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Exercises

  1. Find the extremal curves $ y = y(x)$ of the following constrained optimization problems, using the method of Lagrange multipliers:
    1. $ \int_0^1\left(y'^{\,2}+x^2\right)\,dx$ , such that $ \int_0^1 y^{\,2}\,dx = 2$ .
    2. $ \int_0^\pi y'^{\,2}\,dx$ , such that $ y(0)=y(\pi)=0$ , and $ \int_0^\pi y^{\,2} \,dx=2$ .
    3. $ \int_0^1y\,dx$ , such that $ y(0)=y(1)=1$ , and $ \int\sqrt{1+y'^{\,2}}\,dx=2\pi/3$ .
  2. Suppose $ P$ and $ Q$ are two points lying in the $ x$ -$ y$ plane, which is orientated vertically such that $ P$ is above $ Q$ . Imagine there is a thin, flexible wire connecting the two points and lying entirely in the $ x$ -$ y$ plane. A frictionless bead travels down the wire, impelled by gravity alone. Show that the shape of the wire that results in the bead reaching the point $ Q$ in the least amount of time is a cycloid, which takes the parametric form

    $\displaystyle x(\theta)$ $\displaystyle =k\left(\theta-\sin\theta\right),$    
    $\displaystyle y(\theta)$ $\displaystyle =k \left(1-\cos\theta\right),$    

    where $ k$ is a constant.
  3. Find the curve $ y(x)$ , in the interval $ 0\leq x\leq p$ , which is of length $ \pi$ , and maximizes

    $\displaystyle \int_0^p y\,dx.
$


next up previous
Next: Bibliography Up: Calculus of Variations Previous: Multi-Function Variation
Richard Fitzpatrick 2016-03-31