1179 

 Equations of motion of non-euclidian classic mechanics. 



Bj "classic" rneclianics in a space with the line-element dl we 

 shall understand mechanics with the fundamental eriuation : 



d dx' 



'nK = ^/^ .--- ') ....... (18) 



dt dt 



where *!' is written for the contravariant fundamental tensor, K 

 for the force and c/x' for (he line-element and where the second 

 differentiation is of course a geodetic one. As because of (18): 



dHd\' rdiyddx' 

 M'l K=:m — — 4 m - -— , (19) 



dt' dl \dtj dl dl '■ 



a free point moves in such mechanics along a geodetic line: 



d dx' 



=z (20) 



dl dl - ^ ^ 



When — is the gradient of a force-fnnction U we have for such 

 m 



mechanics the variation theorem : 



for : 



u 'i 



dt=z 



dt dt 



U -^ Ü^\\dt = ^ (21 



dtj 



h u 



/ fdx' \ r \„ d dx' ] 



/ W^ J J I -^^ ^M 



and according to (18) we have now : 



d dx' 



=l'iV6" = . (23) 



dt dt 



1) A force being a covariant vector and an acceleration a contravariant one, a 

 force and an acceleration become quantities of a different kind, as soon as the two 

 kinds may no longer be identified In that case the mass becomes a quantity 

 with the mode of orientation of the covariant fundamental tensor. 



2) Because we have for the geodetic differentiations d and ^, ^dx' = d^x', see 

 A.R. p 57 form. (103). 



