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Specialize problem 2 in the case when the particle moves on the unit sphere, from (0, 0, 1) to (0, 0, −1), in time T . 3. Determine the equation of the shortest arc in the ﬁrst quadrant, which passes through π the points (0, 0) and (1, 0) and encloses a prescribed area A with the x-axis, where A ≤ . 8 π 4. Finish the example on page 51. What if L = ? 2 5. Solve the following variational problem by ﬁnding extremals satisfying the conditions J(y1 , y2) = y1 (0) = 1, y1 π 4 0 π 4 4y12 + y22 + y1 y2 dx = 0, y2 (0) = 0, y2 π 4 = 1.
7. Derive a necessary condition for the isoparametric problem Minimize b I(y1 , y2 ) = L(x, y1 , y2, y1 , y2 )dx a subject to b a G(x, y1 , y2, y1 , y2 )dx = C and y1 (a) = A1 , y2 (a) = A2 , y1 (b) = B1 , where C, A1 , A2 , B1 , and B2 are constants. 8. Use the results of the previous problem to maximize 57 y2 (b) = B2 t1 I(x, y) = t0 (xy˙ − y x)dt ˙ subject to t1 x˙ 2 + y˙ 2dt = 1. t0 Show that I represents the area enclosed by a curve with parametric equations x = x(t), y = y(y) and the contraint ﬁxes the length of the curve.
The equations x = a(t) + b(t)(u + sin u) , 32 y = α + b(t)(1 + cos u) (u3 (t) ≤ u ≤ u4 (t)) (38) deﬁne a one-parameter family of cycloid segments y34 when a, b, u3 , u4 are functions of a parameter t as indicated in the equations. If t varies, the end-points 3 and 4 of this segment describe the two curves C and D whose equations in parametric form with t as independent variable are found by substituting u3 (t) and u4 (t), respectively, in (38). These curves and two of the cycloid segments joining them are shown in Figure 15.