On the stability problem for the $\mathfrak{so}(5)$ free rigid body

Details On the stability problem for the $\mathfrak{so}(5)$ free rigid body On the stability problem for the $\mathfrak{so}(5)$ free rigid body

2010-09-13

arxiv.org/abs/1009.2387v2

Mathematics

Dynamical Systems

Preprint of an article submitted for consideration in International Journal of Geometric Methods in Modern Physics \copyright

Scientific paper

10.1142/S0219887811005610

In the general case of the $\mathfrak{so}(n)$ free rigid body we give a list of integrals of motion, which generate the set of Mishchenko's integrals. In the case of $\mathfrak{so}(5)$ we prove that there are fifteen coordinate type Cartan subalgebras which on a regular adjoint orbit give fifteen Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body on $\mathfrak{so}(3)$. The nonlinear stability and instability of these equilibria is analyzed. In addition to these equilibria there are ten other continuous families of equilibria.

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