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
(8.15) |
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
Steven M LaValle 2020-08-14