Stability in the Presence of First Integrals

Abstract

The principal drawback of Liapunov’s direct method is that no general procedure is known to construct auxiliary functions suiting specific theorems. That is why, in stability problems, one should a priori neglect no available information concerning the solutions. In particular, the first integrals will often be helpful, either to facilitate the search for auxiliary functions or to eliminate part of the variables and thus decrease the number of equations to examine. Both points of view will be developed later, in Sections 3 and 4 respectively. Section 5 deals with an important case where first integrals are known, namely the stationary motions of mechanical systems with ignorable coordinates. Section 6 studies a particular motion of this type: the orbiting particles in the betatron. And the last section gathers practical criteria concerning the various methods of constructing positive definite functions.