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Conservative dissipation: How important is the Jacobi identity in the dynamics?

C E Caligan1, C Chandre2

  • 1School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA.

Chaos (Woodbury, N.Y.)
|June 3, 2016
PubMed
Summary

The Jacobi identity is crucial for structuring phase space in conservative flows. This study compares Hamiltonian, almost-Poisson, and metriplectic dynamics to highlight its importance.

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Area of Science:

  • Mathematical Physics
  • Dynamical Systems Theory

Background:

  • Hamiltonian dynamics are defined by a Hamiltonian function and a Poisson bracket.
  • The Poisson bracket's anti-symmetry ensures Hamiltonian conservation.
  • The Jacobi identity, a property of the Poisson bracket, is often less understood than Hamiltonian conservation.

Purpose of the Study:

  • To investigate the significance of the Jacobi identity in conservative dynamical systems.
  • To compare the dynamics of Hamiltonian, almost-Poisson, and metriplectic flows in ℝ(3).

Main Methods:

  • Analysis of three distinct conservative flow types in three-dimensional Euclidean space (ℝ(3)).
  • Comparative study focusing on the role of the Jacobi identity in each flow type.

Main Results:

  • The Jacobi identity plays a fundamental role in shaping the phase space structure.
  • Differences in dynamics arise from variations in the properties of the bracket structures.

Conclusions:

  • The Jacobi identity is essential for understanding the geometric and dynamic properties of conservative flows.
  • Comparing different bracket structures elucidates the impact of the Jacobi identity on phase space organization.