Gyroscopic systems

FIGURE 7.19. Heavy metal rings for gyroscope systems. By courtesy of Plansee AG, Austria. 

For north finding by means of gyrotheodolites, normally methods for observation of discrete oscillation points (reversing point methods) and transit methods (time difference methods) are used. In case of gyroscopic systems which automatically orient themselves northwards, generally the whole unit is pivoted until the gyroscope s and the whole theodolite s orientation are coincident. 

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). 

Then on the hypersurface (ff = A) C T (M Y ) there exist no analytic functions which are Erst integrals of the gyroscopic system. 

Proposition 5.2.2. Let the manifold Af be compact and let n 2x(Af) and r 0. Then on T (Af there exist no functions of class polynomial in momentum, which are Erst integrals of the gyroscopic system and are independent of the energy integral. 

---