Calculus o f Variation s 11 VI. Importan t disclaime r We cannot overemphasiz e tha t w e have not accomplishe d a s muc h a s we would like. I n all the above we proceed under th e assumptio n tha t our problems actually possess solutions. Th e techniques we have given then enabl e u s to deriv e significant informatio n abou t th e solutions . The problem of existence of solutions requires more-or-less sophis- ticated technique s o f geometry an d especiall y analysis . Fo r instance , in ou r exampl e o f geodesie s i n th e plane , i t i s rathe r elementar y t o prove directl y tha t a straight lin e segment i s indeed th e uniqu e curv e joining tw o given points . Non e of what w e have discussed i s require d for suc h a proof. To illustrat e tha t som e significan t analysi s ma y perhap s b e re - quired, conside r th e isoperimetri c proble m w e have just talke d about . It woul d see m tha t fo r a genera l close d curv e i n th e plane , A -= - L2 is maximize d i n th e cas e o f a circle , fo r whic h A -f- L2 equal s irr 2 - r (27rr)2 = l/(47r) . Thu s w e should imagin e that fo r a general curv e i n the plane , 4nA L 2 . This i s th e famou s isoperimetric inequality. However , yo u surel y re - alize tha t w e are no t eve n clos e to provin g suc h a n inequalit y i n thi s talk. Th e inequality is indeed valid, but a n actual proof requires much different kind s o f analysis . In summary , th e forma l calculu s o f variation s w e hav e talke d about lead s to ver y interestin g mathematica l object s an d ofte n pave s the wa y t o knowin g wha t t o expec t t o b e true .
Previous Page Next Page