Tess Perrin

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The Brachistochrone ProblemAnalyzing a Second
Variation

• Tess Perrin
• Professor Junping Shi

MATH 441
1 Dec 2009
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History of Brachistochrone

• Problem asks for the curve between two points
  down which an object will travel which minimizes
  the time it takes an object to slide, under the
  force of gravity and neglecting friction
• brachistos the shortest, chronos time
• Johann Bernoulli challenged contemporary
  mathematicians reading Acta Eruditorum with this
  problem in June 1696, receiving solutions from
  his brother Jacob, Gottfried Leibniz, and Isaac
  Newton
• Galileo tried to solve a similar problem for the
  path of the fastest descent from a point to a
  wall in his Two New Sciences in 1638. He draws
  the conclusion that the arc of a circle is faster
  than any of its chords. Upon reviewing Galileos
  work, the actual solution to the problem is half
  a cycloid. Galileo studied the cycloid and gave
  it its name, but the connection between it and
  his problem had to wait for advances in
  mathematics.
• Various solutions have been posed over the past
  300 years geometric arguments, calculus of
  variations, graph theory

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History of Brachistochrone

• A cycloid is the curve defined by the path of a
  point on the edge of a circular wheel as the
  wheel rolls along a straight line. It is an
  example of a roulette, a curve generated by a
  curve rolling on another curve. We consider a
  circle rolling on the ceiling.
• The cycloid was first studied by Nicholas of Cusa
  and later by Mersenne. It was named by Galileo in
  1599. In 1634 G.P. de Roberval showed that the
  area under a cycloid is three times the area of
  its generating circle. In 1658 Christopher Wren
  showed that the length of a cycloid is four times
  the diameter of its generating circle. The
  cycloid has been called "The Helen of Geometers"
  as it caused frequent quarrels among 17th century
  mathematicians.

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Abstract

• Fundamental independent variable of the
  brachistochrone problem is r the ratio between
  the horizontal to vertical displacement between
  the two points
• If r exceeds p/2 then the least time attains an
  absolute minimum below the terminal point of the
  trajectory

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Abstract

• http//faculty.matcmadison.edu/alehnen/brach/MathF
  est2008-Brachistochrone.html
• Solution of the brachistochrone problem
• Boundary conditions determine the value of k
• and the maximum value of theta
• In terms of theta In terms of r

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Abstract

• Graph of the minimizing cycloid

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Proving the existence of Brachistochrone
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Cycloid
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Euler-Lagrange Method
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Second Variation
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History of E-L Calculus of Variations
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Time comparison
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Maple Program