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This study explores particle motion in a 2D harmonic oscillator with an elliptical boundary. It reveals unique conditions for degenerate trajectories with identical energy but different motion characteristics.

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Area of Science:

  • Classical mechanics
  • Mathematical physics
  • Dynamical systems

Background:

  • The isotropic two-dimensional harmonic oscillator is a fundamental model in physics.
  • Confining potentials and boundaries introduce complexities to classical dynamics.
  • Elliptic boundaries combined with circular potentials present unique mathematical challenges.

Purpose of the Study:

  • To analyze the classical dynamics of a particle in a 2D isotropic harmonic oscillator confined by an elliptic hard wall.
  • To investigate the interplay between the harmonic potential and the elliptical boundary on particle motion.
  • To classify and characterize the types of motion, including rotational and librational trajectories.

Main Methods:

  • Utilizing elliptic coordinates to maintain separability despite the asymmetric confinement.
  • Analyzing equimomentum surfaces to understand parameter space dependencies.
  • Calculating winding numbers and periods analytically.
  • Verifying analytical results through numerical simulations.

Main Results:

  • The system remains separable in elliptic coordinates, but exhibits nontrivial energy and momentum dependencies.
  • Classification of particle motion into distinct types based on parameter space analysis.
  • Analytical calculation and numerical verification of winding numbers and periods for rotational and librational trajectories.
  • Discovery of degenerate rotational trajectories: same energy, different second constant of motion, caustics, and periods.
  • Identification of conditions for obtaining two different rotational trajectories with the same period and second constant of motion but different energy.

Conclusions:

  • The combination of a circular harmonic potential and an elliptical boundary leads to complex yet analytically tractable dynamics.
  • The study identifies specific conditions for degenerate trajectories, offering insights into the system's rich dynamical behavior.
  • The findings contribute to the understanding of classical billiards with combined potential and boundary symmetries.