Bernoulli's lightray solution for the brachistochrone problem from ...

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Reconstruction IICycloids recovered- I = p x = n 2 (y)ẋ ⇒ẋ =1n 2 (y)(5)I = −2Igy =2EI cos2 (gIτ) = E I (1+cos(2gIτ))- And sox = x 0 + E2gI2(2gIτ +sin(2gIτ))y = − E2gI 2(1+cos(2gIτ))Cycloid radius̺ = E/2gI 2 , rolling angleϕ = 2gIτInclination α = gIτ (proportionality)
Scholium I- Reminds of how Newton ‘principia’ yield theKeplerian orbits in central force field- Nowadays brachistochrone problem excercise incourse calculus of variations:Look for curvey = y(x)arclengthds = √ dx 2 +dy 2 = √ 1+(y ′ ) 2 dxenergy conservation in gravitational fieldso have to optimize... dsdt = √ −2gy
Page 1 and 2: Bernoulli’s lightray solutionfor
Page 3 and 4: IntroductionJohann Bernoulli (1667
Page 5 and 6: Fermat and Snelln 1 sinα = n 2 sin
Page 7 and 8: Bernoulli and Snell- Snell’s law:
Page 9 and 10: A few high school computationsLemma
Page 11 and 12: Christiaan Huygens (1629-1695)Chris
Page 13 and 14: William Rowan Hamilton- Many contri
Page 15 and 16: Calculus of variations- If q = (x,y
Page 17 and 18: Laws, properties- Energy conservati
Page 19 and 20: Gravitational refraction indexBack
Page 21: Reconstruction ITime parametrizatio
Page 25 and 26: Scholium II: lightrays geodesicsGeo
Page 27 and 28: Arclength cycloidarclengths = s(ϕ)

Bernoulli's lightray solution for the brachistochrone problem from ...

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