*(English)*Zbl 0671.15005

Authors summary: It is well known that though the vanishing of the Wronskian W[${\Phi}$ ] of a set $\{$ ${\Phi}$ $\}$ of functions on an interval I is a necessary condition for it to be linearly dependent, it is not a sufficient one. Since Peano in 1889 expressed an interest in finding classes of functions for which W[${\Phi}$ ]$\equiv 0$ is sufficient for dependence and offered one such example himself, others (M. Bocher, D. R. Curtiss, W. Hurewicz, and G. H. Meisters) have provided related results.

Here the author gives a final answer to the question by first generalizing Peano’s result using the order of a critical point, thereby obtaining a dense set of intervals of dependence. He then shows that W[${\Phi}$ ]$\equiv 0$ together with a condition that the intersection of certain subspaces of ${E}^{n}$ is nontrivial is equivalent to the linear dependence of $\{$ ${\Phi}$ $\}$ on I. The above results are used to establish the dynamical theorem that motion of a particle under the action of a central force field is planar so long as the particle is restricted from the origin. The author provides a counterexample for the case in which the particle passes through the origin.