Calculus of Variations
This concise text offers both professionals and students an introduction to the fundamentals and standard methods of the calculus of variations. In addition to surveys of problems with fixed and movable boundaries, it explores highly practical direct methods for the solution of variational problems.
Topics include the method of variation in problems with fixed boundaries; variational problems with movable boundaries and other problems; sufficiency conditions for an extremum; variational problems of constrained extrema; and direct methods of solving variational problems. Each chapter features numerous illustrative problems, and solutions appear at the end.
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absolute value admissible curves approximate solution arbitrary constants assume boundary conditions boundary point brachistochrone problem calculus of variations called central field Consequently consider constrained extrema constraining relations continuous function coordinate functions cusp cycloid depend derivative determined differential equation direct methods domain end points Euler equation examine the extrema EXAMPLE extremal curve extremum can occur ﬁeld field of extremals ﬁnd ﬁnding ﬁrst first-order ﬁxed follows func function F fundamental lemma fundamental necessary condition given gives an extremum hence homogeneous function increment instance integral curves integrand isoperimetric problem Jacobi condition Johann Bernoulli mean value theorem movable boundaries neighbouring curves obtain Ostrogradski equation parameter polygonal curve problem of extrema problem with movable respect Ritz’s method solving straight lines strong minimum surface system of Euler takes the form tangent theorem tion tional transversality condition vanishes variational problem weak extremum Weak minimum