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Area of Science:

  • Physics
  • Applied Mathematics
  • Dynamical Systems

Background:

  • Hamiltonian systems exhibit symplecticity, preserving phase space volume.
  • Discrete symplectic trajectories are generated by time-incremented transition matrices.
  • It was previously assumed a unique Hamiltonian exists for small time-increments.

Purpose of the Study:

  • To investigate the uniqueness of Hamiltonians for discrete symplectic dynamics.
  • To explore the existence of real and complex-valued Hamiltonians for harmonic oscillators.
  • To analyze specific cases of transition matrices and their Hamiltonian solutions.

Main Methods:

  • Analysis of discrete symplectic dynamics for a harmonic oscillator.
  • Mathematical derivation of Hamiltonians from transition matrices.
  • Examination of Jordan normal forms for transition matrices.

Main Results:

  • Demonstrated infinite real-valued Hamiltonians for small time-increments (τ).
  • Showcased infinite complex-valued Hamiltonians for large time-increments (τ).
  • Identified unique Hamiltonian solutions for specific Jordan normal forms (diagonal elements of 1) and no solutions for others (diagonal elements of -1).

Conclusions:

  • The assumption of a unique Hamiltonian for discrete symplectic dynamics is challenged.
  • The number and type (real/complex) of Hamiltonians depend on time-increment and transition matrix properties.
  • Specific matrix structures, like Jordan normal forms, dictate Hamiltonian solution existence.