri Poincare
New methods of celestial mechanics
Ed. and tr. by Daniel L. Goroff
American Institute of Physics, 1993

"T. Essel"