A common misconception is that W = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions x2 and |x|x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0. There are several extra conditions which ensure that the vanishing of the Wronskian in an interval implies linear dependence. Peano (1889) observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent. Bochner (1901) gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of n functions is identically zero and the n Wronskians of n–1 of them do not all vanish at any point then the functions are linearly dependent. Wolsson (1989a) gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.

Oops! repeated roots fall into this category, and although Lagrangian mechanics is useful and reliable in most circumstances, the Russian Academy of Sciences have at last published some excellent old B&W film footage of ‘clockwork’ inertial drive systems. Disallowed by the convenient ‘circulus in probando’ argument presented in Lagrangian Mechanics. The demonstations conclusively prove that there is not always an equal and opposite reaction, especially if we are dealing with the forces of centripetal acceleration. Very handy for manoevering Soviet Military Satellites, so Russia’s recent publication into the Public Domain, is an exceptionally generous gift to the scientific community.

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