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We conclude therefore the following fact: By a ~ansformauon qi=fi(~,···,im) (i=l, ... the formula Ldt = Ldt. In the reversible case we have I* = Lt = 0, and thus £, 2V LoL dt 2£' ds, ~ s = ~ 39 II. VARIATIONAL PRINCIPLES where d s! = L o Lt (d t)! is the squared element of arc on a surface with c06rdinates ql, .. " qm. Thus in the reversible case with fixed energy constant the cm'ves . of motion may be interpreted as geodesics on the in-dimensional surfar-e with squared element of m'c ds! = Lo L!

IV. REVERSIBILITY. 11Uy equally be described in the J'et:ersed order of time. The meaning here is that the above relations continue to hold when t is replaced by - f. leaving the coOrdinates qi' and the accelerations qi' unaltered. We infer that the functions bij must be lacking. Hence we may write DYNAMICAL SYSTEMS Qi - ... 1: av qJ' + 1: bijk qj q" + hi j=1 j,k=1 where av, bijk - ~ikj, bi involve only the co~rdinates. All of the properties I to IV so far employed are invariant under a change of coOrdinates qi and have to do with the nature of the external forces in the neighborhood of a set of values~, "', q~.

PHYSICAL ASPECTS been reduced to oij at the origin by this preliminary transformation, so that we have m Qi= qi' + 1: j,k=l bijk qj qk + b~ (i 1, "', m) at the ongm. Now suppose that we write further where the constants b~jk have the values specified in the equations above. We find by differentiation that the equations (i = 1, "', m) hold at the ongm. 1t is then found at once that in these variables q" the formula for Qi has the stated form at the origin. V. CONSERVATION OF ENERGY. The dynamical system is conkervative.

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