Physics 3210 Spring 2008Problem Set 51. Recall in our derivation of the geodesic equation that extremizingZds =Zdsdtdtis e quivalent to extremizingZdsdt2dt.Show that the Euler-L agrange equations for a geodesic in plane polarcoordinates are the same as the coupled equations given on pag e 10 of thenotes that were derived from the general geodesic equatio n.2. (Taylor, Problem 6.18). Using plane polar coordinates, show that a geodesicin the plane is a straig ht line. [Hint: Show that the function to be min-imized is I =R√1 + r2θ′2dr (where θ′= dθ/dr), which is independentof θ. The integralRc/r√r2− c2dr is easily done with the substitutionr = c/ cos u, yielding an equation for r as a function of θ −θ0where θ0isa constant of integration. Now show that this is the equation of a straightline perpe ndicula r to the direction θ0.]3. Fermat’s principle sta tes : If the velocity of light is given by the continuousfunction u = u(y), the actual light path connecting the points (x1, y1) and(x2, y2) in a plane is one which extremizes the time integralI =Zx2,y2)(x1,y1)dsu.Derive Snell’s law from Ferma t’s principle; that is, prove that sin θ/u =const, where θ is the angle shown below.••xy(x1, y1)(x2, y2)y(x)θ[Hint: Use the se c ond form of Euler’s equation given on page 13 of thenotes.]14. Inverse Isoperimetric Problem (“iso areametric” problem). Prove that ofall simple closed curves enclosing a given area A, the least perimeter ispossessed by the circle. What is the value of the Lagrange multiplier λ interms of A?5. We have seen that the geodesic equa tion isd2xldt2+ Γlijdxidtdxjdt= 0.Show that under a change of parameter t → σ = f (t) the geodesic equationbecomesd2xldσ2+ Γlijdxidσdxjdσ= −¨f˙f2dxldσ.Thus the geodesic equation retains its form only under affine changes ofparameter t → f(t) = a t + b.6. Show that the geodesic equation implies that the quantitygijdxidtdxjdtis a constant along the geodesic, i.e.,ddtgijdxidtdxjdt= 0.[Hint: Don’t forg e t to use the geodesic equation. Also recall the formulafor the Christoffel symbolsΓlij=12glk∂gkj∂xi+∂gki∂xj−∂gij∂xk.7. Using the geodesic equation, show that the geodesics on the surface of asphere in R3are great circles. (Thornton & Marion solve this problemin their Example 6.4. I think their s olution is much more difficult tointerpret.)8. Referring to the notation used on pages 17 – 19 of the notes, suppose thatwe want to extremize the functionalI(ε) =Zt1t0f(˜x,˙˜x,¨˜x, t) dtwhere all other conditions (such as fixed end points and the continuity ofderivatives) are as in the notes. Derive the Euler-Lagrange equation forf. Be sure to clearly state any additional boundary conditions you needto specify in order to obtain your result.29. In the notes it is shown that as a conseq uence of Green’s theorem, thearea enclos e d by a simple closed curve is given byA =12Z(x ˙y − ˙xy) dt.Show by changing to plane polar coordinates that this is the same asA =12Zr2dθ.10. (Taylor, Problem 6.22). Suppose you have a string of length l with oneend fix e d at the origin of the xy-plane, and the other end on the x-axissuch that the area enclosed between the string and the x-axis is a max-imum. Show that the shape of the string is a semicircle. [Hint: Fromelementary calculus the area is given byRy dx. Show that this is thesame asRl0yp1 − y′2ds where s is the distance along the string startingat the origin, and y′= dy/ds. Use the second form o f Euler’s equation tofind y as a function of s. Now use this to find x as a function of s, andthen combine these equations for x and y to eliminate s.]11. (Bonus) (Taylor, Problem 6.25 or Thornton & Marion, Problem 6-6). It isshown in both texts that the equation of the path of least time of transitof a particle falling from one point to another under the force of gravity gis a cycloid. The para metr ic eq uations of a point (x, y) on the curve ar ex = a(θ − sin θ)y = a(1 − cos θ)xxyy2πaaθaθaπaShow that the time required for the particle to fall (starting at rest) fromany point P0= (x0, y0) to the low point P on the curve is πpa/g. (Notethis is independent of P0.) [Hint : First show that along the curve we haveds = ap2(1 − cos θ) dθ. If the point P0has parameter θ0, show (usingconservation of ener gy) that the time to fall is given byt =ZPP0dsv=Zπθ0ap2(1 − cos θ)p2ga(cos θ0− cos θ)dθ.3Now make the substitution θ = π −2α and use the identity cos(A ±B) =cos A cos B ∓ sin A sin B to show thatt = 2ragZα00cos αpsin2α0− sin2αdα.Next, make the substitution sin α = u followed by u/u0= w to obtaint = 2ragZ10dw√1 − w2.Finally, evaluate this integral with the substitution w = sin φ.]12. (Thornton & Marion, Problem 6-8(a)). Find the dimensions of the paral-lelepiped of maximum volume ins cribed in a sphere of radius