Calculus of Variations
This concise text offers an introduction to the fundamentals and standard methods of the calculus of variations. In addition to surveys of problems with fixed and movable boundaries, its subjects include practical direct methods for solution of variational problems. Each chapter features numerous illustrative problems, with solutions. 1961 edition.
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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 closeness of order 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 of extremals ﬁnd ﬁnding ﬁrst ﬁxed follows func function F fundamental lemma fundamental necessary condition given gives an extremum hence homogeneous homogeneous function increment instance integral curves integrand interval isoperimetric problem Jacobi condition Johann Bernoulli mean value theorem minima movable boundaries neighbouring curves obtain Ostrogradski equation parameter polygonal curve problem of extrema respect Ritz’s method solving straight lines strong minimum surface system of Euler takes the form tangent theorem tion tional transversality condition unknown functions vanishes variational problem weak extremum Weak minimum