Abstract
The concept of gradient of smooth functions is generalized for their sums with concave functions. An existence, uniqueness, and continuous dependence theorem for increasing time is formulated and proved for solutions of an ordinary differential equation the right-hand side of which is the gradient of the sum of a concave and a smooth function. With the use of this result a physically natural motion of particles, well defined even at discontinuities of the velocity field, is constructed in the variational problem of the minimal mechanical action in a space of arbitrary dimension. For such a motion of particles in the plane all typical cases of the birth and the interaction of point clusters of positive mass are described.
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