TAILIEUCHUNG - Classical Mechanics - 3rd ed. - Goldstein, Poole & Safk Episode 1 Part 3

Tham khảo tài liệu 'classical mechanics - 3rd ed. - goldstein, poole & safk episode 1 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Some Techniques of the Calculus of Variations 41 i J-n dx V1 r or xy a x l V-2 where a is some constant of integration clearly smaller than the minimum value of X. Squaring the above equation and factoring terms we have y2 x2 fl2 fl2 or solving dy _ a dx v 2 _ ai The general solution of this differential equation in light of the nature of fl is dx X b a arc cosh - b s x2 a2 a or X a cosh y-b a which is the equation of a catenary. Again the two constants of integration fl and b are determined in principle by the requirements that the curve pass through the two given end points as shown in Fig. . Curves satisfying the preceding equation all scale as x a and y a with one independent parameter b a. This suggests that when the solutions are examined in detail they turn out to be a great deal more complicated than these considera FIGURE General catenary solution for minimum surface of revolution. 42 Chapter 2 Variational Principles and Lagrange s Equations tions suggest. For some paừs of end points unique constants of integration a and b can be found. But for other end points it is possible to draw two different catenary curves through the end points while for additional cases no possible values can be found for a and b. Further recall that Eq. represents a condition for finding curves y x continuous through the second derivative that render the integral stationary. The catenary solutions therefore do not always represent minimum values but may represent inflection points where the length of the curve is stationary but not minimum. For certain combinations of end points an example is Xi and X2 both positive and both much smaller than y 2 yi the absolute minimum in the surface of revolution is provided cf. Exercise 8 by a curve composed of straight line segments from the first end point parallel to the X axis until the y axis is reached then along the y axis until the point 0 yi and then out in a straight line to the second end point corresponding to the

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