Lorenz integrable system moves à la Poinsot

Details Lorenz integrable system moves à la Poinsot Lorenz integrable system moves à la Poinsot

2002-09-22

arxiv.org/abs/nlin/0209043v2

J. Math. Phys. 44 (2003) pp. 4107-4118

Nonlinear Sciences

Exactly Solvable and Integrable Systems

14 pages, 6 figures

Scientific paper

10.1063/1.1599955

A transformation is derived which takes Lorenz integrable system into the well-known Euler equations of a free-torque rigid body with a fixed point, i.e. the famous motion \`a la Poinsot. The proof is based on Lie group analysis applied to two third order ordinary differential equations admitting the same two-dimensional Lie symmetry algebra. Lie's classification of two-dimensional symmetry algebra in the plane is used. If the same transformation is applied to Lorenz system with any value of parameters, then one obtains Euler equations of a rigid body with a fixed point subjected to a torsion depending on time and angular velocity. The numerical solution of this system yields a three-dimensional picture which looks like a "tornado" whose cross-section has a butterfly-shape. Thus, Lorenz's {\em butterfly} has been transformed into a {\em tornado}.

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