section contains some more advanced examples that are of interest to
juniors/senior physics majors.
problem of paddling a canoe across a river with a downstream current that
varies across the width of the river is considered. The path taken by the
canoe is determined by solving a non-linear differential equation. In its simplest
form (uniform current) this problem is equivalent to a predator-prey problem.
a mass (or exerting a force) on a fixed line leads to interesting static and
dynamic behavior. A fixed line is a line that is fixed at both ends.
static behavior involves mechanical advantage if the stiffness (i.e. the
value of the equivalent spring constant k) is large enough.
dynamic behavior involves crossovers between harmonic and anharmonic behavior when a mass is attached to the line