By Hajime Urakawa

This publication offers a large view of the calculus of adaptations because it performs an important position in quite a few parts of arithmetic and technological know-how. Containing many examples, open difficulties, and routines with entire strategies, the e-book will be appropriate as a textual content for graduate classes in differential geometry, partial differential equations, and variational equipment. the 1st a part of the booklet is dedicated to explaining the inspiration of (infinite-dimensional) manifolds and comprises many examples. An creation to Morse conception of Banach manifolds is supplied, besides an evidence of the life of minimizing services lower than the Palais-Smale situation. the second one half, that could be learn independently of the 1st, offers the speculation of harmonic maps, with a cautious calculation of the 1st and moment diversifications of the power. numerous purposes of the second one version and category theories of harmonic maps are given.

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**Example text**

Hidt+ az h h = (hI, h2, h3) is any smooth function with h(t0) = h(t1) = 0. Therefore, the equation which the curve o(t) = x(t) satisfying (4) should obey is 8L d OL 8 x; dt 8z; i = 1, 2, 3, 0, (5) which is called the Euler-Lagrange equation of motion. Moreover, since the function L is given by (2), 8L _ M x- 8L 8xj x 8V xj ' = 1, 2, 3, which are substituted into (5), and (1) is obtained. As above, Euler and Lagrange reconstructed Newton's motion equation through the method of variations. R. Hamilton called p := m Y k, the momentum, and considered the function H on R3 x R3 (called the Hamilton function) H(x, p) := 2m IIPII2+ V(x), (x, P) = (XI , x2 , x3 , pI , p2 , p3) E R3 x R3, (6) and the equation for a curve (x(t),p(t))=(xI(t),x2(t),x3(t),p1(t),p2(t),p3(t)) in R3 x R3 in time t, d =--, (7) i=1,2,3, i which is called Hamilton's canonical equation.

21) C°° functions. Define inductively, if dkf : U - Lk(E; F) is defined and differentiable at p, then dk+Ifp = d(dkf)p E L(E; Lk(E; F)) = Lk+I (E; F) is called (k + I )th differentiation. dk+I fp : E x xE k+I - F. ,k+l}, dk+I fp(Xa(I), ... , Xa(k+I)) = dk+1fp(XI , ... , xk+I), X. E E. -+ dk+Ifp ELk+I (E; F) is continuous, then f is said to be Ck+I . If all k > 1, f is Ck on U, then f is said to be C°° on U. §1. CONTINUITY, DIFFERENTIATION, AND INTEGRATION 33 If f is Ck, then for all x1 , ... , xk E E, it follows that dk fo(xl , ...

This enables us to understand the Palais-Smales condition (C) which is a key to the theory of the calculus of variations. Even if it is difficult to solve problems of the calculus of variations, it is very important to imagine a 3-dimensional geometrical figure. The concept of a manifold in a differential geometry is to guarantee this magic. Of course, one needs some effort to get such magic. 1. For a smooth function u on [0, L], we put J(u) := J L uxD dx (0 < p < oo), E(u) :=1 L ux2 dx, CLASSICAL MECHANICS 21 where ux=u(X)=27.