By Max Born

Max Born (1882 1970) was once a physicist and mathematician who received the 1954 Nobel Prize in Physics for his examine within the zone of quantum mechanics. initially released in 1943, this publication provides a touch accelerated model of an tackle Born gave to the Durham Philosophical Society and the natural technology Society, King's collage, at Newcastle-upon-Tyne on twenty first may possibly 1943. The textual content presents an exam of the mutual dating among concept and scan within the improvement of physics. This ebook could be of worth to an individual with an curiosity within the works of Born, physics and the heritage of science."

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E. 1 gives conditions for the existence of such a solution. If (19) is not satisfied with T replaced by T/k (k E R, k> 2), T is called the minimal period of u. 3. Subharmonics of Non-Autonomous Convex Hamiltonian Systems 51 for some k > 2 where minimal period is strictly greater than T. Such solutions are called subharmonic solutions or simply subharmonics. 3. 2. Assume that H(t, u)/Iu12 - 0 (20) H(t, u) --* +00 (21) and as Jul , oo uniformly in t E R. Then, for each k E N \ {0}, there exists a kT-periodic solution Uk of (18), such that (22) IIukiio - oo and such that the minimal period Tk of Uk tends to +oo when k -* oo.

Proof. Let cl = maxtER IH(t, )l. By condition (21) there exists R > 0 such that H(t, u) > 1 + ci for all t E R and u with IuI > R. By convexity we have, for all (t, u) E RxR 2N with Jul > R, 1 + ci < H (t,ju) < IRI H(t, u) + (1- RI) H(t, 0) IRIH(t,u)+cl and hence there is (3 > 0 and d > 0 such that H(t, u) > blul - 0 (23) for all (t, u) E R x R2N If k E N \ {0} is fixed, condition (20) implies that there exists a E ]0, 2ir/kT[ and -t > 0 such that H(t, u) < (a/2)lul2 +'Y (24) for all (t, u) E R x R2N.

The Direct Method of the Calculus of Variations 20 Proof. 1. If u is a solution of (5) then, integrating the differential equation over (0, T) and using the boundary conditions, we get T VF(t, u(t)) dt = 0. J0 (22) Let u = u + u where u = (11T) fo u(t) dt and define the strictly convex functions G and G on RN by G(x) = J T F(t, x) dt; 0 G(x) = f F(t, x + u(l)) dt. 5 implies that G(x) -+ +oo as jxj --} oo. ), we have G(x) < (1/2) J0 T F(t, 2x) dt + (1/2) J0 T F(t, 2u(t)) dt = (1/2)G(2x) + C. (24) It then follows from (23) and (24) that G(x) -+ +oo as jxj --r oo.