By George D. Birkhoff

His study in dynamics constitutes the center interval of Birkhoff's medical occupation, that of adulthood and maximum energy. --Yearbook of the yank Philosophical Society The author's nice e-book ... is widely known to all, and the varied energetic smooth advancements in arithmetic which were encouraged through this quantity undergo the main eloquent testimony to its caliber and impression. --Zentralblatt MATH In 1927, G. D. Birkhoff wrote a outstanding treatise at the conception of dynamical structures that might encourage many later mathematicians to do nice paintings. To a wide quantity, Birkhoff was once writing approximately his personal paintings at the topic, which used to be itself strongly prompted via Poincare's method of dynamical platforms. With this e-book, Birkhoff additionally proven that the topic was once a stunning conception, even more than a compendium of person effects. The impact of this paintings are available in lots of fields, together with differential equations, mathematical physics, or even what's referred to now as Morse conception. the current quantity is the revised 1966 reprinting of the e-book, together with a brand new addendum, a few footnotes, references extra by way of Jurgen Moser, and a unique preface via Marston Morse. even supposing dynamical platforms has thrived within the a long time on account that Birkhoff's publication used to be released, this treatise maintains to provide perception and idea for nonetheless extra generations of mathematicians.

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**Example text**

Suppose for a· moment that such a stationary T exists, so that we have at q~, .. , (i,j,k= 1,"',m). 31 1. ~ ~ CVk qj"qk. ]i j,k=l It is to be observed that the form of Qi employed holds for any eoordinate system. Now if these forces Qi are to be independent of the velocities, we must have Ci,jk = 0 for all i, j, k in the special cdordinate system and thus in the most general system by the known. law of ,transformation of the, terms Ri. Consequently the desired Lagrangian form of external forces is obtained.

H, ... , qm have these assigned constant values, while the corresponding qi, qi' vanish of course. Then it is clear that L = L, oL oL oqi ' 0 qi (i = 1, " '. ,II). Hence we have the relations Qi = d dt (oL) 0 qi - of 0 qi + Ri (i = 1, ... , ft), where Qi and R i are defined as usual. But the original external forces Qi may be decomposed into a sum (i=l, .. ·,m), in which the 'forces of constraint' Pi can do no work for any possible displacement subject to the constraints. It follows that the functions Pi"", Pp.

Let us confine attention to the case where the energy constant c has a specified value, say c - O. Hence we have L 1 = Lo along the motion qi = q~ (t), (i = 1, -. " m), con~ sidered. Now define 1* as follows' 1* = I-rl(V~-VLo)1 dt = jfl(2VLoL1 +L1 )dt. -oVLo) dt. o Accordingly, if the we shall have q~(t) satisfy the assumed energy condition J1* = JI for a-ll variations of the qi' Hence if the q~ in addition satisfy the Lagrangian equations, so that J I = 0, we shall have J 1* = 0 also. II. VARIATIONAL PRINCIPLES 37 The integrand of I* is positively homogeneous of dimensions unity in the derivatives qi.