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A condition eqivalent to linear dependence for functions with vanishing Wronskian. (English) Zbl 0671.15005

Authors summary: It is well known that though the vanishing of the Wronskian W[${\Phi }$ ] of a set $\left\{$ ${\Phi }$ $\right\}$ 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 $\left\{$ ${\Phi }$ $\right\}$ 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.

Reviewer: G.P.Barker
##### MSC:
 15A03 Vector spaces, linear dependence, rank 15A24 Matrix equations and identities