Mathematical Methods for Physics and Engineering - Matematica.NET

CALCULUS OF VARIATIONS22.8 Derive the differential equations for the plane-polar coordinates, r and φ, ofaparticle of unit mass moving in a field of potential V (r). Find the form of V ifthe path of the particle is given by r = a sin φ.22.9 You are provided with a line of length πa/2 and negligible mass and some leadshot of total mass M. Use a variational method to determine how the lead shotmust be distributed along the line if the loaded line is to hang in a circular arcof radius a when its ends are attached to two points at the same height. Measurethe distance s along the line from its centre.22.10 Extend the result of subsection 22.2.2 to the case of several dependent variablesy i (x), showing that, if x does not appear explicitly in the integrand, then a firstintegral of the Euler–Lagrange equations isn∑F − y ′ ∂Fi =constant.∂y ′ i=1 i22.11 A general result is that light travels through a variable medium by a path whichminimises the travel time (this is an alternative formulation of Fermat’s principle).With respect to a particular cylindrical polar coordinate system (ρ, φ, z), the speedof light v(ρ, φ) is independent of z. If the path of the light is parameterised asρ = ρ(z),φ= φ(z), use the result of the previous exercise to show thatv 2 (ρ ′2 + ρ 2 φ ′2 +1)is constant along the path.For the particular case when v = v(ρ) =b(a 2 +ρ 2 ) 1/2 , show that the two Euler–Lagrange equations have a common solution in which the light travels along ahelical path given by φ = Az + B, ρ = C, provided that A has a particular value.22.12 Light travels in the vertical xz-plane through a slab of material which lies betweenthe planes z = z 0 and z =2z 0 , and in which the speed of light v(z) =c 0 z/z 0 .Using the alternative formulation of Fermat’s principle, given in the previousquestion, show that the ray paths are arcs of circles.Deduce that, if a ray enters the material at (0,z 0 ) at an angle to the vertical,π/2 − θ, ofmorethan30 ◦ , then it does not reach the far side of the slab.22.13 A dam of capacity V (less than πb 2 h/2) is to be constructed on level ground nextto a long straight wall which runs from (−b, 0) to (b, 0).Thisistobeachievedbyjoining the ends of a new wall, of height h, to those of the existing wall. Showthat, in order to minimise the length L of new wall to be built, it should formpart of a circle, and that L is then given by∫ bdx−b (1 − λ 2 x 2 ) , 1/2where λ is found fromVhb = sin−1 µ− (1 − µ2 ) 1/22 µ 2 µand µ = λb.22.14 In the brachistochrone problem of subsection 22.3.4 show that if the upper endpointcan lie anywhere on the curve h(x, y) = 0, then the curve of quickest descenty(x) meetsh(x, y) = 0 at right angles.22.15 The Schwarzchild metric for the static field of a non-rotating spherically symmetricblack hole of mass M is given by((ds) 2 = c 2 1 − 2GM )(dt) 2 −c 2 r(dr) 21 − 2GM/(c 2 r) − r2 (dθ) 2 − r 2 sin 2 θ (dφ) 2 .Considering only motion confined to the plane θ = π/2, and assuming that the798