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Mathematical Physics

On The Geometrical Description of Dynamical Stability II

Profile image of Ramis MovassaghRamis Movassagh

2006, Eprint Arxiv Math Ph 0610084

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4 pages

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Abstract

Geometrization of dynamics using (non)-affine parametization of arc length with time is investigated. The two archetypes of such parametrizations, the Eisenhart and the Jacobi metrics, are applied to a system of linear harmonic oscillators. Application of the Jacobi metric results in positive values of geometrical lyapunov exponent. The non-physical instabilities are shown to be due to a non-affine parametrization. In addition the degree of instability is a monotonically increasing function of the fluctuations in the kinetic energy. We argue that the Jacobi metric gives equivalent results as Eisenhart metric for ergodic systems at equilibrium, where number of degrees of freedom $N\to\infty$. We conclude that, in addition to being computationally more expensive, geometrization using the Jacobi metric is meaningful only when the kinetic energy of the system is a positive constant.

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References (10)

  1. L. P. Eisenhart, Ann. Math. 30, 591 (1929).
  2. R. H. Rand, Int. J. Non-Linear Mech. 8, 161-168 (1973).
  3. H. Poincaré, Les Méthodes Nouvelles de la Méchanique Celeste Blanchard, Paris, vol. 3 (1987).
  4. T. Levi-Civita, Ann. Math. 97, 291, (1926).
  5. M. Pettini, Phys. Rev. E 47, 828 (1993).
  6. M. Cerruti-Sola and M. Pettini, Phys. Rev. E 53, 179-188 (1996).
  7. M. Cerruti-Sola and M. Pettini, Phys. Rev. E 51, 53-64 (1995).
  8. E. Cuervo-Reyes, R. Movassagh, Submitted paper.
  9. Marek Szydlowski, J. Math. Phys 40 3519-3539 (1999)
  10. L. D. Landau and E. M. Lifshitz, Mechanics (1976).

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