Average-Case Polynomial-Time Computability of Hamiltonian Dynamics

Authors Akitoshi Kawamura, Holger Thies, Martin Ziegler

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Author Details

Akitoshi Kawamura
  • Kyushu University, Fukuoka, Japan
Holger Thies
  • University of Tokyo, Tokyo, Japan
Martin Ziegler
  • KAIST, Daejeon, Republic of Korea

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Akitoshi Kawamura, Holger Thies, and Martin Ziegler. Average-Case Polynomial-Time Computability of Hamiltonian Dynamics. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We apply average-case complexity theory to physical problems modeled by continuous-time dynamical systems. The computational complexity when simulating such systems for a bounded time-frame mainly stems from trajectories coming close to complex singularities of the system. We show that if for most initial values the trajectories do not come close to singularities the simulation can be done in polynomial time on average. For Hamiltonian systems we relate this to the volume of "almost singularities" in phase space and give some general criteria to show that a Hamiltonian system can be simulated efficiently on average. As an application we show that the planar circular-restricted three-body problem is average-case polynomial-time computable.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Computable Analysis
  • Real computation
  • Dynamical systems
  • Average-case complexity
  • Computation in physics


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