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Mather β-Function for Ellipses and Rigidity.

Michael Bialy1

  • 1School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv 6997801, Israel.

Entropy (Basel, Switzerland)
|November 11, 2022
PubMed
Summary

This study derives an explicit formula for the rotation number and Mather β-function for ellipses using a novel generating function for billiard problems. The findings simplify derivations and offer applications in rigidity problems.

Keywords:
Birkhoff billiardaction minimizers: rigidity, integrable billiardsinvariant curve

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Area of Science:

  • Dynamical Systems
  • Geometric Mechanics

Background:

  • The Mather β-function is crucial in understanding dynamical systems, particularly in celestial mechanics and Hamiltonian systems.
  • Calculating the Mather β-function and rotation number for elliptical systems traditionally involves complex mathematical approaches.

Purpose of the Study:

  • To derive an explicit formula for the rotation number and Mather β-function specifically for elliptical billiards.
  • To demonstrate a simplified derivation method using a non-standard generating function.
  • To explore the application of the Mather β-function in rigidity problems.

Main Methods:

  • Utilized a non-standard generating function derived from the billiard problem.
  • Applied this generating function to obtain explicit formulas for the rotation number and Mather β-function.
  • Investigated the implications of these functions in the context of rigidity theory.

Main Results:

  • An explicit formula for the rotation number of elliptical billiards was successfully obtained.
  • A simplified method for deriving the Mather β-function for ellipses was established.
  • The study provides a foundation for applying these results to rigidity problems.

Conclusions:

  • The non-standard generating function offers a more straightforward approach to calculating key dynamical invariants for elliptical systems.
  • The derived formulas and methods enhance the analytical tractability of problems in dynamical systems and rigidity.
  • This work contributes to a deeper understanding of the Mather β-function and its applications.