Inverted pendulum driven by a horizontal random force: statistics of the neverfalling trajectory and supersymmetry
Abstract
We study stochastic dynamics of an inverted pendulum subject to a random force in the horizontal direction (Whitney's problem). Considered on the entire time axis, the problem admits a unique solution that always remains in the upper half plane. We formulate the problem of statistical description of this neverfalling trajectory and solve it by a fieldtheoretical technique assuming a whitenoise driving. In our approach based on the supersymmetric formalism of Parisi and Sourlas, statistic properties of the neverfalling trajectory are expressed in terms of the zero mode of the corresponding transfermatrix Hamiltonian. The emerging mathematical structure is similar to that of the FokkerPlanck equation, which however is written for the "square root" of the probability distribution function. Our results for the statistics of the nonfalling trajectory are in perfect agreement with direct numerical simulations of the stochastic pendulum equation. In the limit of strong driving (no gravitation), we obtain an exact analytical solution for the instantaneous joint probability distribution function of the pendulum's angle and its velocity.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.13819
 Bibcode:
 2020arXiv200613819S
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Mathematical Physics;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 25 pages, 5 figures