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

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Scholium I: ctd- ... timedt =√1+(y′ ) 2√ −2gy L(y,y ′ ) =√− 1+(y′ ) 22gy- Euler–Lagrange equations ( 1+(y ′ ) 2) y = constantsubstitution of(x,y) = (x(θ),y(θ))again gives cycloid resultV.I. Arnold, Mathematical methods of classical mechanics. GTM 60, Springer 1978, 1989H. Goldstein, Classical mechanics 2nd ed. Addison Wesley 1950, 1980
Scholium II: lightrays geodesicsGeorg Friedrich Bernhard Riemann 1826-1866- Riemannian metricG q (˙q 1 , ˙q 2 ) = n 2 (q)〈˙q 1 , ˙q 2 〉,where latter brackets are EuclideannoteL(q, ˙q) = 1 2 G q(˙q, ˙q)- lightrays are geodesics of metricGM. Spivak, Differential geometry, Vols. I, II, III. Publish or Perish 1970
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 and 22: Reconstruction ITime parametrizatio
Page 23: Scholium I- Reminds of how Newton
Page 27 and 28: Arclength cycloidarclengths = s(ϕ)

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

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