Breakdown of Stability of Motion in Superquadratic Potentials |
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Authors: | Vadim Zharnitsky |
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Institution: | (1) Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, US |
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Abstract: | Based on the KAM theory, investigation of the equation of motion of a classical particle in a one-dimensional superquadratic
potential well, under the influence of an external time-periodic forcing, raised a hope that all the solutions are bounded.
Indeed, due to the superquadraticity of the potential the frequency of oscillations of the solutions in the system tends to
infinity as the amplitude increases. Therefore, because of this relationship between the frequency and the amplitude, intuitively
one might expect that all resonances that could cause the accumulation of energy would be destroyed, and thus all solutions
would stay bounded for all time. More formally, according to Moser's twist theorem, this could mean the existence of invariant
tubes in the extended phase space and therefore would result in the boundedness of the solutions.
Actually, the boundedness results have been established for a large class of superquadratic potentials, but in general, the
above intuition turns out to be wrong. Littlewood showed it by creating a superquadratic potential in which an unbounded motion
occurs in the presence of some particular piecewise constant forcing. Moreover, Littlewood's result holds for a larger class
of forcings.
Here it is proven for the continuous time-periodic forcing. For this purpose a new averaging technique for the forced motions
in superquadratic potentials with rather weak assumptions on the differentiability of the potentials has been developed.
Received: 1 February 1995 / Accepted: 15 March 1997 |
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