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Extra info for Analytical dynamics: being a synopsis of leading topics

Sample text

81 LEMMA 3 . 1 . 18) E, P r o o f . As in the p r o o f of t h e c o r o l l a r y of Theorem 2 . 39) a r e c o n n e c t e d w i t h the f i r s t v a r i a t i o n s 6 I j . Hence, by Eqs. ( 3 . 16) we have -~0. 19) 821. 19), we obtain the assertion of the lemma. Since at minima all A(e) must be positive while P(E) changes sign on passing across a forbidden zone, the second term must also change sign (if the contour of integration does not intersect this zone). 20) is equal to zero. 21 ) V R ( e ) ( e - ~ ) = o.

8, Leningrad Univ. (1976), pp. 70-92. V. A. Kozel and V. P. Kotlyarov, "Almost-periodic solutions of the equation utt -- Uxx + sinu = 0," Dokl. Akad. Nauk Ukr. SSR, A, No. 10, 878-881 (1976). I. M. Krichever, "Methods of algebraic geometry in the theory of nonlinear equations," Usp. Mat. Nauk, 32, No. 6, 180-208 (1977). I. M. Krichever, "Integration of nonlinear equations by methods of algebraic geometry," Funkts. Anal. , 11, No. I, 15-31 (1977). I. M. Krichever, "Commutative rings of ordinary linear differential operators," Funkts.

The magnitude of the jump has order ~ g e x p (--aq). A number of physical considerations and results of machine computations [9, 50] indicate that degeneracy of the base state is also connected with questions of commensurability (to which a great deal of attention has recently been devoted in the physics literature (see, for example, the survey )). We shall prove that for irrational p and 0 ~< g ~< gp the base state is indeed degenerate. 35) with g = 0 (which was considered as the basic example in the preceding section) will be denoted by ~0.