Nonintegrability of Several Gyroscopic Systems

We present the results obtained by developing the ideas expressed in 2.1 and 2.2. 

To apply the variational calculus methods, one should know when a gyroscopic system is Lagrangian, i.e is described by the Lagrange equations with the Lagrange function L on TAf. For this, it is necessary and sufficient that the form of the gyroscopic forces be exact, that is, F = dA where A is the differential 1-form on Af. 

The results formulated below were obtained by Bolotin. We may assume without loss of generality that the configuration space Af is an analytic manifold. Bearing in mind applications to celestial mechanics, we assume the potential energy V to have singularities at the points of a finite set C Af. Then the phase space of the gyroscopic system is T (Af 2). 

In the case n = 2x(Af), the presence of nontrivial gyroscopic forces (for instance, F 0) hampers the existence of first integrals of class on T M polynomial with respect to momentum and independent of H. Here x ) 

Euler characteristic of Af. Exact formulations are presented hereafter. 

---