TABLE OF CONTENTS Some kind of insects and amoeba are lured by special chemical substances of their own. Such a character is called chemotaxis in biology. For its formulation, some mathematical terminologies and notions are necessary. This chapter is devoted to geometric objects.
Movement of a mass point is indicated by the position vector x = x( t) ? R 3 depending on the time variable t ? R. If m and F denote its mass and the force acting on it, respectively, Newton's equation of motion assures the relation
where 
stands for the acceleration vector. If n points x i = x i( t) ( i = 1, 2, , n) are interacting, then they are subject to the system
simply written as
with x = ( x 1, x 2, , x n) ? R 3 n,
It is sometimes referred to as the deterministic principle of Newton, and under reasonable assumptions on f, say, continuity in all variables 
and the Lipschitz continuity in 
, there is a unique solution x = x( t) to (1.2) locally in time with the prescribed initial position x(0) = x 0 and the initial velocity 
. At this occasion, let us recall that the initial values
provide equation (1.2) with the Cauchy problem
