Classical Mechanics - Taylor. Prob 6.8, 6.14: The brachistochrone скачать в хорошем качестве

Classical Mechanics - Taylor. Prob 6.8, 6.14: The brachistochrone

A typo here: Eq. (1) in Prob 6.14 should be x = a(theta - sin(theta)), y = a(1 - cos(theta)). Classical Mechanics - John R. Taylor Chapter 6: Calculus of Variations 6.3: Applications of Euler-Lagrange Equation Prob 6.8: Verify that the speed of the roller coaster car in Example 6.2 (page 222) is sqrt{2gy}. (Assume the wheels have negligible mass and neglect friction.) Prob 6.14: (a) Prove that the brachistochrone curve (6.26) is indeed a cycloid, that is, the curve traced by a point on the circumference of a wheel of radius a rolling along the underside of the x axis. (b) Although the cycloid repeats itself indefinitely in a succession of loops, only one loop is relevant to the brachistochrone problem. Sketch a single loop for three different values of a (all with the same starting point 1) and convince yourself that for any point 2 (with positive coordinates x2, y2) there is exactly one value of a for which the loop goes through the point 2. (c) To find the value of a for a given point x2, y2 usually requires solution of a transcendental equation. Here are two cases where you can do it more simply: For x2 = pib, y = 2b and agin for x2 = 2pib, y2 = 0 find the value of a for which the cycloid goes through the point 2 and find the corresponding minimum times.