By Lev D. Elsgolc

This concise textual content deals either execs and scholars an creation to the basics and conventional equipment of the calculus of adaptations. as well as surveys of issues of mounted and movable obstacles, it explores hugely functional direct equipment for the answer of variational problems.

Topics contain the strategy of version in issues of mounted barriers; variational issues of movable limitations and different difficulties; sufficiency stipulations for an extremum; variational difficulties of restricted extrema; and direct tools of fixing variational difficulties. each one bankruptcy positive aspects quite a few illustrative difficulties, and recommendations look on the end.

Reprint of the Pergamon Press, Oxford, 1961 version.

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**Sample text**

IhJ, where k is a certain coefficient, and the potential energy of the whole string is iu;. dz. ) azdt. t1 v= 0 0 The equation of motion of the string is an Ostrogradski equation for the functional v. __az (k au) = o. - = o. Suppose now that there is a,n. external force f (t, z) acting on the string. This force is assumed to be perpendicular to the string in its equilibrium position and to be proportional to the mass of an element of string. It is easy to see that such an external force acting on an element of the string does work which can be expressed in the form e/ (t, z)udz.

Zl v(y(z)) = J (y+ zy')dz, or v(y(:r:)} Zo Y (Zo) = Yo, = [<11tlz+zdy), Zo Y (z1) = Y1 • The Euler equation is the identity I == I. The integrand is an exact differential and hence the integral z1 v(y(z)) = Jd(zy) = Z1Y1-ZoYo, Zo does not depend on the curve of integration. The variational problem is of no interest. (3) F depends on y' only: = F (y'). The Euler equation is Jl'11 ,11 ,y" = O, for F 11 = FZ'll' = ll'w, = O. Consequently y" = 0 or F'll''ll' = 0. If y" = o, then F y = 0 1 m+ 0 2 is a.

Consequently, z { J i k u= 02 ""/are'- }2 [I+ (au/az)2]3/2 dz. 0 Suppose the deviation from the resting position small, so that the term (a"'1/az) 2 in denominator can be dropped. fzk(ar dai. :l dzdt. 0 0 Therefore the equation of motion of free vibrations of an elastic rod is a 02 8t (Q""t) + afl,2 (kUzz) = o. If the rod is homogeneous, then e and k are constants, and the equation of homogeneous vibrating rod turns into e a2 "" at2 +k a4"" az' = o. If there is an external force I (z, t), acting on the rod, then one shall make allowance for the potential of this force (cf.