Vector fields as velocity fields

We now give a particular interpretation to vector fields. A vector field expressed using (8.10) can be used to define a set of first-order differential equations as

\begin{displaymath}\begin{split}\frac{dx_1}{dt} = f_1(x_1,\ldots,x_n)  \frac{d...
...uein}  \frac{dx_n}{dt} = f_n(x_1,\ldots,x_n) .  \end{split}\end{displaymath} (8.11)

Each equation represents the derivative of one coordinate with respect to time. For any point $ x \in {\mathbb{R}}^n$, a velocity vector is defined as

$\displaystyle \frac{dx}{dt} = \left[\frac{dx_1}{dt} \;\; \frac{dx_2}{dt} \;\; \cdots \;\; \frac{dx_n}{dt} \right].$ (8.12)

This enables $ f$ to be interpreted as a velocity field.

It is customary to use the short notation $ {\dot x}= dx/dt$. Each velocity component can be shortened to $ {\dot x}_i = dx_i/dt$. Using $ f$ to denote the vector of functions $ f_1$, $ \ldots $, $ f_n$, (8.11) can be shorted to

$\displaystyle {\dot x}= f(x) .$ (8.13)

The use of $ f$ here is an intentional coincidence with the use of $ f$ for the state transition equation. In Part IV, we will allow vector fields to be parameterized by actions. This leads to a continuous-time state transition equation that looks like $ {\dot x}=
f(x,u)$ and is very similar to the transition equations defined over discrete stages in Chapter 2.

The differential equations expressed in (8.11) are often referred to as autonomous or stationary because $ f$ does not depend on time. A time-varying vector field could alternatively be defined, which yields $ {\dot x}=
f(x(t),t)$. This will not be covered, however, in this chapter.

Steven M LaValle 2012-04-20