Abstract
We consider the types of critical cases arising in the general equations of a holonomic scleronomous system in independent coordinates. We examine the system's first-approximation matrix and we study the elementary divisors corresponding to this matrix. We prove a theorem on the stability of the trivial solution in one specific critical case when we use a function which is sign-definite in a part of the variables. After Liapunov's original work 1,2 the critical cases in the general problem of stability of motion were considered in [3]. The algebraic unsolvability of stability problems in sufficiently complex critical cases was pointed out in [4].
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