Euler, Lagrange, and Kovalevskaya Cases

1) The Euler case. The case is characterized by the fact that a body of an arbitrary shape is supposed to be fixed in its centre of mass, that is, o o = yo = 0 = 0. In this case, the Euler equations take the form 

Multiplying them respectively by Ap, Bq, Cr and summing up, we obviously obtain the required fourth integral A p + B g + = C4(= const). The Euler 

The Euler equations have two first integrals Ap -h Bg -f Cr = const and A p -h B g + = const. To investigate the trajectories of the system, it is 

If the Cartesian coordinates of the vector of the angular momentum K are denoted by (kiyk2,ks), then the Euler equations take the form 

Then the indicated two first quadratic integrals of the Euler equations are written as 

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