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     This
    section contains some more advanced examples that are of interest to
    juniors/senior physics majors.  
      
    The
    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. 
      
    Placing
    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. 
      
    Interesting
    static behavior involves mechanical advantage if the stiffness (i.e. the
    value of the equivalent spring constant k) is large enough. 
      
    Interesting
    dynamic behavior involves crossovers between harmonic and anharmonic behavior when a mass is attached to the line
    and dropped. 
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