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Appendix B: Mechanics and Geometry

  1. Hans Christian Öttinger

Published Online: 27 JAN 2005

DOI: 10.1002/0471727903.app2

Beyond Equilibrium Thermodynamics

Beyond Equilibrium Thermodynamics

How to Cite

Öttinger, H. C. (2005) Appendix B: Mechanics and Geometry, in Beyond Equilibrium Thermodynamics, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/0471727903.app2

Publication History

  1. Published Online: 27 JAN 2005
  2. Published Print: 2 MAR 2005

ISBN Information

Print ISBN: 9780471666585

Online ISBN: 9780471727903

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Keywords:

  • manifolds;
  • tensors;
  • Lie algebras;
  • differential forms;
  • symplectic structures;
  • Poisson structures;
  • Dirac structures;
  • Jacobi identity;
  • Lie-Poisson reduction;
  • momentum maps;
  • Casimir functions;
  • particle relabeling symmetry;
  • body tensors

Summary

For the distinction between reversible and irreversible contributions to dynamics, one should have the intuitive notion that reversible means “under mechanistic control,” whereas irreversible is associated with “the uncontrollable rest.” It is therefore important to have clear ideas, condensed into a well-defined structure, associated with the notion of “mechanistic,” a structure that can be generally recognized in mechanical and other reversible systems, and which is so restrictive and specific that irreversible dynamics could never share it. The fundamental structure associated with the notion of “under mechanistic control” is well-established in the mathematical literature on mechanics, and it relies on elegant and powerful geometrical concepts: The theory of Hamiltonian dynamical systems. Three different arenas are relevant to mechanics, which allow us to introduce dynamical systems in terms of Hamiltonians, are sketched and related: Symplectic, Poisson, and Dirac structures. The reduction procedure for constructing geometrical structures for particular applications on the basis of a particle relabeling symmetry is discussed in considerable detail.