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Destructive relativity.

Maria Przybylska1, Wojciech Szumiński1, Andrzej J Maciejewski2

  • 1Institute of Physics, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra, Poland.

Chaos (Woodbury, N.Y.)
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Summary
This summary is machine-generated.

We derived new, strong conditions for the integrability of relativistic Hamilton equations for high-energy particles. These conditions, applicable in weak fields, simplify analysis using linear algebra and are stricter than non-relativistic counterparts.

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Area of Science:

  • Theoretical Physics
  • Classical Mechanics
  • Special Relativity

Background:

  • Describing high-energy particle dynamics necessitates the framework of special relativity.
  • Analyzing the properties of relativistic equations of motion is crucial for theoretical physics.

Purpose of the Study:

  • To formulate and analyze necessary integrability conditions for relativistic Hamilton equations in a weak external field limit.
  • To establish a relationship between the integrability of relativistic and non-relativistic systems.

Main Methods:

  • Analysis of Hamilton equations of motion under the condition 2V(q)≪mc².
  • Formulation of integrability conditions for potentials that are homogeneous functions of coordinates.
  • Investigating eigenvalues of the scaled Hessian matrix for Liouville integrability.

Main Results:

  • Derived strong, general necessary conditions for the integrability of relativistic systems.
  • Showed that these conditions are significantly stronger than those for non-relativistic systems.
  • Demonstrated that calculations simplify to linear algebra, making conditions easy to apply.

Conclusions:

  • The derived conditions provide a powerful tool for analyzing relativistic Hamiltonian systems.
  • For polynomial homogeneous potentials, only systems dependent on a single coordinate or of a radial form appear to be integrable in the relativistic domain.