If a vector field is given, then a velocity vector is defined at
each point using (8.10). Imagine a point that starts
at some
at time and then moves according to
the velocities expressed in . Where should it travel? Its
*trajectory* starting from can be expressed as a function
, in which the domain is a time
interval,
. A trajectory represents an *integral
curve* (or *solution trajectory*) of the differential equations
with initial condition
if

for every time . This is sometimes expressed in integral form as

(8.15) |

and is called a solution to the differential equations in the

A basic result from differential equations is that a unique integral
curve exists to
if is smooth. An alternative
condition is that a unique solution exists if satisfies a
*Lipschitz condition*. This means that there exists some constant
such that

for all , and denotes the Euclidean norm (vector magnitude). The constant is often called a

Steven M LaValle 2012-04-20