Inverted pendulum driven by a horizontal random force: statistics of the never-falling trajectory and supersymmetry

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. Assuming a white-noise driving, we develop a field-theoretical approach to statistical description of this never-falling trajectory based on the supersymmetric formalism of Parisi and Sourlas. The emerging mathematical structure is similar to that of the Fokker-Planck equation, which however is written for the ``square root'' of the probability distribution function. In the limit of strong driving, we obtain an exact analytical solution for the instantaneous joint distribution function of the pendulum's angle and its velocity.

• Publication year

  2020

• Venue

  SciPost Physics

• Publication date

  2020-06-23

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.21468/SciPostPhys.13.2.021arXiv 2006.13819
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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