Transformation of CDS to Phase Velocity Unit Vector

Abstract

In many problems that arise in the study of a CDS what interests us in the space {q} is the trajectory itself and not the law of motion of the representative point q along this trajectory. This is the case when, for example, one has to deal with controllability problems. In this case the existence of the trajectory joining, say, two given points has to be guaranteed. But the law of motion itself of the representative point along the trajectory during this time is not always of interest to us. Of course, here it has to be ensured that the time taken to move from the initial point to the final point is finite. In this case it proves more convenient to transform the original CDS to a new CDS with unit velocity vector: |q| = 1. In effect, this transformation performs a “separation of variables” characterising the form of the trajectory in {q} on one hand and, on the other, the law of motion along the trajectory during this time.