By Erwin Schrödinger
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Additional resources for Collected Papers on Wave Mechanics (Second Edition)
Banker in Chartres; from 1841 to 1851 professor at the École Polytechnique; after 1846 professor at the Sorbonne in Paris. Chasles is independently of J. Steiner one of the founders of the synthetic geometry. His Aperçu historique by far surpassed the older representations of the development of geometry and stimulated new geometrical research in his age. W. 1007/978-3-642-03434-3_4, © Springer-Verlag Berlin Heidelberg 2010 41 42 4 Degrees of Freedom Fig. 1. Chasles’ theorem: The translation vector depends on the rotation, and vice versa We now consider the rigid body with one point fixed in space.
3b) z¨ = −g + 2ω sin λy. 5b) z˙ = −gt + 2ω sin λy. 3b) yields y¨ + 4ω2 y = 2ωg sin λt ≡ ct. , y= c t + A sin 2ωt + B cos 2ωt. 3 Exact Solution The initial conditions at the time t = 0 are x = y = 0, z = h, and x˙ = y˙ = z˙ = 0. , y= sin 2ωt g sin λ t− . 5a) yields x˙ = g sin λ cos λ t − sin 2ωt . 2ω From the initial conditions, it follows that x = g sin λ cos λ t 2 1 − cos 2ωt . 5c) yields z˙ = −gt + 2ω sin λ g sin λ sin 2ωt t− 2ω 2ω z˙ = −gt + g sin2 λ t − , sin 2ωt , 2ω and integration with the initial conditions yields g t 2 1 − cos 2ωt z = − t 2 + g sin2 λ − 2 2 4ω2 + h.
Solution. For the earth, we have m d 2r = −mge3 − 2mω × v with ω = −ω sin λe1 + ω cos λe3 . dt 2 The flow velocity is v = −v0 e1 , and hence, ω × v = −ωv0 sin ϕe2 . Then the force is m¨r = F = −mge3 + 2mωv0 sin ϕe2 = F3 e3 + F2 e2 . F must be perpendicular to the water surface (see Fig. 3). 3, determine H = D sin α and sin α = F2 /F . For the desired height H one obtains H =D 2ωv0 sin ϕ 4ω2 v02 sin2 ϕ + g 2 ≈ 2Dωv0 sin ϕ . g Fig. 3. 9 cm. 4 Difference of Sea Depth at the Pole and Equator Problem.